On second-order mimetic and conservative finite-difference discretization schemes
Rojas, S.; J. M. Guevara-Jordan
2008-01-01
Although the scheme could be derived on the grounds of a relatively new numerical discretization methodology known as Mimetic Finite-Difference Approach, the derivation of a second-order mimetic finite difference discretization scheme will be presented in a more intuitive way, using Taylor expansions. Since students become familiar with Taylor expansions in earlier calculus and mathematical methods for physicist courses, one finds this approach of presenting this new discretization scheme to ...
Chalupecký, Vladimír
2011-01-01
We propose a semi-discrete finite difference multiscale scheme for a concrete corrosion model consisting of a system of two-scale reaction-diffusion equations coupled with an ode. We prove energy and regularity estimates and use them to get the necessary compactness of the approximation estimates. Finally, we illustrate numerically the behavior of the two-scale finite difference approximation of the weak solution.
On second-order mimetic and conservative finite-difference discretization schemes
Directory of Open Access Journals (Sweden)
S Rojas
2008-12-01
Full Text Available Although the scheme could be derived on the grounds of a relatively new numerical discretization methodology known as Mimetic Finite-Difference Approach, the derivation of a second-order mimetic finite difference discretization scheme will be presented in a more intuitive way, using Taylor expansions. Since students become familiar with Taylor expansions in earlier calculus and mathematical methods for physicist courses, one finds this approach of presenting this new discretization scheme to be more easily handled in courses on numerical computations of both undergraduate and graduated programs. The robustness of the resulting discretized equations will be illustrated by finding the numerical solution of an essentially hard-to-solve, one-dimensional, boundary-layer-like problem, based on the steady diffusion equation. Moreover, given that the presented mimetic discretization scheme attains second-order accuracy in the entire computational domain (including the boundaries, as a comparative exercise the discretized equations can be readily applied in solving examples commonly found in texbooks on applied numerical methods and solved numerically via other discretization schemes (including some of the standard finite-diffence discretization schemesAunque la derivación del esquema se puede realizar usando la reciente metodología de discretización numérica conocida como Diferencias Finitas Miméticas, estaremos presentando la derivación de un esquema de discretización mimético en diferencias finitas de segundo orden en una forma mas intuitiva, mediante el uso de expansiones de Taylor. Considerando que los estudiantes se familiarizan con expansiones de Taylor en los primeros cursos de cálculo y métodos matemáticos para físicos, pensamos que la presente alternativa de presentar este nuevo esquema de discretización es más favorable de ser asimilada en cursos de computación numérica tanto de pregrado como de postgrado. La robusticidad del esquema será ilustrada encontrando la solución numérica de un problema unidimensional del tipo capa límite difícil de resolver en forma numérica y que se basa en la ecuación de difusión estacionaria. Más aun, dado que el esquema de discretización alcanza segundo orden de precisión en todo el dominio computacional (incluyendo las fronteras, como ejercicio comparativo el mismo puede ser rápidamente aplicado para resolver ejemplos comúnmente encontrados en textos sobre métodos numéricos aplicados y que se resuelven usando otras metodologías numéricas (incluyendo algunos esquemas de discretización en diferencias finitas
On second-order mimetic and conservative finite-difference discretization schemes
Scientific Electronic Library Online (English)
S, Rojas; J.M, Guevara-Jordan.
2008-12-01
Full Text Available Aunque la derivación del esquema se puede realizar usando la reciente metodología de discretización numérica conocida como Diferencias Finitas Miméticas, estaremos presentando la derivación de un esquema de discretización mimético en diferencias finitas de segundo orden en una forma mas intuitiva, m [...] ediante el uso de expansiones de Taylor. Considerando que los estudiantes se familiarizan con expansiones de Taylor en los primeros cursos de cálculo y métodos matemáticos para físicos, pensamos que la presente alternativa de presentar este nuevo esquema de discretización es más favorable de ser asimilada en cursos de computación numérica tanto de pregrado como de postgrado. La robusticidad del esquema será ilustrada encontrando la solución numérica de un problema unidimensional del tipo capa límite difícil de resolver en forma numérica y que se basa en la ecuación de difusión estacionaria. Más aun, dado que el esquema de discretización alcanza segundo orden de precisión en todo el dominio computacional (incluyendo las fronteras), como ejercicio comparativo el mismo puede ser rápidamente aplicado para resolver ejemplos comúnmente encontrados en textos sobre métodos numéricos aplicados y que se resuelven usando otras metodologías numéricas (incluyendo algunos esquemas de discretización en diferencias finitas) Abstract in english Although the scheme could be derived on the grounds of a relatively new numerical discretization methodology known as Mimetic Finite-Difference Approach, the derivation of a second-order mimetic finite difference discretization scheme will be presented in a more intuitive way, using Taylor expansion [...] s. Since students become familiar with Taylor expansions in earlier calculus and mathematical methods for physicist courses, one finds this approach of presenting this new discretization scheme to be more easily handled in courses on numerical computations of both undergraduate and graduated programs. The robustness of the resulting discretized equations will be illustrated by finding the numerical solution of an essentially hard-to-solve, one-dimensional, boundary-layer-like problem, based on the steady diffusion equation. Moreover, given that the presented mimetic discretization scheme attains second-order accuracy in the entire computational domain (including the boundaries), as a comparative exercise the discretized equations can be readily applied in solving examples commonly found in texbooks on applied numerical methods and solved numerically via other discretization schemes (including some of the standard finite-diffence discretization schemes)
Implicit finite difference schemes for the magnetic induction equations
Koley, U
2011-01-01
We describe high order accurate and stable fully-discrete finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the stability as well as high order of accuracy of the schemes
On nonstandard finite difference schemes in biosciences
Anguelov, R.; Dumont, Y.; Lubuma, J. M.-S.
2012-10-01
We design, analyze and implement nonstandard finite difference (NSFD) schemes for some differential models in biosciences. The NSFD schemes are reliable in three directions. They are topologically dynamically consistent for onedimensional models. They can replicate the global asymptotic stability of the disease-free equilibrium of the MSEIR model in epidemiology whenever the basic reproduction number is less than 1. They preserve the positivity and boundedness property of solutions of advection-reaction and reaction-diffusion equations.
Applications of nonstandard finite difference schemes
Mickens, Ronald E
2000-01-01
The main purpose of this book is to provide a concise introduction to the methods and philosophy of constructing nonstandard finite difference schemes and illustrate how such techniques can be applied to several important problems. Chapter 1 gives an overview of the subject and summarizes previous work. Chapters 2 and 3 consider in detail the construction and numerical implementation of schemes for physical problems involving convection-diffusion-reaction equations that arise in groundwater pollution and scattering of electromagnetic waves using Maxwell's equations. Chapter 4 examines certain
Efficient discretization in finite difference method
Rozos, Evangelos; Koussis, Antonis; Koutsoyiannis, Demetris
2015-04-01
Finite difference method (FDM) is a plausible and simple method for solving partial differential equations. The standard practice is to use an orthogonal discretization to form algebraic approximate formulations of the derivatives of the unknown function and a grid, much like raster maps, to represent the properties of the function domain. For example, for the solution of the groundwater flow equation, a raster map is required for the characterization of the discretization cells (flow cell, no-flow cell, boundary cell, etc.), and two raster maps are required for the hydraulic conductivity and the storage coefficient. Unfortunately, this simple approach to describe the topology comes along with the known disadvantages of the FDM (rough representation of the geometry of the boundaries, wasted computational resources in the unavoidable expansion of the grid refinement in all cells of the same column and row, etc.). To overcome these disadvantages, Hunt has suggested an alternative approach to describe the topology, the use of an array of neighbours. This limits the need for discretization nodes only for the representation of the boundary conditions and the flow domain. Furthermore, the geometry of the boundaries is described more accurately using a vector representation. Most importantly, graded meshes can be employed, which are capable of restricting grid refinement only in the areas of interest (e.g. regions where hydraulic head varies rapidly, locations of pumping wells, etc.). In this study, we test the Hunt approach against MODFLOW, a well established finite difference model, and the Finite Volume Method with Simplified Integration (FVMSI). The results of this comparison are examined and critically discussed.
Finite-difference schemes for anisotropic diffusion
Energy Technology Data Exchange (ETDEWEB)
Es, Bram van, E-mail: es@cwi.nl [Centrum Wiskunde and Informatica, P.O. Box 94079, 1090GB Amsterdam (Netherlands); FOM Institute DIFFER, Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM (Netherlands); Koren, Barry [Eindhoven University of Technology (Netherlands); Blank, Hugo J. de [FOM Institute DIFFER, Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM (Netherlands)
2014-09-01
In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10{sup 12} times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretization schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.
Finite-difference schemes for anisotropic diffusion
International Nuclear Information System (INIS)
In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 1012 times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretization schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid
Higher order finite difference schemes for the magnetic induction equations
Koley, Ujjwal; Risebro, Nils Henrik; Svärd, Magnus
2011-01-01
We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the stability as well as high order of accuracy of the schemes.
Explicit and implicit finite difference schemes for fractional Cattaneo equation
Ghazizadeh, H. R.; Maerefat, M.; Azimi, A.
2010-09-01
In this paper, the numerical solution of fractional (non-integer)-order Cattaneo equation for describing anomalous diffusion has been investigated. Two finite difference schemes namely an explicit predictor-corrector and totally implicit schemes have been developed. In developing each scheme, a separate formulation approach for the governing equations has been considered. The explicit predictor-corrector scheme is the fractional generalization of well-known MacCormack scheme and has been called Generalized MacCormack scheme. This scheme solves two coupled low-order equations and simultaneously computes the flux term with the main variable. Fully implicit scheme however solves a single high-order undecomposed equation. For Generalized MacCormack scheme, stability analysis has been studied through Fourier method. Through a numerical test, the experimental order of convergency of both schemes has been found. Then, the domain of applicability and some numerical properties of each scheme have been discussed.
An upwind finite difference scheme for meshless solvers
International Nuclear Information System (INIS)
In this paper, we present a new upwind finite difference scheme for meshless solvers. This new scheme, capable of working on any type of grid (structure, unstructured or even a random distribution of points) produces superior results. A means to construct schemes of specified order of accuracy is discussed. Numerical computations for different types of flow over a wide range of Mach numbers are presented. Also, these results were compared with those obtained using a cell vertex finite volume code on the same grids and with theoretical values wherever possible. The present framework has the flexibility to choose between various upwind flux formulas
Mean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs
Mohammed, W. W.; Sohaly, M. A.; El-bassiouny, A. H.; Elnagar, K. A.
2014-01-01
Stochastic partial differential equations (SPDEs) describe the dynamics of stochastic processes depending on space-time continuum. These equations have been widely used to model many applications in engineering and mathematical sciences. In this paper we use three finite difference schemes in order to approximate the solution of stochastic parabolic partial differential equations. The conditions of the mean square convergence of the numerical solution are studied. Some ...
Reprint of Variable-step finite difference schemes for the solution of Sturm-Liouville problems
Amodio, Pierluigi; Settanni, Giuseppina
2015-04-01
We discuss the solution of regular and singular Sturm-Liouville problems by means of High Order Finite Difference Schemes. We describe a method to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are considered to emphasize the behavior of a code based on the proposed algorithm. The methods solve any regular or singular Sturm-Liouville problem, providing high accuracy and computational efficiency thanks to the powerful strategy of stepsize variation.
Variable-step finite difference schemes for the solution of Sturm-Liouville problems
Amodio, Pierluigi; Settanni, Giuseppina
2015-03-01
We discuss the solution of regular and singular Sturm-Liouville problems by means of High Order Finite Difference Schemes. We describe a method to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are considered to emphasize the behavior of a code based on the proposed algorithm. The methods solve any regular or singular Sturm-Liouville problem, providing high accuracy and computational efficiency thanks to the powerful strategy of stepsize variation.
ADI finite difference schemes for option pricing in the Heston model with correlation
Hout, K J in 't
2008-01-01
This paper deals with the numerical solution of the Heston partial differential equation that plays an important role in financial option pricing, Heston (1993, Rev. Finan. Stud. 6). A feature of this time-dependent, two-dimensional convection-diffusion-reaction equation is the presence of a mixed spatial-derivative term, which stems from the correlation between the two underlying stochastic processes for the asset price and its variance. Semi-discretization of the Heston PDE, using finite difference schemes on a non-uniform grid, gives rise to large systems of stiff ordinary differential equations. For the effective numerical solution of these systems, standard implicit time-stepping methods are often not suitable anymore, and tailored time-discretization methods are required. In the present paper, we investigate four splitting schemes of the Alternating Direction Implicit (ADI) type: the Douglas scheme, the Craig & Sneyd scheme, the Modified Craig & Sneyd scheme, and the Hundsdorfer & Verwer sch...
Asynchronous finite-difference schemes for partial differential equations
Donzis, Diego A.; Aditya, Konduri
2014-10-01
Current trends in massively parallel computing systems suggest that the number of processing elements (PEs) used in simulations will continue to grow over time. A known problem in this context is the overhead associated with communication and/or synchronization between PEs as well as idling due to load imbalances. Simulation at extreme levels of parallelism will then require an elimination, or at least a tight control of these overheads. In this work, we present an analysis of common finite difference schemes for partial differential equations (PDEs) when no synchronization between PEs is enforced. PEs are allowed to continue computations regardless of messages status and are thus asynchronous. We show that while stability is conserved when these schemes are used asynchronously, accuracy is greatly degraded. Since message arrivals at PEs are essentially random processes, so is the behavior of the error. Within a statistical framework we show that average errors drop always to first-order regardless of the original scheme. The value of the error is found to depend on both grid spacing as well as characteristics of the computing system including number of processors and statistics of the delays. We propose new schemes that are robust to asynchrony. The analytical results are compared against numerical simulations.
Noor, A. K.; Stephens, W. B.
1973-01-01
Several finite difference schemes are applied to the stress and free vibration analysis of homogeneous isotropic and layered orthotropic shells of revolution. The study is based on a form of the Sanders-Budiansky first-approximation linear shell theory modified such that the effects of shear deformation and rotary inertia are included. A Fourier approach is used in which all the shell stress resultants and displacements are expanded in a Fourier series in the circumferential direction, and the governing equations reduce to ordinary differential equations in the meridional direction. While primary attention is given to finite difference schemes used in conjunction with first order differential equation formulation, comparison is made with finite difference schemes used with other formulations. These finite difference discretization models are compared with respect to simplicity of application, convergence characteristics, and computational efficiency. Numerical studies are presented for the effects of variations in shell geometry and lamination parameters on the accuracy and convergence of the solutions obtained by the different finite difference schemes. On the basis of the present study it is shown that the mixed finite difference scheme based on the first order differential equation formulation and two interlacing grids for the different fundamental unknowns combines a number of advantages over other finite difference schemes previously reported in the literature.
Christlieb, Andrew J.; Rossmanith, James A.; Tang, Qi
2013-01-01
In this work we develop a class of high-order finite difference weighted essentially non-oscillatory (FD-WENO) schemes for solving the ideal magnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this work is to use efficient high-order WENO spatial discretizations with high-order strong stability-preserving Runge-Kutta (SSP-RK) time-stepping schemes. Numerical results have shown that with such methods we are able to resolve solution structures that are only vis...
Computational Aero-Acoustic Using High-order Finite-Difference Schemes
DEFF Research Database (Denmark)
Zhu, Wei Jun Technical University of Denmark,
2007-01-01
In this paper, a high-order technique to accurately predict flow-generated noise is introduced. The technique consists of solving the viscous incompressible flow equations and inviscid acoustic equations using a incompressible/compressible splitting technique. The incompressible flow equations are solved using the in-house flow solver EllipSys2D/3D which is a second-order finite volume code. The acoustic solution is found by solving the acoustic equations using high-order finite difference schemes. The incompressible flow equations and the acoustic equations are solved at the same time levels where the pressure and the velocities obtained from the incompressible equations form the input to the acoustic equations. To achieve low dissipation and dispersion errors, either Dispersion-Relation-Preserving (DRP) schemes or optimized compact finite difference schemes are used for spatial discretizations of the acoustic equations. The classical fourth-order Runge-Kutta time scheme is applied to the acoustic equationsfor time discretization.
Chapwanya, Michael; Lubuma, Jean M. -s; Mickens, Ronald E.
2012-01-01
We consider the basic SIR epidemiological model with the Michaelis-Menten formulation of the contact rate. From the study of the Michaelis-Menten basic enzymatic reaction, we design two types of Nonstandard Finite Difference (NSFD) schemes for the SIR model: Exact-related schemes based on the Lambert W function and schemes obtained by using Mickens’s rules of more complex denominator functions for discrete derivatives and nonlocal approximations of nonlinear terms. We compare and investigat...
Converged accelerated finite difference scheme for the multigroup neutron diffusion equation
International Nuclear Information System (INIS)
Computer codes involving neutron transport theory for nuclear engineering applications always require verification to assess improvement. Generally, analytical and semi-analytical benchmarks are desirable, since they are capable of high precision solutions to provide accurate standards of comparison. However, these benchmarks often involve relatively simple problems, usually assuming a certain degree of abstract modeling. In the present work, we show how semi-analytical equivalent benchmarks can be numerically generated using convergence acceleration. Specifically, we investigate the error behavior of a 1D spatial finite difference scheme for the multigroup (MG) steady-state neutron diffusion equation in plane geometry. Since solutions depending on subsequent discretization can be envisioned as terms of an infinite sequence converging to the true solution, extrapolation methods can accelerate an iterative process to obtain the limit before numerical instability sets in. The obtained results have been compared to the analytical solution to the 1D multigroup diffusion equation when available, using FORTRAN as the computational language. Finally, a slowing down problem has been solved using a cascading source update, showing how a finite difference scheme performs for ultra-fine groups (104 groups) in a reasonable computational time using convergence acceleration. (authors)
A finite difference scheme for time dependent eddy currents in tokamaks
International Nuclear Information System (INIS)
The paper reports a finite difference scheme for time dependent eddy currents, which result from plasma disruption, in tokamaks. The potential formulation employed in the present work is that of the T - ? approach, and the numerical scheme is based on a finite difference solution of the potential equations in toroidal co-ordinates. The code is written in ALgol 68. Preliminary results using the scheme are presented. (UK)
Higher order finite difference schemes for the magnetic induction equations with resistivity
Koley, U; Risebro, N H; Svard, And M
2011-01-01
In this paper, we design high order accurate and stable finite difference schemes for the initial-boundary value problem, associated with the magnetic induction equation with resistivity. We use Summation-By-Parts (SBP) finite difference operators to approximate spatial derivatives and a Simultaneous Approximation Term (SAT) technique for implementing boundary conditions. The resulting schemes are shown to be energy stable. Various numerical experiments demonstrating both the stability and the high order of accuracy of the schemes are presented.
Christlieb, Andrew J; Tang, Qi
2013-01-01
In this work we develop a class of high-order finite difference weighted essentially non-oscillatory (FD-WENO) schemes for solving the ideal magnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this work is to use efficient high-order WENO spatial discretizations with high-order strong stability-preserving Runge-Kutta (SSP-RK) time-stepping schemes. Numerical results have shown that with such methods we are able to resolve solution structures that are only visible at much higher grid resolutions with lower-order schemes. The key challenge in applying such methods to ideal MHD is to control divergence errors in the magnetic field. We achieve this by augmenting the base scheme with a novel high-order constrained transport approach that updates the magnetic vector potential. The predicted magnetic field from the base scheme is replaced by a divergence-free magnetic field that is obtained from the curl of this magnetic potential. The non-conservative weakly hyperbolic system that the magnetic vecto...
High-Order Entropy Stable Finite Difference Schemes for Nonlinear Conservation Laws: Finite Domains
Fisher, Travis C.; Carpenter, Mark H.
2013-01-01
Developing stable and robust high-order finite difference schemes requires mathematical formalism and appropriate methods of analysis. In this work, nonlinear entropy stability is used to derive provably stable high-order finite difference methods with formal boundary closures for conservation laws. Particular emphasis is placed on the entropy stability of the compressible Navier-Stokes equations. A newly derived entropy stable weighted essentially non-oscillatory finite difference method is used to simulate problems with shocks and a conservative, entropy stable, narrow-stencil finite difference approach is used to approximate viscous terms.
Nikkar, Samira; Nordström, Jan
2015-06-01
A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.
Development and application of a third order scheme of finite differences centered in mesh
International Nuclear Information System (INIS)
In this work the development of a third order scheme of finite differences centered in mesh is presented and it is applied in the numerical solution of those diffusion equations in multi groups in stationary state and X Y geometry. Originally this scheme was developed by Hennart and del Valle for the monoenergetic diffusion equation with a well-known source and they show that the one scheme is of third order when comparing the numerical solution with the analytical solution of a model problem using several mesh refinements and boundary conditions. The scheme by them developed it also introduces the application of numeric quadratures to evaluate the rigidity matrices and of mass that its appear when making use of the finite elements method of Galerkin. One of the used quadratures is the open quadrature of 4 points, no-standard, of Newton-Cotes to evaluate in approximate form the elements of the rigidity matrices. The other quadrature is that of 3 points of Radau that it is used to evaluate the elements of all the mass matrices. One of the objectives of these quadratures are to eliminate the couplings among the Legendre moments 0 and 1 associated to the left and right faces as those associated to the inferior and superior faces of each cell of the discretization. The other objective is to satisfy the particles balance in weighed form in each cell. In this work it expands such development to multiplicative means considering several energy groups. There are described diverse details inherent to the technique, particularly those that refer to the simplification of the algebraic systems that appear due to the space discretization. Numerical results for several test problems are presented and are compared with those obtained with other nodal techniques. (Author)
High-order flux correction/finite difference schemes for strand grids
Katz, Aaron; Work, Dalon
2015-02-01
A novel high-order method combining unstructured flux correction along body surfaces and high-order finite differences normal to surfaces is formulated for unsteady viscous flows on strand grids. The flux correction algorithm is applied in each unstructured layer of the strand grid, and the layers are then coupled together via a source term containing derivatives in the strand direction. Strand-direction derivatives are approximated to high-order via summation-by-parts operators for first derivatives and second derivatives with variable coefficients. We show how this procedure allows for the proper truncation error canceling properties required for the flux correction scheme. The resulting scheme possesses third-order design accuracy, but often exhibits fourth-order accuracy when higher-order derivatives are employed in the strand direction, especially for highly viscous flows. We prove discrete conservation for the new scheme and time stability in the absence of the flux correction terms. Results in two dimensions are presented that demonstrate improvements in accuracy with minimal computational and algorithmic overhead over traditional second-order algorithms.
A finite difference scheme for a degenerated diffusion equation arising in microbial ecology
Directory of Open Access Journals (Sweden)
Hermann J. Eberl
2007-02-01
Full Text Available A finite difference scheme is presented for a density-dependent diffusion equation that arises in the mathematical modelling of bacterial biofilms. The peculiarity of the underlying model is that it shows degeneracy as the dependent variable vanishes, as well as a singularity as the dependent variable approaches its a priori known upper bound. The first property leads to a finite speed of interface propagation if the initial data have compact support, while the second one introduces counter-acting super diffusion. This squeezing property of this model leads to steep gradients at the interface. Moving interface problems of this kind are known to be problematic for classical numerical methods and introduce non-physical and non-mathematical solutions. The proposed method is developed to address this observation. The central idea is a non-local (in time representation of the diffusion operator. It can be shown that the proposed method is free of oscillations at the interface, that the discrete interface satisfies a discrete version of the continuous interface condition and that the effect of interface smearing is quantitatively small.
Christlieb, Andrew J.; Rossmanith, James A.; Tang, Qi
2014-07-01
In this work we develop a class of high-order finite difference weighted essentially non-oscillatory (FD-WENO) schemes for solving the ideal magnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this work is to use efficient high-order WENO spatial discretizations with high-order strong stability-preserving Runge-Kutta (SSP-RK) time-stepping schemes. Numerical results have shown that with such methods we are able to resolve solution structures that are only visible at much higher grid resolutions with lower-order schemes. The key challenge in applying such methods to ideal MHD is to control divergence errors in the magnetic field. We achieve this by augmenting the base scheme with a novel high-order constrained transport approach that updates the magnetic vector potential. The predicted magnetic field from the base scheme is replaced by a divergence-free magnetic field that is obtained from the curl of this magnetic potential. The non-conservative weakly hyperbolic system that the magnetic vector potential satisfies is solved using a version of FD-WENO developed for Hamilton-Jacobi equations. The resulting numerical method is endowed with several important properties: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as point values on the same mesh (i.e., there is no mesh staggering); (2) both the spatial and temporal orders of accuracy are fourth-order; (3) no spatial integration or multidimensional reconstructions are needed in any step; and (4) special limiters in the magnetic vector potential update are used to control unphysical oscillations in the magnetic field. Several 2D and 3D numerical examples are presented to verify the order of accuracy on smooth test problems and to show high-resolution on test problems that involve shocks.
Sun, Zhen-sheng; Luo, Lei; Ren, Yu-xin; Zhang, Shi-ying
2014-08-01
The dispersion and dissipation properties of a scheme are of great importance for the simulation of flow fields which involve a broad range of length scales. In order to improve the spectral properties of the finite difference scheme, the authors have previously proposed the idea of optimizing the dispersion and dissipation properties separately and a fourth order scheme based on the minimized dispersion and controllable dissipation (MDCD) technique is thus constructed [29]. In the present paper, we further investigate this technique and extend it to a sixth order finite difference scheme to solve the Euler and Navier-Stokes equations. The dispersion properties of the scheme is firstly optimized by minimizing an elaborately designed integrated error function. Then the dispersion-dissipation condition which is newly derived by Hu and Adams [30] is introduced to supply sufficient dissipation to damp the unresolved wavenumbers. Furthermore, the optimized scheme is blended with an optimized Weighted Essentially Non-Oscillation (WENO) scheme to make it possible for the discontinuity-capturing. In this process, the approximation-dispersion-relation (ADR) approach is employed to optimize the spectral properties of the nonlinear scheme to yield the true wave propagation behavior of the finite difference scheme. Several benchmark test problems, which include broadband fluctuations and strong shock waves, are solved to validate the high-resolution, the good discontinuity-capturing capability and the high-efficiency of the proposed scheme.
Stability of finite difference schemes for generalized von Foerster equations with renewal
Directory of Open Access Journals (Sweden)
Henryk Leszczy?ski
2014-01-01
Full Text Available We consider a von Foerster-type equation describing the dynamics of a population with the production of offsprings given by the renewal condition. We construct a finite difference scheme for this problem and give sufficient conditions for its stability with respect to \\(l^1\\ and \\(l^\\infty\\ norms.
Localized solutions for the finite difference semi-discretization of the wave equation
Zuazua E.; Marica A.
2010-01-01
We study the propagation properties of the solutions of the finite-difference space semi-discrete wave equation on an uniform grid of the whole Euclidean space. We provide a construction of high frequency wave packets that propagate along the corresponding bi-characteristic rays of Geometric Optics with a group velocity arbitrarily close to zero. Our analysis is motivated by control theoretical issues. In particular, the continuous wave equation has the so-called observabili...
Single-cone real-space finite difference schemes for the Dirac von Neumann equation
Schreilechner, Magdalena
2015-01-01
Two finite difference schemes for the numerical treatment of the von Neumann equation for the (2+1)D Dirac Hamiltonian are presented. Both utilize a single-cone staggered space-time grid which ensures a single-cone energy dispersion to formulate a numerical treatment of the mixed-state dynamics within the von Neumann equation. The first scheme executes the time-derivative according to the product rule for "bra" and "ket" indices of the density operator. It therefore directly inherits all the favorable properties of the difference scheme for the pure-state Dirac equation and conserves positivity. The second scheme proposed here performs the time-derivative in one sweep. This direct scheme is investigated regarding stability and convergence. Both schemes are tested numerically for elementary simulations using parameters which pertain to topological insulator surface states. Application of the schemes to a Dirac Lindblad equation and real-space-time Green's function formulations are discussed.
A multigrid algorithm for the cell-centered finite difference scheme
Ewing, Richard E.; Shen, Jian
1993-01-01
In this article, we discuss a non-variational V-cycle multigrid algorithm based on the cell-centered finite difference scheme for solving a second-order elliptic problem with discontinuous coefficients. Due to the poor approximation property of piecewise constant spaces and the non-variational nature of our scheme, one step of symmetric linear smoothing in our V-cycle multigrid scheme may fail to be a contraction. Again, because of the simple structure of the piecewise constant spaces, prolongation and restriction are trivial; we save significant computation time with very promising computational results.
A new finite difference scheme for a dissipative cubic nonlinear Schrödinger equation
International Nuclear Information System (INIS)
This paper considers the one-dimensional dissipative cubic nonlinear Schrödinger equation with zero Dirichlet boundary conditions on a bounded domain. The equation is discretized in time by a linear implicit three-level central difference scheme, which has analogous discrete conservation laws of charge and energy. The convergence with two orders and the stability of the scheme are analysed using a priori estimates. Numerical tests show that the three-level scheme is more efficient. (general)
Gao, Guang-Hua; Sun, Hai-Wei; Sun, Zhi-Zhong
2015-01-01
This paper is devoted to the construction and analysis of finite difference methods for solving a class of time-fractional subdiffusion equations. Based on the certain superconvergence at some particular points of the fractional derivative by the traditional first-order Grünwald-Letnikov formula, some effective finite difference schemes are derived. The obtained schemes can achieve the global second-order numerical accuracy in time, which is independent of the values of anomalous diffusion exponent ? (0 governing equation. The spatial second-order scheme and the spatial fourth-order compact scheme, respectively, are established for the one-dimensional problem along with the strict analysis on the unconditional stability and convergence of these schemes by the discrete energy method. Furthermore, the extension to the two-dimensional case is also considered. Numerical experiments support the correctness of the theoretical analysis and effectiveness of the new developed difference schemes.
Chew, C. S.; Yeo, K. S.; Shu, C.
2006-11-01
A scheme using the mesh-free generalized finite differencing (GFD) on flows past moving bodies is proposed. The aim is to devise a method to simulate flow past an immersed moving body that avoids the intensive remeshing of the computational domain and minimizes data interpolation associated with the established computational fluid methodologies; as such procedures are time consuming and are a significant source of error in flow simulation. In the present scheme, the moving body is embedded and enveloped by a cloud of mesh-free nodes, which convects with the motion of the body against a background of Cartesian nodes. The generalized finite-difference (GFD) method with weighted least squares (WLS) approximation is used to discretize the two-dimensional viscous incompressible Navier-Stokes equations at the mesh-free nodes, while standard finite-difference approximations are applied elsewhere. The convecting motion of the mesh-free nodes is treated by the Arbitrary Lagrangian-Eulerian (ALE) formulation of the flow equations, which are solved by a second-order Crank-Nicolson based projection method. The proposed numerical scheme was tested on a number of problems including the decaying-vortex flow, external flows past moving bodies and body-driven flows in enclosures.
Application of the symplectic finite-difference time-domain scheme to electromagnetic simulation
International Nuclear Information System (INIS)
An explicit fourth-order finite-difference time-domain (FDTD) scheme using the symplectic integrator is applied to electromagnetic simulation. A feasible numerical implementation of the symplectic FDTD (SFDTD) scheme is specified. In particular, new strategies for the air-dielectric interface treatment and the near-to-far-field (NFF) transformation are presented. By using the SFDTD scheme, both the radiation and the scattering of three-dimensional objects are computed. Furthermore, the energy-conserving characteristic hold for the SFDTD scheme is verified under long-term simulation. Numerical results suggest that the SFDTD scheme is more efficient than the traditional FDTD method and other high-order methods, and can save computational resources
Directory of Open Access Journals (Sweden)
??
2013-05-01
Full Text Available ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????Fractional order differential equations are generalizations of classical differential equations. They are widely used in the fields of diffusive transport, finance, nonlinear dynamics, signal processing and others. In this paper, an implicit finite difference method for a class of initial-boundary value space-time fractional two-sided space partial differential equations with variable coefficients on a finite domain is established. The stability and convergence order are analyzed for the resulted implicit scheme. With mathematical induction skills, the scheme is proved to be unconditionally stable and convergent.
Directory of Open Access Journals (Sweden)
C. Bommaraju
2005-01-01
Full Text Available Numerical methods are extremely useful in solving real-life problems with complex materials and geometries. However, numerical methods in the time domain suffer from artificial numerical dispersion. Standard numerical techniques which are second-order in space and time, like the conventional Finite Difference 3-point (FD3 method, Finite-Difference Time-Domain (FDTD method, and Finite Integration Technique (FIT provide estimates of the error of discretized numerical operators rather than the error of the numerical solutions computed using these operators. Here optimally accurate time-domain FD operators which are second-order in time as well as in space are derived. Optimal accuracy means the greatest attainable accuracy for a particular type of scheme, e.g., second-order FD, for some particular grid spacing. The modified operators lead to an implicit scheme. Using the first order Born approximation, this implicit scheme is transformed into a two step explicit scheme, namely predictor-corrector scheme. The stability condition (maximum time step for a given spatial grid interval for the various modified schemes is roughly equal to that for the corresponding conventional scheme. The modified FD scheme (FDM attains reduction of numerical dispersion almost by a factor of 40 in 1-D case, compared to the FD3, FDTD, and FIT. The CPU time for the FDM scheme is twice of that required by the FD3 method. The simulated synthetic data for a 2-D P-SV (elastodynamics problem computed using the modified scheme are 30 times more accurate than synthetics computed using a conventional scheme, at a cost of only 3.5 times as much CPU time. The FDM is of particular interest in the modeling of large scale (spatial dimension is more or equal to one thousand wave lengths or observation time interval is very high compared to reference time step wave propagation and scattering problems, for instance, in ultrasonic antenna and synthetic scattering data modeling for Non-Destructive Testing (NDT applications, where other standard numerical methods fail due to numerical dispersion effects. The possibility of extending this method to staggered grid approach is also discussed. The numerical FD3, FDTD, FIT, and FDM results are compared against analytical solutions.
C. Bommaraju; R. Marklein; P. K. Chinta
2005-01-01
Numerical methods are extremely useful in solving real-life problems with complex materials and geometries. However, numerical methods in the time domain suffer from artificial numerical dispersion. Standard numerical techniques which are second-order in space and time, like the conventional Finite Difference 3-point (FD3) method, Finite-Difference Time-Domain (FDTD) method, and Finite Integration Technique (FIT) provide estimates of the error of discretized numerical operators rather than th...
DEFF Research Database (Denmark)
Fuhrmann, David R.; Bingham, Harry B.
2004-01-01
This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly nonlinear and extremely dispersive water waves. The analysis demonstrates the near-equivalence of classical linear Fourier (von Neumann) techniques with matrix-based methods for formulations in both one and two horizontal dimensions. The matrix-based method is also extended to show the local de-stabilizing effects of the nonlinear terms, as well as the stabilizing effects of numerical dissipation. A comparison of the relative stability of rotational and irrotational formulations in two horizontal dimensions provides evidence that the irrotational formulation has significantly better stability properties when the deep-water nonlinearity is high, particularly on refined grids. Computation of matrix pseudospectra shows that the system is only moderately non-normal, suggesting that the eigenvalues are likely suitable for analysis purposes. Numerical experiments demonstrate excellent agreement with the linear analysis, and good qualitative agreement with the local nonlinear analysis. The various methods of analysis combine to provide significant insight into into the numerical behavior of this rather complicated system of nonlinear PDEs.
Hammer, René; Arnold, Anton
2013-01-01
A finite difference scheme is presented for the Dirac equation in (1+1)D. It can handle space- and time-dependent mass and potential terms and utilizes exact discrete transparent boundary conditions (DTBCs). Based on a space- and time-staggered leap-frog scheme it avoids fermion doubling and preserves the dispersion relation of the continuum problem for mass zero (Weyl equation) exactly. Considering boundary regions, each with a constant mass and potential term, the associated DTBCs are derived by first applying this finite difference scheme and then using the Z-transform in the discrete time variable. The resulting constant coefficient difference equation in space can be solved exactly on each of the two semi-infinite exterior domains. Admitting only solutions in $l_2$ which vanish at infinity is equivalent to imposing outgoing boundary conditions. An inverse Z-transformation leads to exact DTBCs in form of a convolution in discrete time which suppress spurious reflections at the boundaries and enforce stabi...
Finite difference elastic wave modeling with an irregular free surface using ADER scheme
Almuhaidib, Abdulaziz M.; Nafi Toksöz, M.
2015-06-01
In numerical modeling of seismic wave propagation in the earth, we encounter two important issues: the free surface and the topography of the surface (i.e. irregularities). In this study, we develop a 2D finite difference solver for the elastic wave equation that combines a 4th- order ADER scheme (Arbitrary high-order accuracy using DERivatives), which is widely used in aeroacoustics, with the characteristic variable method at the free surface boundary. The idea is to treat the free surface boundary explicitly by using ghost values of the solution for points beyond the free surface to impose the physical boundary condition. The method is based on the velocity-stress formulation. The ultimate goal is to develop a numerical solver for the elastic wave equation that is stable, accurate and computationally efficient. The solver treats smooth arbitrary-shaped boundaries as simple plane boundaries. The computational cost added by treating the topography is negligible compared to flat free surface because only a small number of grid points near the boundary need to be computed. In the presence of topography, using 10 grid points per shortest shear-wavelength, the solver yields accurate results. Benchmark numerical tests using several complex models that are solved by our method and other independent accurate methods show an excellent agreement, confirming the validity of the method for modeling elastic waves with an irregular free surface.
Mickens, Ronald E.
1989-01-01
A family of conditionally stable, forward Euler finite difference equations can be constructed for the simplest equation of Schroedinger type, namely u sub t - iu sub xx. Generalization of this result to physically realistic Schroedinger type equations is presented.
Carlos Duque-Daza; Duncan Lockerby; Carlos Galeano
2011-01-01
We present a computational study of the solution of the Falkner-Skan equation (a thirdorder boundary value problem arising in boundary-layer theory) using high-order and high-order-compact finite differences schemes. There are a number of previously reported solution approaches that adopt a reduced-order system of equations, and numerical methods such as: shooting, Taylor series, Runge-Kutta and other semi-analytic methods. Interestingly, though, methods that solve the original non-reduced th...
Fisher, Travis C.; Carpenter, Mark H.; Nordstroem, Jan; Yamaleev, Nail K.; Swanson, R. Charles
2011-01-01
Simulations of nonlinear conservation laws that admit discontinuous solutions are typically restricted to discretizations of equations that are explicitly written in divergence form. This restriction is, however, unnecessary. Herein, linear combinations of divergence and product rule forms that have been discretized using diagonal-norm skew-symmetric summation-by-parts (SBP) operators, are shown to satisfy the sufficient conditions of the Lax-Wendroff theorem and thus are appropriate for simulations of discontinuous physical phenomena. Furthermore, special treatments are not required at the points that are near physical boundaries (i.e., discrete conservation is achieved throughout the entire computational domain, including the boundaries). Examples are presented of a fourth-order, SBP finite-difference operator with second-order boundary closures. Sixth- and eighth-order constructions are derived, and included in E. Narrow-stencil difference operators for linear viscous terms are also derived; these guarantee the conservative form of the combined operator.
Energy Technology Data Exchange (ETDEWEB)
Karlsen, Kenneth Hvistendal; Risebro, Nils Henrik
2000-09-01
We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a ''rough'' coefficient function k(x). we show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, k' is in BV, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general L{sup p} compactness criterion. (author)
International Nuclear Information System (INIS)
To explore the behavior of electromagnetic waves in cold magnetized plasma, a three-dimensional cylindrical hybrid finite-difference time-domain model is developed. The full discrete dispersion relation is derived and compared with the exact solutions. We establish an analytical proof of stability in the case of nonmagnetized plasma. We demonstrate that in the case of nonmagnetized cold plasma the maximum stable Courant number of the hybrid method coincides with the vacuum Courant condition. In the case of magnetized plasma the stability of the applied numerical scheme is investigated by numerical simulation. In order to determine the utility of the applied difference scheme we complete the analysis of the numerical method demonstrating the limit of the reliability of the numerical results. (paper)
Lie Symmetry Preservation by Finite Difference Schemes for the Burgers Equation
Marx Chhay; Aziz Hamdouni
2010-01-01
Invariant numerical schemes possess properties that may overcome the numerical properties of most of classical schemes. When they are constructed with moving frames, invariant schemes can present more stability and accuracy. The cornerstone is to select relevant moving frames. We present a new algorithmic process to do this. The construction of invariant schemes consists in parametrizing the scheme with constant coefficients. These coefficients are determined in order to satisfy a fixed order...
Xiong, Tao; Qiu, Jing-Mei; Xu, Zhengfu
2013-11-01
In Xu (2013) [14], a class of parametrized flux limiters is developed for high order finite difference/volume essentially non-oscillatory (ENO) and Weighted ENO (WENO) schemes coupled with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration for solving scalar hyperbolic conservation laws to achieve strict maximum principle preserving (MPP). In this paper, we continue along this line of research, but propose to apply the parametrized MPP flux limiter only to the final stage of any explicit RK method. Compared with the original work (Xu, 2013) [14], the proposed new approach has several advantages: First, the MPP property is preserved with high order accuracy without as much time step restriction; Second, the implementation of the parametrized flux limiters is significantly simplified. Analysis is performed to justify the maintenance of third order spatial/temporal accuracy when the MPP flux limiters are applied to third order finite difference schemes solving general nonlinear problems. We further apply the limiting procedure to the simulation of the incompressible flow: the numerical fluxes of a high order scheme are limited toward that of a first order MPP scheme which was discussed in Levy (2005) [3]. The MPP property is guaranteed, while designed high order of spatial and temporal accuracy for the incompressible flow computation is not affected via extensive numerical experiments. The efficiency and effectiveness of the proposed scheme are demonstrated via several test examples.
Marcus, Sherman W.; Degani, David
1996-07-01
Tropospheric wave propagation can be described by a parabolic differential equation which may be solved using implicit finite differences. The large grid systems which can be required in such solutions make parallelization expedient. This can be accomplished by decomposing the computational domain into several subdomains over which the solution is calculated in parallel at each step in a marching procedure. A Schwarz-type algorithm, which has become popular in computational fluid dynamics applications, is investigated for accomplishing this domain decomposition. It is found to be unconditionally unstable for the wave propagation parabolic equation when the subdomains overlap, while for nonoverlapping subdomains, its accuracy depends strongly on the method for establishing the boundary conditions on the pseudoboundaries of the subdomains.
Scientific Electronic Library Online (English)
Carlos, Duque-Daza; Duncan, Lockerby; Carlos, Galeano.
2011-12-01
Full Text Available We present a computational study of the solution of the Falkner-Skan equation (a thirdorder boundary value problem arising in boundary-layer theory) using high-order and high-order-compact finite differences schemes. There are a number of previously reported solution approaches that adopt a reduced- [...] order system of equations, and numerical methods such as: shooting, Taylor series, Runge-Kutta and other semi-analytic methods. Interestingly, though, methods that solve the original non-reduced third-order equation directly are absent from the literature. Two high-order schemes are presented using both explicit (third-order) and implicit compact-difference (fourth-order) formulations on a semi-infinite domain; to our knowledge this is the first time that high-order finite difference schemes are presented to find numerical solutions to the non-reduced-order Falkner-Skan equation directly. This approach maintains the simplicity of Taylor-series coefficient matching methods, avoiding complicated numerical algorithms, and in turn presents valuable information about the numerical behaviour of the equation. The accuracy and effectiveness of this approach is established by comparison with published data for accelerating, constant and decelerating flows; excellent agreement is observed. In general, the numerical behaviour of formulations that seek an optimum physical domain size (for a given computational grid) is discussed. Based on new insight into such methods, an alternative optimisation procedure is proposed that should increase the range of initial seed points for which convergence can be achieved.
A convergent 2D finite-difference scheme for the Dirac–Poisson system and the simulation of graphene
International Nuclear Information System (INIS)
We present a convergent finite-difference scheme of second order in both space and time for the 2D electromagnetic Dirac equation. We apply this method in the self-consistent Dirac–Poisson system to the simulation of graphene. The model is justified for low energies, where the particles have wave vectors sufficiently close to the Dirac points. In particular, we demonstrate that our method can be used to calculate solutions of the Dirac–Poisson system where potentials act as beam splitters or Veselago lenses
Yurkin, Maxim A; Hoekstra, Alfons G; Brock, R Scott; Lu, Jun Q
2007-12-24
We compare the discrete dipole approximation (DDA) and the finite difference time domain (FDTD) method for simulating light scattering of spheres in a range of size parameters x up to 80 and refractive indices m up to 2. Using parallel implementations of both methods, we require them to reach a certain accuracy goal for scattering quantities and then compare their performance. We show that relative performance sharply depends on m. The DDA is faster for smaller m, while the FDTD for larger values of m. The break-even point lies at m = 1.4. We also compare the performance of both methods for a few particular biological cells, resulting in the same conclusions as for optically soft spheres. PMID:19551085
A family of energy stable, skew-symmetric finite difference schemes on collocated grids
Reiss, Julius
2014-01-01
A simple scheme for incompressible, constant density flows is presented, which avoids odd-even decoupling for the Laplacian on a collocated grids. Energy stability is implied by maintaining strict energy conservation. Momentum is conserved. Arbitrary order in space and time can easily be obtained. The conservation properties hold on transformed grids.
A finite difference scheme for the equilibrium equations of elastic bodies
Phillips, T. N.; Rose, M. E.
1984-01-01
A compact difference scheme is described for treating the first-order system of partial differential equations which describe the equilibrium equations of an elastic body. An algebraic simplification enables the solution to be obtained by standard direct or iterative techniques.
Directory of Open Access Journals (Sweden)
Carlos Duque-Daza
2011-12-01
Full Text Available We present a computational study of the solution of the Falkner-Skan equation (a thirdorder boundary value problem arising in boundary-layer theory using high-order and high-order-compact finite differences schemes. There are a number of previously reported solution approaches that adopt a reduced-order system of equations, and numerical methods such as: shooting, Taylor series, Runge-Kutta and other semi-analytic methods. Interestingly, though, methods that solve the original non-reduced third-order equation directly are absent from the literature. Two high-order schemes are presented using both explicit (third-order and implicit compact-difference (fourth-order formulations on a semi-infinite domain; to our knowledge this is the first time that high-order finite difference schemes are presented to find numerical solutions to the non-reduced-order Falkner-Skan equation directly. This approach maintains the simplicity of Taylor-series coefficient matching methods, avoiding complicated numerical algorithms, and in turn presents valuable information about the numerical behaviour of the equation. The accuracy and effectiveness of this approach is established by comparison with published data for accelerating, constant and decelerating flows; excellent agreement is observed. In general, the numerical behaviour of formulations that seek an optimum physical domain size (for a given computational grid is discussed. Based on new insight into such methods, an alternative optimisation procedure is proposed that should increase the range of initial seed points for which convergence can be achieved.
Fully Discrete Wavelet Galerkin Schemes
Harbrecht, Helmut; Konik, Michael; Schneider, Reinhold
2006-01-01
The present paper is intended to give a survey of the developments of the wavelet Galerkin boundary element method. Using appropriate wavelet bases for the discretization of boundary integral operators yields numerically sparse system matrices. These system matrices can be compressed to O(N_j) nonzero matrix entries without loss of accuracy of the underlying Galerkin scheme. Herein, O(N_j) denotes the number of unknowns. As we show in the present paper, the ...
DEFF Research Database (Denmark)
Fuhrman, David R.; Bingham, Harry B.
2004-01-01
This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly non-linear and extremely dispersive water waves. The analysis demonstrates the near-equivalence of classical linear Fourier (von Neumann) techniques with matrix-based methods for formulations in both one and two horizontal dimensions. The matrix-based method is also extended to show the local de-stabilizing effects of the non-linear terms, as well as the stabilizing effects of numerical dissipation. A comparison of the relative stability of rotational and irrotational formulations in two horizontal dimensions provides evidence that the irrotational formulation has significantly better stability properties when the deep-water non-linearity is high, particularly on refined grids. Computation of matrix pseudospectra shows that the system is only moderately non-normal, suggesting that the eigenvalues are likely suitable for analysis purposes. Numerical experiments demonstrate excellent agreement with the linear analysis, and good qualitative agreement with the local non-linear analysis. The various methods of analysis combine to provide significant insight into the numerical behaviour of this rather complicated system of non-linear PDEs.
An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation
International Nuclear Information System (INIS)
The pure P-wave equation for modelling and migration in tilted transversely isotropic (TTI) media has attracted more and more attention in imaging seismic data with anisotropy. The desirable feature is that it is absolutely free of shear-wave artefacts and the consequent alleviation of numerical instabilities generally suffered by some systems of coupled equations. However, due to several forward–backward Fourier transforms in wavefield updating at each time step, the computational cost is significant, and thereby hampers its prevalence. We propose to use a hybrid pseudospectral (PS) and finite-difference (FD) scheme to solve the pure P-wave equation. In the hybrid solution, most of the cost-consuming wavenumber terms in the equation are replaced by inexpensive FD operators, which in turn accelerates the computation and reduces the computational cost. To demonstrate the benefit in cost saving of the new scheme, 2D and 3D reverse-time migration (RTM) examples using the hybrid solution to the pure P-wave equation are carried out, and respective runtimes are listed and compared. Numerical results show that the hybrid strategy demands less computation time and is faster than using the PS method alone. Furthermore, this new TTI RTM algorithm with the hybrid method is computationally less expensive than that with the FD solution to conventional TTI coupled equations. (paper)
A New Scheme for Discrete HJB Equations
Directory of Open Access Journals (Sweden)
Zhanyong Zou
2014-10-01
Full Text Available In this paper we propose a relaxation scheme for solving discrete HJB equations based on scheme II [1] of Lions and Mercier. The convergence of the new scheme has been established. Numerical example shows that the scheme is efficient.
International Nuclear Information System (INIS)
The results of numerical simulation of fluid flow and heat transfer in the rod bundle with geometrical disturbance are presented. The geometry of the rod bundle was chosen according to the benchmark problem for 9th IAHR Working Group Meeting (April 7-9, 1998, Grenoble, France). For such a case, experimental data for local velocity and wall shear stress distributions were obtained by group of F. Mantlic at NRI (Czech Republic). Another series of the experiments which provide a data on the wall temperature profiles had been done at the IPPE (Russia). Both experiments provide complete set of data for comparison with the results of numerical simulation. Reynolds equation for axial velocity component has been simulated in two dimensions. Turbulent shear stresses have been simulated by turbulent eddy viscosity with anisotropy defined for radial and azimuth components. Secondary flows have not been taken into consideration. The averaged energy conservation equation closed with anisotropic turbulent conductivity coefficients was simulated. Reynolds and energy conservation equations have been discretized by the Efficient Finite-Difference (EFD) scheme based on the 'locally exact' analytical solution. The comparison of the accuracy of the EFD method and traditional central-difference scheme has been performed. The benchmark problem has been simulated using components of the Computational Object-Oriented Library for Fluid Dynamics (COOLFD) which is a new-generation programming tool aimed to improve the development of the CFD application for complex calculation areas such as rod bundle of nuclear reactor. Comparison of calculated results and experimental data is presented for the local shear stress, axial velocity and the wall temperature distributions in the 'geometrically disturbed' region around dislocated rod. (author)
Mingalev, V. S.; Mingalev, I. V.; Mingalev, O. V.; Oparin, A. M.; Orlov, K. G.
2010-05-01
A generalization of the explicit hybrid monotone second-order finite difference scheme for the use on unstructured 3D grids is proposed. In this scheme, the components of the momentum density in the Cartesian coordinates are used as the working variables; the scheme is conservative. Numerical results obtained using an implementation of the proposed solution procedure on an unstructured 3D grid in a spherical layer in the model of the global circulation of the Titan’s (a Saturn’s moon) atmosphere are presented.
El-Amin, M.F.
2011-05-14
In this paper, a finite difference scheme is developed to solve the unsteady problem of combined heat and mass transfer from an isothermal curved surface to a porous medium saturated by a non-Newtonian fluid. The curved surface is kept at constant temperature and the power-law model is used to model the non-Newtonian fluid. The explicit finite difference method is used to solve simultaneously the equations of momentum, energy and concentration. The consistency of the explicit scheme is examined and the stability conditions are determined for each equation. Boundary layer and Boussinesq approximations have been incorporated. Numerical calculations are carried out for the various parameters entering into the problem. Velocity, temperature and concentration profiles are shown graphically. It is found that as time approaches infinity, the values of wall shear, heat transfer coefficient and concentration gradient at the wall, which are entered in tables, approach the steady state values.
A Finite Difference Interpretation of the Lattice Boltzmann Method
Junk, Michael
1999-01-01
Compared to conventional techniques in computational fluid dynamics, the lattice Boltzmann method (LBM) seems to be a completely different approach to solve the incompressible Navier-Stokes equations. The aim of this article is to correct this impression by showing the close relation of LBM to two standard methods: relaxation schemes and explicit finite difference discretizations. As a side effect, new starting points for a discretization of the incompressible Navier-Stokes equations are obta...
International Nuclear Information System (INIS)
In this paper, we use the staggered grid, the auxiliary grid, the rotated staggered grid and the non-staggered grid finite-difference methods to simulate the wavefield propagation in 2D elastic tilted transversely isotropic (TTI) and viscoelastic TTI media, respectively. Under the stability conditions, we choose different spatial and temporal intervals to get wavefront snapshots and synthetic seismograms to compare the four algorithms in terms of computational accuracy, CPU time, phase shift, frequency dispersion and amplitude preservation. The numerical results show that: (1) the rotated staggered grid scheme has the least memory cost and the fastest running speed; (2) the non-staggered grid scheme has the highest computational accuracy and least phase shift; (3) the staggered grid has less frequency dispersion even when the spatial interval becomes larger. (paper)
Caplan, Ronald M
2011-01-01
Linearized numerical stability bounds for solving the nonlinear time-dependent Schr\\"odinger equation (NLSE) are shown. The bounds are computed for the fourth-order Runge-Kutta scheme in time and both second-order and fourth-order central differencing in space. Results are given for Dirichlet, modulus-squared Dirichlet, Laplacian-zero, and periodic boundary conditions for one, two, and three dimensions. Our approach is to use standard Runge-Kutta linear stability theory, treating the nonlinearity of the NLSE as a constant. The required bounds on the eigenvalues of the scheme matrices are found analytically when possible, and otherwise estimated using the Gershgorin circle theorem.
Gao, YingJie; Yang, HongWei
2014-01-01
An explicit high-order, symplectic, finite-difference time-domain (SFDTD) scheme is applied to a bioelectromagnetic simulation using a simple model of a pregnant woman and her fetus. Compared to the traditional FDTD scheme, this scheme maintains the inherent nature of the Hamilton system and ensures energy conservation numerically and a high precision. The SFDTD scheme is used to predict the specific absorption rate (SAR) for a simple model of a pregnant female woman (month 9) using radio frequency (RF) fields from 1.5 T and 3 T MRI systems (operating at approximately 64 and 128 MHz, respectively). The results suggest that by using a plasma protective layer under the 1.5 T MRI system, the SAR values for the pregnant woman and her fetus are significantly reduced. Additionally, for a 90 degree plasma protective layer, the SAR values are approximately equal to the 120 degree layer and the 180 degree layer, and it is reduced relative to the 60 degree layer. This proves that using a 90 degree plasma protective layer is the most effective and economical angle to use. PMID:25493433
Finite-Difference Algorithms For Computing Sound Waves
Davis, Sanford
1993-01-01
Governing equations considered as matrix system. Method variant of method described in "Scheme for Finite-Difference Computations of Waves" (ARC-12970). Present method begins with matrix-vector formulation of fundamental equations, involving first-order partial derivatives of primitive variables with respect to space and time. Particular matrix formulation places time and spatial coordinates on equal footing, so governing equations considered as matrix system and treated as unit. Spatial and temporal discretizations not treated separately as in other finite-difference methods, instead treated together by linking spatial-grid interval and time step via common scale factor related to speed of sound.
Energy Technology Data Exchange (ETDEWEB)
Delfin L, A.; Alonso V, G. [ININ, 52045 Ocoyoacac, Estado de Mexico (Mexico); Valle G, E. del [IPN-ESFM, 07738 Mexico D.F. (Mexico)]. e-mail: adl@nuclear.inin.mx
2003-07-01
In this work the development of a third order scheme of finite differences centered in mesh is presented and it is applied in the numerical solution of those diffusion equations in multi groups in stationary state and X Y geometry. Originally this scheme was developed by Hennart and del Valle for the monoenergetic diffusion equation with a well-known source and they show that the one scheme is of third order when comparing the numerical solution with the analytical solution of a model problem using several mesh refinements and boundary conditions. The scheme by them developed it also introduces the application of numeric quadratures to evaluate the rigidity matrices and of mass that its appear when making use of the finite elements method of Galerkin. One of the used quadratures is the open quadrature of 4 points, no-standard, of Newton-Cotes to evaluate in approximate form the elements of the rigidity matrices. The other quadrature is that of 3 points of Radau that it is used to evaluate the elements of all the mass matrices. One of the objectives of these quadratures are to eliminate the couplings among the Legendre moments 0 and 1 associated to the left and right faces as those associated to the inferior and superior faces of each cell of the discretization. The other objective is to satisfy the particles balance in weighed form in each cell. In this work it expands such development to multiplicative means considering several energy groups. There are described diverse details inherent to the technique, particularly those that refer to the simplification of the algebraic systems that appear due to the space discretization. Numerical results for several test problems are presented and are compared with those obtained with other nodal techniques. (Author)
Explicit finite difference methods for the delay pseudoparabolic equations.
Amirali, I; Amiraliyev, G M; Cakir, M; Cimen, E
2014-01-01
Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown. PMID:24688392
Mimetic finite difference method
Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail
2014-01-01
The mimetic finite difference (MFD) method mimics fundamental properties of mathematical and physical systems including conservation laws, symmetry and positivity of solutions, duality and self-adjointness of differential operators, and exact mathematical identities of the vector and tensor calculus. This article is the first comprehensive review of the 50-year long history of the mimetic methodology and describes in a systematic way the major mimetic ideas and their relevance to academic and real-life problems. The supporting applications include diffusion, electromagnetics, fluid flow, and Lagrangian hydrodynamics problems. The article provides enough details to build various discrete operators on unstructured polygonal and polyhedral meshes and summarizes the major convergence results for the mimetic approximations. Most of these theoretical results, which are presented here as lemmas, propositions and theorems, are either original or an extension of existing results to a more general formulation using polyhedral meshes. Finally, flexibility and extensibility of the mimetic methodology are shown by deriving higher-order approximations, enforcing discrete maximum principles for diffusion problems, and ensuring the numerical stability for saddle-point systems.
On the accuracy and efficiency of finite difference solutions for nonlinear waves
DEFF Research Database (Denmark)
Bingham, Harry B.
2006-01-01
We consider the relative accuracy and efficiency of low- and high-order finite difference discretizations of the exact potential flow problem for nonlinear water waves. The continuous differential operators are replaced by arbitrary order finite difference schemes on a structured but non-uniform grid. Time-integration is performed using a fourth-order Runge-Kutta scheme. The linear accuracy, stability and convergence properties of the method are analyzed in two-dimensions, and high-order schemes with a stretched vertical grid are found to be advantageous relative to second-order schemes on an even grid. Comparison with highly accurate periodic solutions shows that these conclusions carry over to nonlinear problems. The combination of non-uniform grid spacing in the vertical and fourth-order schemes is suggested as providing an optimal balance between accuracy and complexity for practical purposes.
Mimetic finite difference methods in image processing
Scientific Electronic Library Online (English)
C., Bazan; M., Abouali; J., Castillo; P., Blomgren.
Full Text Available We introduce the use of mimetic methods to the imaging community, for the solution of the initial-value problems ubiquitous in the machine vision and image processing and analysis fields. PDE-based image processing and analysis techniques comprise a host of applications such as noise removal and res [...] toration, deblurring and enhancement, segmentation, edge detection, inpainting, registration, motion analysis, etc. Because of their favorable stability and efficiency properties, semi-implicit finite difference and finite element schemes have been the methods of choice (in that order of preference). We propose a new approach for the numerical solution of these problems based on mimetic methods. The mimetic discretization scheme preserves the continuum properties of the mathematical operators often encountered in the image processing and analysis equations. This is the main contributing factor to the improved performance of the mimetic method approach, as compared to both of the aforementioned popular numerical solution techniques. To assess the performance of the proposed approach, we employ the Catté-Lions-Morel-Coll model to restore noisy images, by solving the PDE with the three numerical solution schemes. For all of the benchmark images employed in our experiments, and for every level of noise applied, we observe that the best image restored by using the mimetic method is closer to the noise-free image than the best images restored by the other two methods tested. These results motivate further studies of the application of the mimetic methods to other imaging problems. Mathematical subject classification: Primary: 68U10; Secondary: 65L12.
Hanert, Emmanuel; Piret, Cécile; International Symposium on Fractional PDEs: Theory, Numerics and Applications
2013-01-01
One of the ongoing issues with fractional-order diffusion models is the design of efficient numerical schemes for the space and time discretizations. Until now, most models have relied on a low-order finite difference (FD) method to discretize both the fractional-order space and time derivatives. Some numerical schemes using low-order finite elements (FE) have also been proposed. Both the FD and FE methods have long been used to solve integer-order partial differential equations. These low-or...
Comparison of Taylor finite difference and window finite difference and their application in FDTD
Xiao, F.; Tang, X. H.; Zhang, X. J.
2006-09-01
The finite difference time domain (FDTD) method is an important tool in numerical electromagnetic simulation. There are many ways to construct a finite difference approximation such as the Taylor series expansion theorem, the filtering theory, etc. This paper aims to provide the comparison between the Taylor finite difference (TFD) scheme based on the Taylor series expansion theorem and the window finite difference (WFD) scheme based on the filtering theory. Their properties have been examined in detail, separately. In addition, the formula of the generalized finite difference (GFD) scheme is presented, which can include both the TFD scheme and the WFD scheme. Furthermore, their application in the numerical solution of Maxwell's equations is presented. The formulas for the stability criterion and the numerical dispersion relation are derived and analyzed. In order to evaluate their performance more accurately, a new definition of error is presented. Upon it, the effect of several factors including the grid resolution, the Courant number and the aspect ratio of the cell on the performance of the numerical dispersion is examined.
High-resolution finite-difference algorithms for conservation laws
International Nuclear Information System (INIS)
A new class of Total Variation Decreasing (TVD) schemes for 2-dimensional scalar conservation laws is constructed using either flux-limited or slope-limited numerical fluxes. The schemes are proven to have formal second-order accuracy in regions where neither u/sub x/ nor y/sub y/ vanishes. A new class of high-resolution large-time-step TVD schemes is constructed by adding flux-limited correction terms to the first-order accurate large-time-step version of the Engquist-Osher scheme. The use of the transport-collapse operator in place of the exact solution operator for the construction of difference schemes is studied. The production of spurious extrema by difference schemes is studied. A simple condition guaranteeing the nonproduction of spurious extrema is derived. A sufficient class of entropy inequalities for a conservation law with a flux having a single inflection point is presented. Finite-difference schemes satisfying a discrete version of each entropy inequality are only first-order accurate
Moufekkir, Fayçal; Moussaoui, Mohammed Amine; Mezrhab, Ahmed; Naji, Hassan
2015-04-01
The coupled double diffusive natural convection and radiation in a tilted and differentially heated square cavity containing a non-gray air-CO2 (or air-H2O) mixtures was numerically investigated. The horizontal walls are insulated and impermeable and the vertical walls are maintained at different temperatures and concentrations. The hybrid lattice Boltzmann method with the multiple-relaxation time model is used to compute the hydrodynamics and the finite difference method to determine temperatures and concentrations. The discrete ordinates method combined to the spectral line-based weighted sum of gray gases model is used to compute the radiative term and its spectral aspect. The effects of the inclination angle on the flow, thermal and concentration fields are analyzed for both aiding and opposing cases. It was found that radiation gas modifies the structure of the velocity and thermal fields by generating inclined stratifications and promoting the instabilities in opposing flows.
International Nuclear Information System (INIS)
There dimensional hydrodynamical calculations with heat transfer for nuclear reactors are complicated and actual tasks, their singularity is high numbers of Reynolds Re ? 106. The offered paper is one of initial development stages programs for problem solving the similar class. Operation contains exposition: mathematical setting of the task for the equations of Navier-Stokes with heat transfer compiling of space difference schemes by a method of check sizes, deriving of difference equations for pressure. The steady explicit methods of a solution of rigid tasks included in DUMKA program, and research of areas of their stability are used. Outcomes of numerical experiments of current of liquid in channels of rectangular cut are reduced. The complete spectrum analysis of the considered task is done (Authors)
Finite Difference Method of Modelling Groundwater Flow
Directory of Open Access Journals (Sweden)
Magnus.U. Igboekwe
2011-03-01
Full Text Available In this study, finite difference method is used to solve the equations that govern groundwater flow to obtain flow rates, flow direction and hydraulic heads through an aquifer. The aim therefore is to discuss the principles of Finite Difference Method and its applications in groundwater modelling. To achieve this, a rectangular grid is overlain an aquifer in order to obtain an exact solution. Initial and boundary conditions are then determined. By discretizing the system into grids and cells that are small compared to the entire aquifer, exact solutions are obtained. A flow chart of the computational algorithm for particle tracking is also developed. Results show that under a steady-state flow with no recharge, pathlines coincide with streamlines. It is also found that the accuracy of the numerical solution by Finite Difference Method is largely dependent on initial particle distribution and number of particles assigned to a cell. It is therefore concluded that Finite Difference Method can be used to predict the future direction of flow and particle location within a simulation domain.
High order discretization schemes for stochastic volatility models
Jourdain, Benjamin
2009-01-01
In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using It\\^o's formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose - a scheme, based on the Milstein discretization of this SDE, with order one of weak trajectorial convergence for the asset price, - a scheme, based on the Ninomiya-Victoir discretization of this SDE, with order two of weak convergence for the asset price. We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an Orstein-Uhlenbeck process. We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles [2008a,b].
High-order Finite Difference Solution of Euler Equations for Nonlinear Water Waves
DEFF Research Database (Denmark)
Christiansen, Torben Robert Bilgrav; Bingham, Harry B.
2012-01-01
The incompressible Euler equations are solved with a free surface, the position of which is captured by applying an Eulerian kinematic boundary condition. The solution strategy follows that of [1, 2], applying a coordinate-transformation to obtain a time-constant spatial computational domain which is discretized using arbitrary-order finite difference schemes on a staggered grid with one optional stretching in each coordinate direction. The momentum equations and kinematic free surface condition are integrated in time using the classic fourth-order Runge-Kutta scheme. Mass conservation is satisfied implicitly, at the end of each time stage, by constructing the pressure from a discrete Poisson equation, derived from the discrete continuity and momentum equations and taking the time-dependent physical domain into account. An efficient preconditionedDefect Correction (DC) solution of the discrete Poisson equation for the pressure is presented, in which the preconditioning step is based on an order-multigrid formulation with a direct solution on the lowest order-level. This ensures fast convergence of the DC method with a computational effort which scales linearly with the problem size. Results obtained with a two-dimensional implementation of the model are compared with highly accurate stream function solutions to the nonlinear wave problem, which show the approximately expected convergence rates and a clear advantage of using high-order finite difference schemes in combination with the Euler equations.
Novel coupling scheme to control dynamics of coupled discrete systems
Shekatkar, Snehal M.; Ambika, G.
2015-08-01
We present a new coupling scheme to control spatio-temporal patterns and chimeras on 1-d and 2-d lattices and random networks of discrete dynamical systems. The scheme involves coupling with an external lattice or network of damped systems. When the system network and external network are set in a feedback loop, the system network can be controlled to a homogeneous steady state or synchronized periodic state with suppression of the chaotic dynamics of the individual units. The control scheme has the advantage that its design does not require any prior information about the system dynamics or its parameters and works effectively for a range of parameters of the control network. We analyze the stability of the controlled steady state or amplitude death state of lattices using the theory of circulant matrices and Routh-Hurwitz criterion for discrete systems and this helps to isolate regions of effective control in the relevant parameter planes. The conditions thus obtained are found to agree well with those obtained from direct numerical simulations in the specific context of lattices with logistic map and Henon map as on-site system dynamics. We show how chimera states developed in an experimentally realizable 2-d lattice can be controlled using this scheme. We propose this mechanism can provide a phenomenological model for the control of spatio-temporal patterns in coupled neurons due to non-synaptic coupling with the extra cellular medium. We extend the control scheme to regulate dynamics on random networks and adapt the master stability function method to analyze the stability of the controlled state for various topologies and coupling strengths.
Choi, S.-J.; Giraldo, F. X.; Kim, J.; Shin, S.
2014-06-01
The non-hydrostatic (NH) compressible Euler equations of dry atmosphere are solved in a simplified two dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss-Lobatto-Legendre (GLL) quadrature points. The FDM employs a third-order upwind biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative terms and quadrature. The Euler equations used here are in a flux form based on the hydrostatic pressure vertical coordinate, which are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma-pressure vertical coordinate is implemented in this model. We verified the model by conducting widely used standard benchmark tests: the inertia-gravity wave, rising thermal bubble, density current wave, and linear hydrostatic mountain wave. The results from those tests demonstrate that the horizontally spectral element vertically finite difference model is accurate and robust. By using the 2-D slice model, we effectively show that the combined spatial discretization method of the spectral element and finite difference method in the horizontal and vertical directions, respectively, offers a viable method for the development of a NH dynamical core.
Directory of Open Access Journals (Sweden)
S.-J. Choi
2014-06-01
Full Text Available The non-hydrostatic (NH compressible Euler equations of dry atmosphere are solved in a simplified two dimensional (2-D slice framework employing a spectral element method (SEM for the horizontal discretization and a finite difference method (FDM for the vertical discretization. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss–Lobatto–Legendre (GLL quadrature points. The FDM employs a third-order upwind biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative terms and quadrature. The Euler equations used here are in a flux form based on the hydrostatic pressure vertical coordinate, which are the same as those used in the Weather Research and Forecasting (WRF model, but a hybrid sigma-pressure vertical coordinate is implemented in this model. We verified the model by conducting widely used standard benchmark tests: the inertia-gravity wave, rising thermal bubble, density current wave, and linear hydrostatic mountain wave. The results from those tests demonstrate that the horizontally spectral element vertically finite difference model is accurate and robust. By using the 2-D slice model, we effectively show that the combined spatial discretization method of the spectral element and finite difference method in the horizontal and vertical directions, respectively, offers a viable method for the development of a NH dynamical core.
Bu, Weiping; Tang, Yifa; Wu, Yingchuan; Yang, Jiye
2015-07-01
In this paper, a class of two-dimensional space and time fractional Bloch-Torrey equations (2D-STFBTEs) are considered. Some definitions and properties of fractional derivative spaces are presented. By finite difference method and Galerkin finite element method, a semi-discrete variational formulation for 2D-STFBTEs is obtained. The stability and convergence of the semi-discrete form are discussed. Then, a fully discrete scheme of 2D-STFBTEs is derived and the convergence is investigated. Finally, some numerical examples based on linear piecewise polynomials and quadratic piecewise polynomials are given to prove the correctness of our theoretical analysis.
International Nuclear Information System (INIS)
The NEWT (NEW Transport algorithm) code is a multi-group discrete ordinates neutral-particle transport code with flexible meshing capabilities. This code employs the Extended Step Characteristic spatial discretization approach using arbitrary polygonal mesh cells. Until recently, the coarse mesh finite difference acceleration scheme in NEWT for fission source iteration has been available only for rectangular domain boundaries because of the limitation to rectangular coarse meshes. Therefore no acceleration scheme has been available for triangular or hexagonal problem boundaries. A conventional and a new partial-current based coarse mesh finite difference acceleration schemes with unstructured coarse meshes have been implemented within NEWT to support any form of domain boundaries. The computational results show that the new acceleration schemes works well, with performance often improved over the earlier two-level rectangular approach.
Discrete unified gas kinetic scheme on unstructured meshes
Zhu, Lianhua; Xu, Kun
2015-01-01
The recently proposed discrete unified gas kinetic scheme (DUGKS) is a finite volume method for deterministic solution of the Boltzmann model equation with asymptotic preserving property. In DUGKS, the numerical flux of the distribution function is determined from a local numerical solution of the Boltzmann model equation using an unsplitting approach. The time step and mesh resolution are not restricted by the molecular collision time and mean free path. To demonstrate the capacity of DUGKS in practical problems, this paper extends the DUGKS to arbitrary unstructured meshes. Several tests of both internal and external flows are performed, which include the cavity flow ranging from continuum to free molecular regimes, a multiscale flow between two connected cavities with a pressure ratio of 10000, and a high speed flow over a cylinder in slip and transitional regimes. The numerical results demonstrate the effectiveness of the DUGKS in simulating multiscale flow problems.
Compatible discrete operator schemes on polyhedral meshes for elliptic and Stokes equations
International Nuclear Information System (INIS)
This thesis presents a new class of spatial discretization schemes on polyhedral meshes, called Compatible Discrete Operator (CDO) schemes and their application to elliptic and Stokes equations In CDO schemes, preserving the structural properties of the continuous equations is the leading principle to design the discrete operators. De Rham maps define the degrees of freedom according to the physical nature of fields to discretize. CDO schemes operate a clear separation between topological relations (balance equations) and constitutive relations (closure laws). Topological relations are related to discrete differential operators, and constitutive relations to discrete Hodge operators. A feature of CDO schemes is the explicit use of a second mesh, called dual mesh, to build the discrete Hodge operator. Two families of CDO schemes are considered: vertex-based schemes where the potential is located at (primal) mesh vertices, and cell-based schemes where the potential is located at dual mesh vertices (dual vertices being in one-to-one correspondence with primal cells). The CDO schemes related to these two families are presented and their convergence is analyzed. A first analysis hinges on an algebraic definition of the discrete Hodge operator and allows one to identify three key properties: symmetry, stability, and P0-consistency. A second analysis hinges on a definition of the discrete Hodge operator using reconstruction operators, and the requirements on these reconstruction operators are identified. In addition, CDO schemes provide a unified vision on a broad class of schemes proposed in the literature (finite element, finite element, mimetic schemes... ). Finally, the reliability and the efficiency of CDO schemes are assessed on various test cases and several polyhedral meshes. (author)
Convergence of the Approximation Scheme to American Option Pricing via the Discrete Morse Semiflow
International Nuclear Information System (INIS)
We consider the approximation scheme to the American call option via the discrete Morse semiflow, which is a minimizing scheme of a time semi-discretized variational functional. In this paper we obtain a rate of convergence of approximate solutions and the convergence of approximate free boundaries. We mainly apply the theory of variational inequalities and that of viscosity solutions to prove our results.
Using the Finite Difference Calculus to Sum Powers of Integers.
Zia, Lee
1991-01-01
Summing powers of integers is presented as an example of finite differences and antidifferences in discrete mathematics. The interrelation between these concepts and their analogues in differential calculus, the derivative and integral, is illustrated and can form the groundwork for students' understanding of differential and integral calculus.…
Discretizations for the Incompressible Navier-Stokes Equations based on the Lattice Boltzmann Method
Junk, Michael; Klar, Axel
1999-01-01
A discrete velocity model with spatial and velocity discretization based on a lattice Boltzmann method is considered in the low Mach number limit. A uniform numerical scheme for this model is investigated. In the limit, the scheme reduces to a finite difference scheme for the incompressible Navier-Stokes equation which is a projection method with a second order spatial discretization on a regular grid. The discretization is analyzed and the method is compared to Chorin's original spatial disc...
Implicit time-dependent finite different algorithm for quench simulation
International Nuclear Information System (INIS)
A magnet in a fusion machine has many difficulties in its application because of requirement of a large operating current, high operating field and high breakdown voltage. A cable-in-conduit (CIC) conductor is the best candidate to overcome these difficulties. However, there remained uncertainty in a quench event in the cable-in-conduit conductor because of a difficulty to analyze a fluid dynamics equation. Several scientists, then, developed the numerical code for the quench simulation. However, most of them were based on an explicit time-dependent finite difference scheme. In this scheme, a discrete time increment is strictly restricted by CFL (Courant-Friedrichs-Lewy) condition. Therefore, long CPU time was consumed for the quench simulation. Authors, then, developed a new quench simulation code, POCHI1, which is based on an implicit time dependent scheme. In POCHI1, the fluid dynamics equation is linearlized according to a procedure applied by Beam and Warming and then, a tridiagonal system can be offered. Therefore, no iteration is necessary to solve the fluid dynamics equation. This leads great reduction of the CPU time. Also, POCHI1 can cope with non-linear boundary condition. In this study, comparison with experimental results was carried out. The normal zone propagation behavior was investigated in two samples of CIC conductors which had different hydraulic diameters. The measured and simulated normal zone propagation length showed relatively good agreement. However, the behavior of the normal voltage shows a little disagreement. These results indicate necessity to improve the treatment of the heat transfer coefficient in the turbulent flow region and the electric resistivity of the copper stabilizer in high temperature and high field region. (author)
Froese, Brittany D
2012-01-01
The theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear Partial Differential Equations (PDEs) such as the elliptic Monge-Amp\\`ere equation. The approximation theory of Barles-Souganidis [Barles and Souganidis, Asymptotic Anal., 4 (1999) 271-283] requires that numerical schemes be monotone (or elliptic in the sense of [Oberman, SIAM J. Numer. Anal, 44 (2006) 879-895]. But such schemes have limited accuracy. In this article, we establish a convergence result for nearly monotone schemes. This allows us to construct finite difference discretizations of arbitrarily high-order. We demonstrate that the higher accuracy is achieved when solutions are sufficiently smooth. In addition, the filtered scheme provides a natural detection principle for singularities. We employ this framework to construct a formally second-order scheme for the Monge-Amp\\`ere equation and present computational results on smooth and singular solutions.
An energy conserving finite-difference model of Maxwell's equations for soliton propagation
Bachiri, H; Vázquez, L
1997-01-01
We present an energy conserving leap-frog finite-difference scheme for the nonlinear Maxwell's equations investigated by Hile and Kath [C.V.Hile and W.L.Kath, J.Opt.Soc.Am.B13, 1135 (96)]. The model describes one-dimensional scalar optical soliton propagation in polarization preserving nonlinear dispersive media. The existence of a discrete analog of the underlying continuous energy conservation law plays a central role in the global accuracy of the scheme and a proof of its generalized nonlinear stability using energy methods is given. Numerical simulations of initial fundamental, second and third-order hyperbolic secant soliton pulses of fixed spatial full width at half peak intensity containing as few as 4 and 8 optical carrier wavelengths, confirm the stability, accuracy and efficiency of the algorithm. The effect of a retarded nonlinear response time of the media modeling Raman scattering is under current investigation in this context.
On the Total Variation of High-Order Semi-Discrete Central Schemes for Conservation Laws
Bryson, Steve; Levy, Doron
2004-01-01
We discuss a new fifth-order, semi-discrete, central-upwind scheme for solving one-dimensional systems of conservation laws. This scheme combines a fifth-order WENO reconstruction, a semi-discrete central-upwind numerical flux, and a strong stability preserving Runge-Kutta method. We test our method with various examples, and give particular attention to the evolution of the total variation of the approximations.
Unifying scheme for generating discrete integrable systems including inhomogeneous and hybrid models
Kundu, Anjan
2002-01-01
A unifying scheme based on an ancestor model is proposed for generating a wide range of integrable discrete and continuum as well as inhomogeneous and hybrid models. They include in particular discrete versions of sine-Gordon, Landau-Lifshitz, nonlinear Schr\\"odinger (NLS), derivative NLS equations, Liouville model, (non-)relativistic Toda chain, Ablowitz-Ladik model etc. Our scheme introduces the possibility of building a novel class of integrable hybrid systems including m...
Chen, W
2001-01-01
This paper is concerned with a few novel RBF-based numerical schemes discretizing partial differential equations. For boundary-type methods, we derive the indirect and direct symmetric boundary knot methods (BKM). The resulting interpolation matrix of both is always symmetric irrespective of boundary geometry and conditions. In particular, the direct BKM applies the practical physical variables rather than expansion coefficients and becomes very competitive to the boundary element method. On the other hand, based on the multiple reciprocity principle, we invent the RBF-based boundary particle method (BPM) for general inhomogeneous problems without a need using inner nodes. The direct and symmetric BPM schemes are also developed. For domain-type RBF discretization schemes, by using the Green integral we develop a new Hermite RBF scheme called as the modified Kansa method (MKM), which differs from the symmetric Hermite RBF scheme in that the MKM discretizes both governing equation and boundary conditions on the...
The mimetic finite difference method for elliptic problems
Veiga, Lourenço Beirão; Manzini, Gianmarco
2014-01-01
This book describes the theoretical and computational aspects of the mimetic finite difference method for a wide class of multidimensional elliptic problems, which includes diffusion, advection-diffusion, Stokes, elasticity, magnetostatics and plate bending problems. The modern mimetic discretization technology developed in part by the Authors allows one to solve these equations on unstructured polygonal, polyhedral and generalized polyhedral meshes. The book provides a practical guide for those scientists and engineers that are interested in the computational properties of the mimetic finite difference method such as the accuracy, stability, robustness, and efficiency. Many examples are provided to help the reader to understand and implement this method. This monograph also provides the essential background material and describes basic mathematical tools required to develop further the mimetic discretization technology and to extend it to various applications.
Symmetry-preserving discrete schemes for some heat transfer equations
Bakirova, Margarita; Dorodnitsyn, Vladimir; Kozlov, Roman
2004-01-01
Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their di...
A parallel adaptive finite difference algorithm for petroleum reservoir simulation
Energy Technology Data Exchange (ETDEWEB)
Hoang, Hai Minh
2005-07-01
Adaptive finite differential for problems arising in simulation of flow in porous medium applications are considered. Such methods have been proven useful for overcoming limitations of computational resources and improving the resolution of the numerical solutions to a wide range of problems. By local refinement of the computational mesh where it is needed to improve the accuracy of solutions, yields better solution resolution representing more efficient use of computational resources than is possible with traditional fixed-grid approaches. In this thesis, we propose a parallel adaptive cell-centered finite difference (PAFD) method for black-oil reservoir simulation models. This is an extension of the adaptive mesh refinement (AMR) methodology first developed by Berger and Oliger (1984) for the hyperbolic problem. Our algorithm is fully adaptive in time and space through the use of subcycling, in which finer grids are advanced at smaller time steps than the coarser ones. When coarse and fine grids reach the same advanced time level, they are synchronized to ensure that the global solution is conservative and satisfy the divergence constraint across all levels of refinement. The material in this thesis is subdivided in to three overall parts. First we explain the methodology and intricacies of AFD scheme. Then we extend a finite differential cell-centered approximation discretization to a multilevel hierarchy of refined grids, and finally we are employing the algorithm on parallel computer. The results in this work show that the approach presented is robust, and stable, thus demonstrating the increased solution accuracy due to local refinement and reduced computing resource consumption. (Author)
Symmetry-preserving discrete schemes for some heat transfer equations
Bakirova, M; Kozlov, R; Bakirova, Margarita; Dorodnitsyn, Vladimir; Kozlov, Roman
2004-01-01
Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their difference counterparts.
Novel Two-Scale Discretization Schemes for Lagrangian Hydrodynamics
International Nuclear Information System (INIS)
In this report we propose novel higher order conservative schemes of discontinuous Galerkin (or DG) type for the equations of gas dynamics in Lagrangian coordinates suitable for general unstructured finite element meshes. The novelty of our approach is in the formulation of two-scale non-oscillatory function recovery procedures utilizing integral moments of the quantities of interest (pressure and velocity). The integral moments are computed on a primary mesh (cells or zones) which defines our original scale that governs the accuracy of the schemes. In the non-oscillatory smooth function recovery procedures, we introduce a finer mesh which defines the second scale. Mathematically, the recovery can be formulated as nonlinear energy functional minimization subject to equality and nonlinear inequality constraints. The schemes are highly accurate due to both the embedded (local) mesh refinement features as well as the ability to utilize higher order integral moments. The new DG schemes seem to offer an alternative to currently used artificial viscosity techniques and limiters since the two-scale recovery procedures aim at resolving these issues. We report on some preliminary tests for the lowest order case, and outline some possible future research directions
International Nuclear Information System (INIS)
The Coarse Mesh Finite Difference (CMFD) acceleration was devised to enhance the computational acceleration for the high order diffusion calculation such as nodal diffusion. And then it could be successfully applied to the acceleration of the transport eigenvalue calculations using Method of Characteristics (MOC). This method is quite effective for the fission source iteration by conserving the reaction rates inside each coarse mesh through the non-linear updating of the interface net currents from the high order transport equation. Fourier analysis for the fixed source and eigenvalue problems showed that the coupling of the high order transport and the low order CMFD calculations is not unconditionally stable. The partial current based CMFD (pCMFD) was devised to consider the interface partial currents for the better performance. Fourier analysis for the eigenvalue problems showed that the coupling of the high order SC (Step Characteristics) transport and the low order pCMFD calculations is unconditionally stable. The NEWT code is a multi-group discrete ordinate neutron transport code with flexible meshing capabilities. This code adopts the Extended Step Characteristic (ESC) approach for the arbitrary polygon meshes. In NEWT an acceleration scheme for the fission source iteration has been available only for the rectangular domain boundaries by using coarse mesh finite difference acceleration method only with rectangular coarse meshes. Therefore no acceleration schemee meshes. Therefore no acceleration scheme could be applied to the wedge, triangle, hexagon and their symmetric domain boundaries. The conventional and the partial current based unstructured CMFD acceleration schemes (uCMFD and upCMFD) with the unstructured coarse meshes were implemented to be used for any domain boundaries
Weighted Average Finite Difference Methods for Fractional Reaction-Subdiffusion Equation
Directory of Open Access Journals (Sweden)
Nasser Hassen SWEILAM
2014-04-01
Full Text Available In this article, a numerical study for fractional reaction-subdiffusion equations is introduced using a class of finite difference methods. These methods are extensions of the weighted average methods for ordinary (non-fractional reaction-subdiffusion equations. A stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. Simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, are given and checked numerically. Numerical test examples, figures, and comparisons have been presented for clarity.doi:10.14456/WJST.2014.50
Accuracy Analysis for Finite-Volume Discretization Schemes on Irregular Grids
Diskin, Boris; Thomas, James L.
2010-01-01
A new computational analysis tool, downscaling test, is introduced and applied for studying the convergence rates of truncation and discretization errors of nite-volume discretization schemes on general irregular (e.g., unstructured) grids. The study shows that the design-order convergence of discretization errors can be achieved even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all. The downscaling test is a general, efficient, accurate, and practical tool, enabling straightforward extension of verification and validation to general unstructured grid formulations. It also allows separate analysis of the interior, boundaries, and singularities that could be useful even in structured-grid settings. There are several new findings arising from the use of the downscaling test analysis. It is shown that the discretization accuracy of a common node-centered nite-volume scheme, known to be second-order accurate for inviscid equations on triangular grids, degenerates to first order for mixed grids. Alternative node-centered schemes are presented and demonstrated to provide second and third order accuracies on general mixed grids. The local accuracy deterioration at intersections of tangency and in flow/outflow boundaries is demonstrated using the DS tests tailored to examining the local behavior of the boundary conditions. The discretization-error order reduction within inviscid stagnation regions is demonstrated. The accuracy deterioration is local, affecting mainly the velocity components, but applies to any order scheme.
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Thompson, K.G.
2000-11-01
In this work, we develop a new spatial discretization scheme that may be used to numerically solve the neutron transport equation. This new discretization extends the family of corner balance spatial discretizations to include spatial grids of arbitrary polyhedra. This scheme enforces balance on subcell volumes called corners. It produces a lower triangular matrix for sweeping, is algebraically linear, is non-negative in a source-free absorber, and produces a robust and accurate solution in thick diffusive regions. Using an asymptotic analysis, we design the scheme so that in thick diffusive regions it will attain the same solution as an accurate polyhedral diffusion discretization. We then refine the approximations in the scheme to reduce numerical diffusion in vacuums, and we attempt to capture a second order truncation error. After we develop this Upstream Corner Balance Linear (UCBL) discretization we analyze its characteristics in several limits. We complete a full diffusion limit analysis showing that we capture the desired diffusion discretization in optically thick and highly scattering media. We review the upstream and linear properties of our discretization and then demonstrate that our scheme captures strictly non-negative solutions in source-free purely absorbing media. We then demonstrate the minimization of numerical diffusion of a beam and then demonstrate that the scheme is, in general, first order accurate. We also note that for slab-like problems our method actually behaves like a second-order method over a range of cell thicknesses that are of practical interest. We also discuss why our scheme is first order accurate for truly 3D problems and suggest changes in the algorithm that should make it a second-order accurate scheme. Finally, we demonstrate 3D UCBL's performance on several very different test problems. We show good performance in diffusive and streaming problems. We analyze truncation error in a 3D problem and demonstrate robustness in a coarsely discretized problem that contains sharp boundary layers. We also examine eigenvalue and fixed source problems with mixed-shape meshes, anisotropic scattering and multi-group cross sections. Finally, we simulate the MOX fuel assembly in the Advance Test Reactor.
International Nuclear Information System (INIS)
In this work, we develop a new spatial discretization scheme that may be used to numerically solve the neutron transport equation. This new discretization extends the family of corner balance spatial discretizations to include spatial grids of arbitrary polyhedra. This scheme enforces balance on subcell volumes called corners. It produces a lower triangular matrix for sweeping, is algebraically linear, is non-negative in a source-free absorber, and produces a robust and accurate solution in thick diffusive regions. Using an asymptotic analysis, we design the scheme so that in thick diffusive regions it will attain the same solution as an accurate polyhedral diffusion discretization. We then refine the approximations in the scheme to reduce numerical diffusion in vacuums, and we attempt to capture a second order truncation error. After we develop this Upstream Corner Balance Linear (UCBL) discretization we analyze its characteristics in several limits. We complete a full diffusion limit analysis showing that we capture the desired diffusion discretization in optically thick and highly scattering media. We review the upstream and linear properties of our discretization and then demonstrate that our scheme captures strictly non-negative solutions in source-free purely absorbing media. We then demonstrate the minimization of numerical diffusion of a beam and then demonstrate that the scheme is, in general, first order accurate. We also note that for slab-like problems our also note that for slab-like problems our method actually behaves like a second-order method over a range of cell thicknesses that are of practical interest. We also discuss why our scheme is first order accurate for truly 3D problems and suggest changes in the algorithm that should make it a second-order accurate scheme. Finally, we demonstrate 3D UCBL's performance on several very different test problems. We show good performance in diffusive and streaming problems. We analyze truncation error in a 3D problem and demonstrate robustness in a coarsely discretized problem that contains sharp boundary layers. We also examine eigenvalue and fixed source problems with mixed-shape meshes, anisotropic scattering and multi-group cross sections. Finally, we simulate the MOX fuel assembly in the Advance Test Reactor
Normal scheme for solving the transport equation independently of spatial discretization
International Nuclear Information System (INIS)
To solve the discrete ordinates neutron transport equation, a general order nodal scheme is used, where nodes are allowed to have different orders of approximation and the whole system reaches a final order distribution. Independence in the election of system discretization and order of approximation is obtained without loss of accuracy. The final equations and the iterative method to reach a converged order solution were implemented in a two-dimensional computer code to solve monoenergetic, isotropic scattering, external source problems. Two benchmark problems were solved using different automatic selection order methods. Results show accurate solutions without spatial discretization, regardless of the initial selection of distribution order. (author)
A unified formalism for spatial discretization schemes of transport equations in slab geometry
International Nuclear Information System (INIS)
It is shown that most of the spatial discretization schemes of transport equations in slab geometry which have been developed recently are particular applications of a general finite element oriented formalism developed by this author and his collaborators for the numerical integration of systems of stiff ordinary differential equations. (author)
Fully discrete Galerkin schemes for the nonlinear and nonlocal Hartree equation
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Walter H. Aschbacher
2009-01-01
Full Text Available We study the time dependent Hartree equation in the continuum, the semidiscrete, and the fully discrete setting. We prove existence-uniqueness, regularity, and approximation properties for the respective schemes, and set the stage for a controlled numerical computation of delicate nonlinear and nonlocal features of the Hartree dynamics in various physical applications.
Total internal reflection microscopy: examination of competitive schemes via discrete sources method
International Nuclear Information System (INIS)
The discrete sources method has been applied to perform a computer simulation analysis of different total internal reflection microscopy schemes. It has been found that the positioning of the objective lens beneath a glass prism can provide a considerable advantage for determination of the particle–film distance
An Efficient Signcryption Scheme based on The Elliptic Curve Discrete Logarithm Problem
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Fatima Amounas
2013-02-01
Full Text Available Elliptic Curve Cryptosystems (ECC have recently received significant attention by researchers due to their performance. Here, an efficient signcryption scheme based on elliptic curve will be proposed, which can effectively combine the functionalities of digital signature and encryption. Since the security of the proposed method is based on the difficulty of solving discrete logarithm over an elliptic curve. The purposes of this paper are to demonstrate how to specify signcryption scheme on elliptic curves over finite field, and to examine the efficiency of such scheme. The results analysis are explained.
Asakura, T; Ishizuka, T; Miyajima, T; Toyoda, M; Sakamoto, S
2014-09-01
Due to limitations of computers, prediction of structure-borne sound remains difficult for large-scale problems. Herein a prediction method for low-frequency structure-borne sound transmissions on concrete structures using the finite-difference time-domain scheme is proposed. The target structure is modeled as a composition of multiple plate elements to reduce the dimensions of the simulated vibration field from three-dimensional discretization by solid elements to two-dimensional discretization. This scheme reduces both the calculation time and the amount of required memory. To validate the proposed method, the vibration characteristics using the numerical results of the proposed scheme are compared to those measured for a two-level concrete structure. Comparison of the measured and simulated results suggests that the proposed method can be used to simulate real-scale structures. PMID:25190384
Young, P.; Hao, S.; Martinsson, P. G.
2012-06-01
A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in R3 is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a surface to a sequence of BIEs defined on a generating curve for the surface. It can handle loads that are not necessarily rotationally symmetric. Nyström discretization is used to discretize the BIEs on the generating curve. The quadrature is a high-order Gaussian rule that is modified near the diagonal to retain high-order accuracy for singular kernels. The reduction in dimensionality, along with the use of high-order accurate quadratures, leads to small linear systems that can be inverted directly via, e.g., Gaussian elimination. This makes the scheme particularly fast in environments involving multiple right hand sides. It is demonstrated that for BIEs associated with the Laplace and Helmholtz equations, the kernel in the reduced equations can be evaluated very rapidly by exploiting recursion relations for Legendre functions. Numerical examples illustrate the performance of the scheme; in particular, it is demonstrated that for a BIE associated with Laplace's equation on a surface discretized using 320,800 points, the set-up phase of the algorithm takes 1 min on a standard laptop, and then solves can be executed in 0.5 s.
A finite-difference lattice Boltzmann approach for gas microflows
Ghiroldi, G P
2013-01-01
Finite-difference Lattice Boltzmann (LB) models are proposed for simulating gas flows in devices with microscale geometries. The models employ the roots of half-range Gauss-Hermite polynomials as discrete velocities. Unlike the standard LB velocity-space discretizations based on the roots of full-range Hermite polynomials, using the nodes of a quadrature defined in the half-space permits a consistent treatment of kinetic boundary conditions. The possibilities of the proposed LB models are illustrated by studying the one-dimensional Couette flow and the two-dimensional driven cavity flow. Numerical and analytical results show an improved accuracy in finite Knudsen flows as compared with standard LB models.
Finite difference heterogeneous multi-scale method for homogenization problems
International Nuclear Information System (INIS)
In this paper, we propose a numerical method, the finite difference heterogeneous multi-scale method (FD-HMM), for solving multi-scale parabolic problems. Based on the framework introduced in [Commun. Math. Sci. 1 (1) 87], the numerical method relies on the use of two different schemes for the original equation, at different grid level which allows to give numerical results at a much lower cost than solving the original equations. We describe the strategy for constructing such a method, discuss generalization for cases with time dependency, random correlated coefficients, non-conservative form and implementation issues. Finally, the new method is illustrated with several test examples
Error Estimate for a Fully Discrete Spectral Scheme for Korteweg-de Vries-Kawahara Equation
Koley, U
2011-01-01
We are concerned with the convergence of spectral method for the numerical solution of the initial-boundary value problem associated to the Korteweg-de Vries-Kawahara equation (in short Kawahara equation), which is a transport equation perturbed by dispersive terms of 3rd and 5th order. This equation appears in several fluid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the standard Fourier- Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L2-error bound of spectral accuracy in space and of second-order accuracy in time.
Energy Technology Data Exchange (ETDEWEB)
Kim, S. [Purdue Univ., West Lafayette, IN (United States)
1994-12-31
Parallel iterative procedures based on domain decomposition techniques are defined and analyzed for the numerical solution of wave propagation by finite element and finite difference methods. For finite element methods, in a Lagrangian framework, an efficient way for choosing the algorithm parameter as well as the algorithm convergence are indicated. Some heuristic arguments for finding the algorithm parameter for finite difference schemes are addressed. Numerical results are presented to indicate the effectiveness of the methods.
Finite difference approximations for a class of non-local parabolic equations
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Hong-Ming Yin
1997-03-01
Full Text Available In this paper we study finite difference procedures for a class of parabolic equations with non-local boundary condition. The semi-implicit and fully implicit backward Euler schemes are studied. It is proved that both schemes preserve the maximum principle and monotonicity of the solution of the original equation, and fully-implicit scheme also possesses strict monotonicity. It is also proved that finite difference solutions approach to zero as tÃ¢Â†Â’Ã¢ÂˆÂž exponentially. The numerical results of some examples are presented, which support our theoretical justifications.
Non-linear analysis of skew thin plate by finite difference method
International Nuclear Information System (INIS)
This paper deals with a discrete analysis capability for predicting the geometrically nonlinear behavior of skew thin plate subjected to uniform pressure. The differential equations are discretized by means of the finite difference method which are used to determine the deflections and the in-plane stress functions of plates and reduced to several sets of linear algebraic simultaneous equations. For the geometrically non-linear, large deflection behavior of the plate, the non-linear plate theory is used for the analysis. An iterative scheme is employed to solve these quasi-linear algebraic equations. Several problems are solved which illustrate the potential of the method for predicting the finite deflection and stress. For increasing lateral pressures, the maximum principal tensile stress occurs at the center of the plate and migrates toward the corners as the load increases. It was deemed important to describe the locations of the maximum principal tensile stress as it occurs. The load-deflection relations and the maximum bending and membrane stresses for each case are presented and discussed
Validation of NSWING, a multi-core finite difference code for tsunami propagation and run-up
Miranda, J. M. A.; Luis, J. M. F.; Reis, C.; Omira, R.; Baptista, M. A.
2014-12-01
We present the finite difference tsunami code NSWING (Non-linear Shallow Water model With Nested Grids), that solves the non-linear shallow water equations using the discretization and explicit leap-frog finite difference scheme, in a Cartesian or Spherical frame, as developed by Liu et al. (1998). An open boundary condition is used on the outward limit of the grid, whenever it does not correspond to land. The model also incorporates Coriolis acceleration, bottom friction and a moving boundary scheme to model run-up. Multiple levels of nesting are possible. NSWING runs on MS windows operating system using more than one core. The code is applied to classical benchmark tests (Synolakis et al., 2007) and to a test case in SW Portugal. It is shown that the code reproduces well the numerical benchmarks, improves its accuracy with increasing resolution and ensures mass conservation. It is also shown that NSWING can efficiently provide inundation modelling for high resolution studies. This work is a contribution to GEONUM project FCT-ANR/MAT-NAN/0122/2012
Elementary introduction to finite difference equations
International Nuclear Information System (INIS)
An elementary description is given of the basic vocabulary and concepts associated with finite difference modeling. The material discussed is biased toward the types of large computer programs used at the Lawrence Livermore Laboratory. Particular attention is focused on truncation error and how it can be affected by zoning patterns. The principle of convergence is discussed, and convergence as a tool for improving calculational accuracy and efficiency is emphasized
Operational Method for Finite Difference Equations
Merino, S.
2011-01-01
In this article I present a fast and direct method for solving several types of linear finite difference equations (FDE) with constant coefficients. The method is based on a polynomial form of the translation operator and its inverse, and can be used to find the particular solution of the FDE. This work raises the possibility of developing new ways to expand the scope of the operational methods.
Finite difference approximations for a fractional diffusion/anti-diffusion equation
Azerad, Pascal
2011-01-01
A class of finite difference schemes for solving a fractional anti-diffusive equation, recently proposed by Andrew C. Fowler to describe the dynamics of dunes, is considered. Their linear stability is analyzed using the standard Von Neumann analysis: stability criteria are found and checked numerically. Moreover, we investigate the consistency and convergence of these schemes.
Young, P; Martinsson, P G
2012-01-01
A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in R^3 is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a surface to a sequence of BIEs defined on a generating curve for the surface. It can handle loads that are not necessarily rotationally symmetric. Nystrom discretization is used to discretize the BIEs on the generating curve. The quadrature is a high-order Gaussian rule that is modified near the diagonal to retain high-order accuracy for singular kernels. The reduction in dimensionality, along with the use of high-order accurate quadratures, leads to small linear systems that can be inverted directly via, e.g., Gaussian elimination. This makes the scheme particularly fast in environments involving multiple right hand sides. It is demonstrated that for BIEs associated with the Laplace and Helmholtz equations, the kernel in the reduced equations can be evaluated very rapidly by exploiting...
A coupled discrete unified gas-kinetic scheme for Boussinesq flows
Wang, Peng; Guo, Zhaoli
2014-01-01
Recently, the discrete unified gas-kinetic scheme (DUGKS) [Z. L. Guo \\emph{et al}., Phys. Rev. E ${\\bf 88}$, 033305 (2013)] based on the Boltzmann equation is developed as a new multiscale kinetic method for isothermal flows. In this paper, a thermal and coupled discrete unified gas-kinetic scheme is derived for the Boussinesq flows, where the velocity and temperature fields are described independently. Kinetic boundary conditions for both velocity and temperature fields are also proposed. The proposed model is validated by simulating several canonical test cases, including the porous plate problem, the Rayleigh-b\\'{e}nard convection, and the natural convection with Rayleigh number up to $10^{10}$ in a square cavity. The results show that the coupled DUGKS is of second order accuracy in space and can well describe the convection phenomena from laminar to turbulent flows. Particularly, it is found that this new scheme has better numerical stability in simulating high Rayleigh number flows compared with the pre...
Asymptotically Correct Finite Difference Schemes for Highly Oscillatory ODEs
International Nuclear Information System (INIS)
We are concerned with the numerical integration of ODE-initial value problems of the form ?2?xx+a(x)? = 0 with given a(x)?a0>0 in the highly oscillatory regime 03h2). As an application we present simulations of a 1D-model for ballistic quantum transport in a MOSFET (metal oxide semiconductor field-effect transistor).
Finite difference methods for coupled flow interaction transport models
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Shelly McGee
2009-04-01
Full Text Available Understanding chemical transport in blood flow involves coupling the chemical transport process with flow equations describing the blood and plasma in the membrane wall. In this work, we consider a coupled two-dimensional model with transient Navier-Stokes equation to model the blood flow in the vessel and Darcy's flow to model the plasma flow through the vessel wall. The advection-diffusion equation is coupled with the velocities from the flows in the vessel and wall, respectively to model the transport of the chemical. The coupled chemical transport equations are discretized by the finite difference method and the resulting system is solved using the additive Schwarz method. Development of the model and related analytical and numerical results are presented in this work.
Optimization of Dengue Epidemics: A Test Case with Different Discretization Schemes
Rodrigues, Helena Sofia; Monteiro, M. Teresa T.; Torres, Delfim F. M.
2009-09-01
The incidence of Dengue epidemiologic disease has grown in recent decades. In this paper an application of optimal control in Dengue epidemics is presented. The mathematical model includes the dynamic of Dengue mosquito, the affected persons, the people's motivation to combat the mosquito and the inherent social cost of the disease, such as cost with ill individuals, educations and sanitary campaigns. The dynamic model presents a set of nonlinear ordinary differential equations. The problem was discretized through Euler and Runge Kutta schemes, and solved using nonlinear optimization packages. The computational results as well as the main conclusions are shown.
Optimization of Dengue Epidemics: a test case with different discretization schemes
Rodrigues, Helena Sofia; Torres, Delfim F M; 10.1063/1.3241345
2010-01-01
The incidence of Dengue epidemiologic disease has grown in recent decades. In this paper an application of optimal control in Dengue epidemics is presented. The mathematical model includes the dynamic of Dengue mosquito, the affected persons, the people's motivation to combat the mosquito and the inherent social cost of the disease, such as cost with ill individuals, educations and sanitary campaigns. The dynamic model presents a set of nonlinear ordinary differential equations. The problem was discretized through Euler and Runge Kutta schemes, and solved using nonlinear optimization packages. The computational results as well as the main conclusions are shown.
Hejranfar, Kazem; Ezzatneshan, Eslam
2014-06-01
In this work, the implementation of a high-order compact finite-difference lattice Boltzmann method (CFDLBM) is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non-uniform grids. The incompressible form of the discrete Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates. Herein, the spatial derivatives in the resulting lattice Boltzmann (LB) equation in the computational plane are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge-Kutta scheme to provide an accurate and efficient incompressible flow solver. A high-order spectral-type low-pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions, flow non-linearities and grid non-uniformity. All boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates. The accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution. Four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are: unsteady Couette flow and steady flow in a 2-D cavity with non-uniform grid and steady and unsteady flows over a circular cylinder and the NACA0012 hydrofoil at different flow conditions. Results obtained for the above test cases are in good agreement with the existing numerical and experimental results. The study shows the present solution methodology based on the implementation of the high-order compact finite-difference Lattice Boltzmann method (CFDLBM) in the generalized curvilinear coordinates is robust, efficient and accurate for solving steady and unsteady incompressible flows over practical geometries.
Verma, Prabal Singh
2015-01-01
The dimensionally split reconstruction method as described by Kurganov et al.\\cite{kurganov-2000} is revisited for better understanding and a simple fourth order scheme is introduced to solve 3D hyperbolic conservation laws following dimension by dimension approach. Fourth order central weighted essentially non-oscillatory (CWENO) reconstruction methods have already been proposed to study multidimensional problems \\cite{lpr4,cs12}. In this paper, it is demonstrated that a simple 1D fourth order CWENO reconstruction method by Levy et al.\\cite{lpr7} provides fourth order accuracy for 3D hyperbolic nonlinear problems when combined with the semi-discrete scheme by Kurganov et al.\\cite{kurganov-2000} and fourth order Runge-Kutta method for time integration.
Discrete unified gas kinetic scheme for all Knudsen number flows. II. Thermal compressible case
Guo, Zhaoli; Wang, Ruijie; Xu, Kun
2015-03-01
This paper is a continuation of our work on the development of multiscale numerical scheme from low-speed isothermal flow to compressible flows at high Mach numbers. In our earlier work [Z. L. Guo et al., Phys. Rev. E 88, 033305 (2013), 10.1103/PhysRevE.88.033305], a discrete unified gas kinetic scheme (DUGKS) was developed for low-speed flows in which the Mach number is small so that the flow is nearly incompressible. In the current work, we extend the scheme to compressible flows with the inclusion of thermal effect and shock discontinuity based on the gas kinetic Shakhov model. This method is an explicit finite-volume scheme with the coupling of particle transport and collision in the flux evaluation at a cell interface. As a result, the time step of the method is not limited by the particle collision time. With the variation of the ratio between the time step and particle collision time, the scheme is an asymptotic preserving (AP) method, where both the Chapman-Enskog expansion for the Navier-Stokes solution in the continuum regime and the free transport mechanism in the rarefied limit can be precisely recovered with a second-order accuracy in both space and time. The DUGKS is an idealized multiscale method for all Knudsen number flow simulations. A number of numerical tests, including the shock structure problem, the Sod tube problem in a whole range of degree of rarefaction, and the two-dimensional Riemann problem in both continuum and rarefied regimes, are performed to validate the scheme. Comparisons with the results of direct simulation Monte Carlo (DSMC) and other benchmark data demonstrate that the DUGKS is a reliable and efficient method for multiscale flow problems.
Directory of Open Access Journals (Sweden)
Hongjie Dong
2005-09-01
Full Text Available We consider degenerate parabolic and elliptic equations of second order with $C^1$ and $C^2$ coefficients. Error bounds for certain types of finite-difference schemes are obtained.
Fourth order compact schemes for variable coefficient parabolic problems with mixed derivatives
Sen, Shuvam
2013-01-01
In this article, we have developed a higher order compact numerical method for variable coefficient parabolic problems with mixed derivatives. The finite difference scheme, presented here for two-dimensional domains, is based on fourth order spatial discretization. The time discretization has been carried out using using second order Crank-Nicolson. The present scheme shows good dispersion relation preserving property and has been thoroughly investigated for stability. The d...
An induced charge readout scheme incorporating image charge splitting on discrete pixels
International Nuclear Information System (INIS)
Top hat electrostatic analysers used in space plasma instruments typically use microchannel plates (MCPs) followed by discrete pixel anode readout for the angular definition of the incoming particles. Better angular definition requires more pixels/readout electronics channels but with stringent mass and power budgets common in space applications, the number of channels is restricted. We describe here a technique that improves the angular definition using induced charge and an interleaved anode pattern. The technique adopts the readout philosophy used on the CRRES and CLUSTER I instruments but has the advantages of the induced charge scheme and significantly reduced capacitance. Charge from the MCP collected by an anode pixel is inductively split onto discrete pixels whose geometry can be tailored to suit the scientific requirements of the instrument. For our application, the charge is induced over two pixels. One of them is used for a coarse angular definition but is read out by a single channel of electronics, allowing a higher rate handling. The other provides a finer angular definition but is interleaved and hence carries the expense of lower rate handling. Using the technique and adding four channels of electronics, a four-fold increase in the angular resolution is obtained. Details of the scheme and performance results are presented
Viscoelastic Finite Difference Modeling Using Graphics Processing Units
Fabien-Ouellet, G.; Gloaguen, E.; Giroux, B.
2014-12-01
Full waveform seismic modeling requires a huge amount of computing power that still challenges today's technology. This limits the applicability of powerful processing approaches in seismic exploration like full-waveform inversion. This paper explores the use of Graphics Processing Units (GPU) to compute a time based finite-difference solution to the viscoelastic wave equation. The aim is to investigate whether the adoption of the GPU technology is susceptible to reduce significantly the computing time of simulations. The code presented herein is based on the freely accessible software of Bohlen (2002) in 2D provided under a General Public License (GNU) licence. This implementation is based on a second order centred differences scheme to approximate time differences and staggered grid schemes with centred difference of order 2, 4, 6, 8, and 12 for spatial derivatives. The code is fully parallel and is written using the Message Passing Interface (MPI), and it thus supports simulations of vast seismic models on a cluster of CPUs. To port the code from Bohlen (2002) on GPUs, the OpenCl framework was chosen for its ability to work on both CPUs and GPUs and its adoption by most of GPU manufacturers. In our implementation, OpenCL works in conjunction with MPI, which allows computations on a cluster of GPU for large-scale model simulations. We tested our code for model sizes between 1002 and 60002 elements. Comparison shows a decrease in computation time of more than two orders of magnitude between the GPU implementation run on a AMD Radeon HD 7950 and the CPU implementation run on a 2.26 GHz Intel Xeon Quad-Core. The speed-up varies depending on the order of the finite difference approximation and generally increases for higher orders. Increasing speed-ups are also obtained for increasing model size, which can be explained by kernel overheads and delays introduced by memory transfers to and from the GPU through the PCI-E bus. Those tests indicate that the GPU memory size and the slow memory transfers are the limiting factors of our GPU implementation. Those results show the benefits of using GPUs instead of CPUs for time based finite-difference seismic simulations. The reductions in computation time and in hardware costs are significant and open the door for new approaches in seismic inversion.
Finite difference analysis of the transient temperature profile within GHARR-1 fuel element
International Nuclear Information System (INIS)
Highlights: • Transient heat conduction for GHARR-1 fuel was developed and simulated by MATLAB. • The temperature profile after shutdown showed parabolic decay pattern. • The recorded temperature of about 411.6 K was below the melting point of the clad. • The fuel is stable and no radioactivity will be released into the coolant. - Abstract: Mathematical model of the transient heat distribution within Ghana Research Reactor-1 (GHARR-1) fuel element and related shutdown heat generation rates have been developed. The shutdown heats considered were residual fission and fission product decay heat. A finite difference scheme for the discretization by implicit method was used. Solution algorithms were developed and MATLAB program implemented to determine the temperature distributions within the fuel element after shutdown due to reactivity insertion accident. The simulations showed a steady state temperature of about 341.3 K which deviated from that reported in the GHARR-1 safety analysis report by 2% error margin. The average temperature obtained under transient condition was found to be approximately 444 K which was lower than the melting point of 913 K for the aluminium cladding. Thus, the GHARR-1 fuel element was stable and there would be no release of radioactivity in the coolant during accident conditions
A Novel Image Encryption Scheme Based on Multi-orbit Hybrid of Discrete Dynamical System
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Ruisong Ye
2014-10-01
Full Text Available A multi-orbit hybrid image encryption scheme based on discrete chaotic dynamical systems is proposed. One generalized Arnold map is adopted to generate three orbits for three initial conditions. Another chaotic dynamical system, tent map, is applied to generate one pseudo-random sequence to determine the hybrid orbit points from which one of the three orbits of generalized Arnold map. The hybrid orbit sequence is then utilized to shuffle the pixels' positions of plain-image so as to get one permuted image. To enhance the encryption security, two rounds of pixel gray values' diffusion is employed as well. The proposed encryption scheme is simple and easy to manipulate. The security and performance of the proposed image encryption have been analyzed, including histograms, correlation coefficients, information entropy, key sensitivity analysis, key space analysis, differential analysis, etc. All the experimental results suggest that the proposed image encryption scheme is robust and secure and can be used for secure image and video communication applications.
Discrete unified gas kinetic scheme for all Knudsen number flows: II. Compressible case
Guo, Zhaoli; Xu, Kun
2014-01-01
This paper is a continuation of our earlier work [Z.L. Guo {\\it et al.}, Phys. Rev. E {\\bf 88}, 033305 (2013)] where a multiscale numerical scheme based on kinetic model was developed for low speed isothermal flows with arbitrary Knudsen numbers. In this work, a discrete unified gas-kinetic scheme (DUGKS) for compressible flows with the consideration of heat transfer and shock discontinuity is developed based on the Shakhov model with an adjustable Prandtl number. The method is an explicit finite-volume scheme where the transport and collision processes are coupled in the evaluation of the fluxes at cell interfaces, so that the nice asymptotic preserving (AP) property is retained, such that the time step is limited only by the CFL number, the distribution function at cell interface recovers to the Chapman-Enskog one in the continuum limit while reduces to that of free-transport for free-molecular flow, and the time and spatial accuracy is of second-order accuracy in smooth region. These features make the DUGK...
The discrete variational derivative method based on discrete differential forms
Yaguchi, Takaharu; Matsuo, Takayasu; Sugihara, Masaaki
2012-05-01
As is well known, for PDEs that enjoy a conservation or dissipation property, numerical schemes that inherit this property are often advantageous in that the schemes are fairly stable and give qualitatively better numerical solutions in practice. Lately, Furihata and Matsuo have developed the so-called “discrete variational derivative method” that automatically constructs energy preserving or dissipative finite difference schemes. Although this method was originally developed on uniform meshes, the use of non-uniform meshes is of importance for multi-dimensional problems. On the other hand, the theories of discrete differential forms have received much attention recently. These theories provide a discrete analogue of the vector calculus on general meshes. In this paper, we show that the discrete variational derivative method and the discrete differential forms by Bochev and Hyman can be combined. Applications to the Cahn-Hilliard equation and the Klein-Gordon equation on triangular meshes are provided as demonstrations. We also show that the schemes for these equations are H1-stable under some assumptions. In particular, one for the nonlinear Klein-Gordon equation is obtained by combination of the energy conservation property and the discrete Poincaré inequality, which are the temporal and spacial structures that are preserved by the above methods.
ON FINITE DIFFERENCES ON A STRING PROBLEM
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J. M. Mango
2014-01-01
Full Text Available This study presents an analysis of a one-Dimensional (1D time dependent wave equation from a vibrating guitar string. We consider the transverse displacement of a plucked guitar string and the subsequent vibration motion. Guitars are known for production of great sound in form of music. An ordinary string stretched between two points and then plucked does not produce quality sound like a guitar string. A guitar string produces loud and unique sound which can be organized by the player to produce music. Where is the origin of guitar sound? Can the contribution of each part of the guitar to quality sound be accounted for, by mathematically obtaining the numerical solution to wave equation describing the vibration of the guitar string? In the present sturdy, we have solved the wave equation for a vibrating string using the finite different method and analyzed the wave forms for different values of the string variables. The results show that the amplitude (pitch or quality of the guitar wave (sound vary greatly with tension in the string, length of the string, linear density of the string and also on the material of the sound board. The approximate solution is representative; if the step width; ?x and ?t are small, that is <0.5.
Iterative solutions of finite difference diffusion equations
International Nuclear Information System (INIS)
The heterogeneous arrangement of materials and the three-dimensional character of the reactor physics problems encountered in the design and operation of nuclear reactors makes it necessary to use numerical methods for solution of the neutron diffusion equations which are based on the linear Boltzmann equation. The commonly used numerical method for this purpose is the finite difference method. It converts the diffusion equations to a system of algebraic equations. In practice, the size of this resulting algebraic system is so large that the iterative methods have to be used. Most frequently used iterative methods are discussed. They include : (1) basic iterative methods for one-group problems, (2) iterative methods for eigenvalue problems, and (3) iterative methods which use variable acceleration parameters. Application of Chebyshev theorem to iterative methods is discussed. The extension of the above iterative methods to multigroup neutron diffusion equations is also considered. These methods are applicable to elliptic boundary value problems in reactor design studies in particular, and to elliptic partial differential equations in general. Solution of sample problems is included to illustrate their applications. The subject matter is presented in as simple a manner as possible. However, a working knowledge of matrix theory is presupposed. (M.G.B.)
Chen, G.; Zheng, Q.; Coleman, M.; Weerakoon, S.
1983-01-01
This paper briefly reviews convergent finite difference schemes for hyperbolic initial boundary value problems and their applications to boundary control systems of hyperbolic type which arise in the modelling of vibrations. These difference schemes are combined with the primal and the dual approaches to compute the optimal control in the unconstrained case, as well as the case when the control is subject to inequality constraints. Some of the preliminary numerical results are also presented.
A finite difference, multipoint flux numerical approach to flow in porous media: Numerical examples
Osman, Hossam
2012-06-17
It is clear that none of the current available numerical schemes which may be adopted to solve transport phenomena in porous media fulfill all the required robustness conditions. That is while the finite difference methods are the simplest of all, they face several difficulties in complex geometries and anisotropic media. On the other hand, while finite element methods are well suited to complex geometries and can deal with anisotropic media, they are more involved in coding and usually require more execution time. Therefore, in this work we try to combine some features of the finite element technique, namely its ability to work with anisotropic media with the finite difference approach. We reduce the multipoint flux, mixed finite element technique through some quadrature rules to an equivalent cell-centered finite difference approximation. We show examples on using this technique to single-phase flow in anisotropic porous media.
On the Stability of the Finite Difference based Lattice Boltzmann Method
El-Amin, M.F.
2013-06-01
This paper is devoted to determining the stability conditions for the finite difference based lattice Boltzmann method (FDLBM). In the current scheme, the 9-bit two-dimensional (D2Q9) model is used and the collision term of the Bhatnagar- Gross-Krook (BGK) is treated implicitly. The implicitness of the numerical scheme is removed by introducing a new distribution function different from that being used. Therefore, a new explicit finite-difference lattice Boltzmann method is obtained. Stability analysis of the resulted explicit scheme is done using Fourier expansion. Then, stability conditions in terms of time and spatial steps, relaxation time and explicitly-implicitly parameter are determined by calculating the eigenvalues of the given difference system. The determined conditions give the ranges of the parameters that have stable solutions.
Fast finite difference solvers for singular solutions of the elliptic Monge-Amp\\'ere equation
Froese, Brittany D
2010-01-01
The elliptic Monge-Amp\\`ere equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. In this article we build a finite difference solver for the Monge-Amp\\'ere equation, which converges even for singular solutions. Regularity results are used to select a priori between a stable, provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain. This allows singular solutions to be computed using a stable method, and regular solutions to be computed more accurately. The resulting nonlinear equations are then solved by Newton's method. Computational results in two and three dimensions validate the claims of accuracy and solution speed. A computational example is presented which demonstrates the nece...
Minimal positive stencils in meshfree finite difference methods for the Poisson equation
Seibold, Benjamin
2008-01-01
Meshfree finite difference methods for the Poisson equation approximate the Laplace operator on a point cloud. Desirable are positive stencils, i.e. all neighbor entries are of the same sign. Classical least squares approaches yield large stencils that are in general not positive. We present an approach that yields stencils of minimal size, which are positive. We provide conditions on the point cloud geometry, so that positive stencils always exist. The new discretization method is compared to least squares approaches.
Minimal positive stencils in meshfree finite difference methods for the Poisson equation
Seibold, Benjamin
2008-01-01
Meshfree finite difference methods for the Poisson equation approximate the Laplace operator on a point cloud. Desirable are positive stencils, i.e. all neighbor entries are of the same sign. Classical least squares approaches yield large stencils that are in general not positive. We present an approach that yields stencils of minimal size, which are positive. We provide conditions on the point cloud geometry, so that positive stencils always exist. The new discretization me...
Energy Technology Data Exchange (ETDEWEB)
Ling Zou; Haihua Zhao; Hongbin Zhang
2015-04-01
The majority of the existing reactor system analysis codes are de- veloped using low-order numerical schemes in both space and time. The disadvantages of using low-order numerical methods have long been realized. However, there have been very few attempts in the thermal-hydraulics field to use advanced numerical schemes to achieve higher-order accuracy. High-resolution spatial schemes, based upon Godunov’s groundbreaking work, have been widely accepted in the research field of computational fluid dynamics. Such schemes pro- vide high-order accuracy in smooth regions and non-oscillation results near discontinuities, both of which are essential features to improve the spatial accuracy of system analysis codes. However, most of these schemes were developed for conservative hyperbolic equations using cell-centered grid arrangement. On the contrary, the non-hyperbolic two-fluid two-phase equations in primitive form are commonly used in the nuclear thermal-hydraulics field. In this work, we adopted the commonly used slope limiter concept into the staggered grid mesh arrangement to achieve spatially high-resolution results. High-order time integration schemes are also desirable in many applications to reduce numerical errors from low-order or operator-splitting type of time integration schemes. The resulted discretization system, using high-resolution spatial discretization and high-order time integration schemes, are highly nonlinear for flow equations. In this work, this nonlinear system has been successfully solved using the Newton-Krylov method. The high-resolution and high-order numerical schemes were applied to several single- and two-phase test problems relevant to nu- clear thermal-hydraulics field. Numerical results clearly demonstrated the advantages to use such high-resolution and high-order numerical schemes to significantly reduce numerical diffusion, and therefore to improve accuracy.
Choi, S.-J.; Giraldo, F. X.; Kim, J.; Shin, S.
2014-11-01
The non-hydrostatic (NH) compressible Euler equations for dry atmosphere were solved in a simplified two-dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. By using horizontal SEM, which decomposes the physical domain into smaller pieces with a small communication stencil, a high level of scalability can be achieved. By using vertical FDM, an easy method for coupling the dynamics and existing physics packages can be provided. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss-Lobatto-Legendre (GLL) quadrature points. The FDM employs a third-order upwind-biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative and integral terms. For temporal integration, a time-split, third-order Runge-Kutta (RK3) integration technique was applied. The Euler equations that were used here are in flux form based on the hydrostatic pressure vertical coordinate. The equations are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma-pressure vertical coordinate was implemented in this model. We validated the model by conducting the widely used standard tests: linear hydrostatic mountain wave, tracer advection, and gravity wave over the Schär-type mountain, as well as density current, inertia-gravity wave, and rising thermal bubble. The results from these tests demonstrated that the model using the horizontal SEM and the vertical FDM is accurate and robust provided sufficient diffusion is applied. The results with various horizontal resolutions also showed convergence of second-order accuracy due to the accuracy of the time integration scheme and that of the vertical direction, although high-order basis functions were used in the horizontal. By using the 2-D slice model, we effectively showed that the combined spatial discretization method of the spectral element and finite difference methods in the horizontal and vertical directions, respectively, offers a viable method for development of an NH dynamical core.
International Nuclear Information System (INIS)
Nuclear reactor behaviour is determined by the relation between the temperature and the effective neutron reaction cross-sections of the materials in the reactor core. In order to accurately calculate the material temperatures, heat transfer to the reactor coolant is of critical importance. To increase the accuracy of heat transfer calculation, this paper presents a second-order accurate convection heat transfer discretization for application in two or more dimensions in a finite-volume discretization. This discretization is compared to a first-order accurate scheme by means of a grid dependence study on representative reactor geometry. To obtain a certain level of accuracy, the use of second-order accurate convection discretization reduced the solution time by up to 60 percent, as the second-order accurate scheme required fewer grid points than the first-order accurate scheme. This is of particular importance for systems simulation reactor models, as they use the minimum number of grid points and have to run at the highest possible simulation speed. This study was done using the systems simulation code Flownex. (authors)
Explicit Finite Difference Solution of Heat Transfer Problems of Fish Packages in Precooling
Directory of Open Access Journals (Sweden)
A. S. Mokhtar
2004-01-01
Full Text Available The present work aims at finding an optimized explicit finite difference scheme for the solution of problems involving pure heat transfer from the surfaces of Pangasius Sutchi fish samples suddenly exposed to a cooling environment. Regular shaped packages in the form of an infinite slab were considered and a generalized mathematical model was written in dimensionless form. An accurate sample of the data set was chosen from the experimental work and was used to seek an optimized scheme of solutions. A fully explicit finite difference scheme has been thoroughly studied from the viewpoint of stability, the required time for execution and precision. The characteristic dimension (half thickness was divided into a number of divisions; n = 5, 10, 20, 50 and 100 respectively. All the possible options of dimensionless time (the Fourier number increments were taken one by one to give the best convergence and truncation error criteria. The simplest explicit finite difference scheme with n = (10 and stability factor (Î?X2/Î?Ï? = 2 was found to be reliable and accurate for prediction purposes."
Jang, Juhi; Li, Fengyan; Qiu, Jing-Mei; Xiong, Tao
2015-01-01
In this paper, we develop a family of high order asymptotic preserving schemes for some discrete-velocity kinetic equations under a diffusive scaling, that in the asymptotic limit lead to macroscopic models such as the heat equation, the porous media equation, the advection-diffusion equation, and the viscous Burgers' equation. Our approach is based on the micro-macro reformulation of the kinetic equation which involves a natural decomposition of the equation to the equilibrium and non-equilibrium parts. To achieve high order accuracy and uniform stability as well as to capture the correct asymptotic limit, two new ingredients are employed in the proposed methods: discontinuous Galerkin (DG) spatial discretization of arbitrary order of accuracy with suitable numerical fluxes; high order globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta scheme in time equipped with a properly chosen implicit-explicit strategy. Formal asymptotic analysis shows that the proposed scheme in the limit of ? ? 0 is a consistent high order discretization for the limiting equation. Numerical results are presented to demonstrate the stability and high order accuracy of the proposed schemes together with their performance in the limit. Our methods are also tested for the continuous-velocity one-group transport equation in slab geometry and for several examples with spatially varying parameters.
Al-Mansoori, Saeed; Kunhu, Alavi
2013-10-01
This paper proposes a blind multi-watermarking scheme based on designing two back-to-back encoders. The first encoder is implemented to embed a robust watermark into remote sensing imagery by applying a Discrete Cosine Transform (DCT) approach. Such watermark is used in many applications to protect the copyright of the image. However, the second encoder embeds a fragile watermark using `SHA-1' hash function. The purpose behind embedding a fragile watermark is to prove the authenticity of the image (i.e. tamper-proof). Thus, the proposed technique was developed as a result of new challenges with piracy of remote sensing imagery ownership. This led researchers to look for different means to secure the ownership of satellite imagery and prevent the illegal use of these resources. Therefore, Emirates Institution for Advanced Science and Technology (EIAST) proposed utilizing existing data security concept by embedding a digital signature, "watermark", into DubaiSat-1 satellite imagery. In this study, DubaiSat-1 images with 2.5 meter resolution are used as a cover and a colored EIAST logo is used as a watermark. In order to evaluate the robustness of the proposed technique, a couple of attacks are applied such as JPEG compression, rotation and synchronization attacks. Furthermore, tampering attacks are applied to prove image authenticity.
International Nuclear Information System (INIS)
This report presents comparisons of results of five implicit and explicit finite difference recession computation techniques with results from a more accurate ''benchmark'' solution applied to a simple one-dimensional nonlinear ablation problem. In the comparison problem a semi-infinite solid is subjected to a constant heat flux at its surface and the rate of recession is controlled by the solid material's latent heat of fusion. All thermal properties are assumed constant. The five finite difference methods include three front node dropping schemes, a back node dropping scheme, and a method in which the ablation problem is embedded in an inverse heat conduction problem and no nodes are dropped. Constancy of thermal properties and the semiinfinite and one-dimensional nature of the problem at hand are not necessary assumptions in applying the methods studied to more general problems. The best of the methods studied will be incorporated into APL's Standard Heat Transfer Program
Optimal variable-grid finite-difference modeling for porous media
Liu, Xinxin; Yin, Xingyao; Li, Haishan
2014-12-01
Numerical modeling of poroelastic waves by the finite-difference (FD) method is more expensive than that of acoustic or elastic waves. To improve the accuracy and computational efficiency of seismic modeling, variable-grid FD methods have been developed. In this paper, we derived optimal staggered-grid finite difference schemes with variable grid-spacing and time-step for seismic modeling in porous media. FD operators with small grid-spacing and time-step are adopted for low-velocity or small-scale geological bodies, while FD operators with big grid-spacing and time-step are adopted for high-velocity or large-scale regions. The dispersion relations of FD schemes were derived based on the plane wave theory, then the FD coefficients were obtained using the Taylor expansion. Dispersion analysis and modeling results demonstrated that the proposed method has higher accuracy with lower computational cost for poroelastic wave simulation in heterogeneous reservoirs.
International Nuclear Information System (INIS)
The sub-library of discrete level schemes and gamma radiation branching ratios (DLS) is translated from the evaluated nuclear structure data file (ENSDF). The data are further checked and corrected. In consideration of the demands for different kinds of research fields most of the evaluated experimental levels and their gamma rays in the ENSDF are kept in DLS data file. the management-retrieval code can provide two retrieving ways. One is a retrieval for a single nucleus (SN), and the other is one for a neutron reaction (NR). The latter contains four kinds of retrieving types corresponding four types of different fast neutron calculation codes. The code can cut off and select the required level and gamma rays from whole discrete level scheme according to user's demands
Calculating photonic Green's functions using a non-orthogonal finite difference time domain method
Ward, A. J.; Pendry, J. B.
1998-01-01
In this paper we shall propose a simple scheme for calculating Green's functions for photons propagating in complex structured dielectrics or other photonic systems. The method is based on an extension of the finite difference time domain (FDTD) method, originally proposed by Yee, also known as the Order-N method, which has recently become a popular way of calculating photonic band structures. We give a new, transparent derivation of the Order-N method which, in turn, enable...
Gupta, A.; Sbragaglia, M.; Scagliarini, A.
2014-01-01
We propose numerical simulations of viscoelastic fluids based on a hybrid algorithm combining Lattice-Boltzmann models (LBM) and Finite Differences (FD) schemes, the former used to model the macroscopic hydrodynamic equations, and the latter used to model the polymer dynamics. The kinetics of the polymers is introduced using constitutive equations for viscoelastic fluids with finitely extensible non-linear elastic dumbbells with Peterlin's closure (FENE-P). The numerical mod...
Finite difference lattice Boltzmann model with flux limiters for liquid-vapor systems
Sofonea, V.; Lamura, A.; Gonnella, G; Cristea, A.
2004-01-01
In this paper we apply a finite difference lattice Boltzmann model to study the phase separation in a two-dimensional liquid-vapor system. Spurious numerical effects in macroscopic equations are discussed and an appropriate numerical scheme involving flux limiter techniques is proposed to minimize them and guarantee a better numerical stability at very low viscosity. The phase separation kinetics is investigated and we find evidence of two different growth regimes depending ...
Ghorai, Anjana P.; Tiwary, R.
2013-01-01
In this present context, mathematical modeling of the propagation of surface waves in a fluid saturated poro-elastic medium under the influence of initial stress has been considered using time dependent higher order finite difference method (FDM). We have proved that the accuracy of this finite-difference scheme is when we use 2nd order time domain finite-difference and 2M-th order space domain finite-difference. It also has been shown that the dispersion curves of Love waves are less dispe...
Scattering by hydraulic fractures: Finite-difference modeling and laboratory data
Energy Technology Data Exchange (ETDEWEB)
Groenenboom, J.; Falk, J.
2000-04-01
Reservoir production can be stimulated by creating hydraulic fractures that effectively facilitate the inflow of hydrocarbons into a well. Considering the effectiveness and safety of the operation, it is desirable to monitor the size and location of the fracture. In this paper, the authors investigate the possibilities of using seismic waves generated by active sources to characterize the fractures. First, the authors must understand the scattering of seismic waves by hydraulic fractures. For that purpose they use a finite-difference modeling scheme. They argue that a mechanically open hydraulic fracture can be represented by a thin, fluid-filled layer. The width or aperture of the fracture is often small compared to the seismic wavelength, which forces one to use a very find grid spacing to define the fracture. Based on equidistant grids, this results in a large number of grid points and hence computationally expensive problems. The authors show that this problem can be overcome by allowing for a variation in grid spacing in the finite-difference scheme to accommodate the large-scale variation in such a model. Second, they show ultrasonic data of small-scale hydraulic fracture experiments in the laboratory. At first sight it is difficult to unravel the interpretation of the various events measured. They use the results of the finite-difference modeling to postulate various possible events that might be present in the data. By comparing the calculated arrival times of these events with the laboratory and finite-difference data, they are able to propose a plausible explanation of the set of scattering events. Based on the laboratory data, they conclude that active seismic sources can potentially be used to determine fracture size and location in the field. The modeling example of fracture scattering illustrates the benefit of the finite-difference technique with a variation in grid spacing for comparing numerical and physical experiments.
Directory of Open Access Journals (Sweden)
M.Vasim babu
2014-05-01
Full Text Available Most of the localization algorithms in past decade are usually based on Monte Carlo, sequential monte carlo and adaptive monte carlo localization method. In this paper we proposed a new scheme called DQMCL which employs the antithetic variance reduction method to improve the localization accuracy. Most existing SMC and AMC based localization algorithm cannot be used in dynamic sensor network but DQMCL can work well even without need of static sensor network with the help of discrete power control method for the entire sensor to improve the average Localization accuracy. Also we analyse a quasi monte carlo method for simulating a discrete time antithetic markov time steps to improve the life time of the sensor node. Our simulation result shows that overall localization accuracy will be more than 88% and localization error is below 35% with synchronization error observed at different discrete time interval.
M.Vasim babu; Dr.A.V.Ramprasad
2014-01-01
Most of the localization algorithms in past decade are usually based on Monte Carlo, sequential monte carlo and adaptive monte carlo localization method. In this paper we proposed a new scheme called DQMCL which employs the antithetic variance reduction method to improve the localization accuracy. Most existing SMC and AMC based localization algorithm cannot be used in dynamic sensor network but DQMCL can work well even without need of static sensor network with the help of discrete power con...
International Nuclear Information System (INIS)
A novel approach is presented in this paper for improving anisotropic diffusion PDE models, based on the Perona–Malik equation. A solution is proposed from an engineering perspective to adaptively estimate the parameters of the regularizing function in this equation. The goal of such a new adaptive diffusion scheme is to better preserve edges when the anisotropic diffusion PDE models are applied to image enhancement tasks. The proposed adaptive parameter estimation in the anisotropic diffusion PDE model involves self-organizing maps and Bayesian inference to define edge probabilities accurately. The proposed modifications attempt to capture not only simple edges but also difficult textural edges and incorporate their probability in the anisotropic diffusion model. In the context of the application of PDE models to image processing such adaptive schemes are closely related to the discrete image representation problem and the investigation of more suitable discretization algorithms using constraints derived from image processing theory. The proposed adaptive anisotropic diffusion model illustrates these concepts when it is numerically approximated by various discretization schemes in a database of magnetic resonance images (MRI), where it is shown to be efficient in image filtering and restoration applications
Jung, Chang-yeol; Nguyen, Thien Binh
2015-01-01
A new adaptive weighted essentially non-oscillatory WENO-$\\theta$ scheme in the context of finite difference is proposed. Depending on the smoothness of the large stencil used in the reconstruction of the numerical flux, a parameter $\\theta$ is set adaptively to switch the scheme between a 5th-order upwind and 6th-order central discretization. A new indicator $\\tau^{\\theta}$ measuring the smoothness of the large stencil is chosen among two candidates which are devised based ...
Finite difference modelling as a practical exploration tool
Energy Technology Data Exchange (ETDEWEB)
Manning, P.M.; Margrave, G.F. [Calgary Univ., AB (Canada)
1999-07-01
The ongoing increase in computer power is useful for the modelling by finite difference methods which requires considerable computer resources. Modelling of surface waves is a possibility for finite difference methods, especially because ray-tracing is useful. Surface waves are a persistent feature on seismic records and have been considered noise in the past, but recent studies have shown that the shallow penetration of surface waves is ideal for near surface investigations, especially for environmental purposes. The large static shifts found on shear wave traces are usually caused by very shallow conditions, and an understanding and interpretation of surface wave anomalies will probably lead to improved shear wave statics, because surface waves are mainly determined by shear wave velocities. Finite difference modelling in Matlab, a finite difference code, can provide very useful insights, especially for surface waves. 3 refs.
Stability analysis of single-phase thermosyphon loops by finite difference numerical methods
International Nuclear Information System (INIS)
In this paper, examples of the application of finite difference numerical methods in the analysis of stability of single-phase natural circulation loops are reported. The problem is here addressed for its relevance for thermal-hydraulic system code applications, in the aim to point out the effect of truncation error on stability prediction. The methodology adopted for analysing in a systematic way the effect of various finite difference discretization can be considered the numerical analogue of the usual techniques adopted for PDE stability analysis. Three different single-phase loop configurations are considered involving various kinds of boundary conditions. In one of these cases, an original dimensionless form of the governing equations is proposed, adopting the Reynolds number as a flow variable. This allows for an appropriate consideration of transition between laminar and turbulent regimes, which is not possible with other dimensionless forms, thus enlarging the field of validity of model assumptions. (author). 14 refs., 8 figs
The representation of absorbers in finite difference diffusion codes
International Nuclear Information System (INIS)
In this paper we present a new method of representing absorbers in finite difference codes utilising the analytical flux solution in the vicinity of the absorbers. Taking an idealised reactor model, numerical comparisons are made between the finite difference eigenvalues and fluxes and results obtained from a purely analytical treatment of control rods in a reactor (the Codd-Rennie method), and agreement is found to be encouraging. The method has been coded for the IBM7090. (author)
Hybrid semi-Lagrangian finite element-finite difference methods for the Vlasov equation
Guo, Wei; Qiu, Jing-Mei
2013-02-01
In this paper, we propose a new conservative hybrid finite element-finite difference method for the Vlasov equation. The proposed methodology uses Strang splitting to decouple the nonlinear high dimensional Vlasov equation into two lower dimensional equations, which describe spatial advection and velocity acceleration/deceleration processes respectively. We then propose to use a semi-Lagrangian (SL) discontinuous Galerkin (DG) scheme (or Eulerian Runge-Kutta (RK) DG scheme with local time stepping) for spatial advection, and use a SL finite difference WENO for velocity acceleration/deceleration. Such hybrid method takes the advantage of DG scheme in its compactness and its ability in handling complicated spatial geometry; while takes the advantage of the WENO scheme in its robustness in resolving filamentation solution structures of the Vlasov equation. The proposed highly accurate methodology enjoys great computational efficiency, as it allows one to use relatively coarse phase space mesh due to the high order nature of the scheme. At the same time, the time step can be taken to be extra large in the SL framework. The quality of the proposed method is demonstrated via basic test problems, such as linear advection and rigid body rotation, and classical plasma problems, such as Landau damping and the two stream instability. Although we only tested 1D1V examples, the proposed method has the potential to be extended to problems with high spatial dimensions and complicated geometry. This constitutes our future research work.
Improvement of the finite difference lattice Boltzmann method for low mach number flows
International Nuclear Information System (INIS)
This paper presents a numerical method to compute flow-acoustic resonance at low Mach number within a reasonable computing time. Light water reactors have experienced flow-acoustic resonance which is attributed to unsteady compressible flows at low Mach number. This phenomenon is undesirable because the induced sound causes loud noise and vibrations of the mechanical structures. However, a numerical simulation of this flow-acoustic resonance at low Mach number requires a large computing time and a highly accurate method in order to simultaneously compute flows and acoustic waves. The finite difference lattice Boltzmann method which is a powerful tool for obtaining computational fluid dynamics has high accuracy for simultaneous calculation of flows and acoustic waves. It becomes an efficient method to compute low Mach number flow if the computing time is shortened. In this paper, the finite difference lattice Boltzmann method was sped up. Three improvements were proposed: development of a new particle model, modification of the governing equation, and employment of an efficient time marching scheme. The computing time of the proposed finite difference lattice Boltzmann model was compared with the conventional finite difference lattice Boltzmann model for the calculation of the cubic cavity flow. The results showed that the computing time of the proposed model is 30% of the time needed by the conventional finite difference lattice Boltzmann model. The flow-acoustic resoce Boltzmann model. The flow-acoustic resonance at low Mach number at the side branch was calculated using the proposed model. The numerical results showed quantitative agreement with the experimental data. (author)
Ghil, M.; Balgovind, R.
1979-01-01
The inhomogeneous Cauchy-Riemann equations in a rectangle are discretized by a finite difference approximation. Several different boundary conditions are treated explicitly, leading to algorithms which have overall second-order accuracy. All boundary conditions with either u or v prescribed along a side of the rectangle can be treated by similar methods. The algorithms presented here have nearly minimal time and storage requirements and seem suitable for development into a general-purpose direct Cauchy-Riemann solver for arbitrary boundary conditions.
Error estimates for finite difference approximations of American put option price
Šiška, David
2011-01-01
Finite difference approximations to multi-asset American put option price are considered. The assets are modelled as a multi-dimensional diffusion process with variable drift and volatility. Approximation error of order one quarter with respect to the time discretisation parameter and one half with respect to the space discretisation parameter is proved by reformulating the corresponding optimal stopping problem as a solution of a degenerate Hamilton-Jacobi-Bellman equation. Furthermore, the error arising from restricting the discrete problem to a finite grid by reducing the original problem to a bounded domain is estimated.
Extending geometric conservation law to cell-centered finite difference methods on stationary grids
Liao, Fei; Ye, Zhengyin; Zhang, Lingxia
2015-03-01
In a wide range of high-order high-resolution schemes, the finite difference method (FDM) is a suitable selection for accurate numerical calculations because it efficiently reduces dispersion and dissipation errors. FDM is easier to perform to obtain high-order capabilities than the finite volume method (FVM). Most FDMs are node-centered; such techniques include weighted essentially non-oscillatory schemes (WENO) [1], weighted compact nonlinear schemes (WCNS) [2,3], dissipative compact schemes (DCS) [4], and compact central schemes [5,6]. WENO represents a class of nonlinear high-order high-resolution shock-capture schemes derived by Shu [1]; this technique can be successfully used in multiscale flow simulation problems. WCNS is another nonlinear high-order shock-capture scheme derived by Deng and Zhang. WCNS uses interpolation and not reconstruction to obtain half-node values and features a better spectral resolution than WENO. Deng et al. [4] further developed linear DCS with a free parameter to control upwind tendency and thus decrease the dissipation of upwind schemes. Furthermore, compact central scheme proposed by Lele [5] and developed by Visbal and Gaitonde [6] plays a dominant role for research on large eddy simulation and direct numerical simulation because of its ultra-high-order and spectral-like resolution.
Higher-order finite-difference formulation of periodic Orbital-free Density Functional Theory
Ghosh, Swarnava
2014-01-01
We present a real-space formulation and higher-order finite-difference implementation of periodic Orbital-free Density Functional Theory (OF-DFT). Specifically, utilizing a local reformulation of the electrostatic and kernel terms, we develop a generalized framework suitable for performing OF-DFT simulations with different variants of the electronic kinetic energy. In particular, we develop a self-consistent field (SCF) type fixed-point method for calculations involving linear-response kinetic energy functionals. In doing so, we make the calculation of the electronic ground-state and forces on the nuclei amenable to computations that altogether scale linearly with the number of atoms. We develop a parallel implementation of this formulation using the finite-difference discretization, using which we demonstrate that higher-order finite-differences can achieve relatively large convergence rates with respect to mesh-size in both the energies and forces. Additionally, we establish that the fixed-point iteration c...
Gaonkar, A. K.; Kulkarni, S. S.
2015-01-01
In the present paper, a method to reduce the computational cost associated with solving a nonlinear transient heat conduction problem is presented. The proposed method combines the ideas of two level discretization and the multilevel time integration schemes with the proper orthogonal decomposition model order reduction technique. The accuracy and the computational efficiency of the proposed methods is discussed. Several numerical examples are presented for validation of the approach. Compared to the full finite element model, the proposed method significantly reduces the computational time while maintaining an acceptable level of accuracy.
Czech Academy of Sciences Publication Activity Database
Komenda, Jan; Masopust, Tomáš; van Schuppen, J. H.
Berlin : The International Federation of Automatic Control, 2010 - (Raisch, J.; Giua, A.; Lafortune, S.; Moor , T.), s. 436-441 ISBN 978-3-902661-79-1. [10th International Workshop on Discrete Event Systems. Berlin (DE), 29.08.2010-01.09.2010] Grant ostatní: EU Projekt(XE) EU.ICT.DISC 224498 Institutional research plan: CEZ:AV0Z10190503 Keywords : discrete-event systems * modular supervisory control * coordinator * conditional controllability Subject RIV: BA - General Mathematics http://www.ifac-papersonline.net/Detailed/42964.html
Serpentine: Finite Difference Methods for Wave Propagation in Second Order Formulation
International Nuclear Information System (INIS)
Wave propagation phenomena are important in many DOE applications such as nuclear explosion monitoring, geophysical exploration, estimating ground motion hazards and damage due to earthquakes, non-destructive testing, underground facilities detection, and acoustic noise propagation. There are also future applications that would benefit from simulating wave propagation, such as geothermal energy applications and monitoring sites for carbon storage via seismic reflection techniques. In acoustics and seismology, it is of great interest to increase the frequency bandwidth in simulations. In seismic exploration, greater frequency resolution enables shorter wave lengths to be included in the simulations, allowing for better resolution in the seismic imaging. In nuclear explosion monitoring, higher frequency seismic waves are essential for accurate discrimination between explosions and earthquakes. When simulating earthquake induced motion of large structures, such as nuclear power plants or dams, increased frequency resolution is essential for realistic damage predictions. Another example is simulations of micro-seismic activity near geothermal energy plants. Here, hydro-fracturing induces many small earthquakes and the time scale of each event is proportional to the square root of the moment magnitude. As a result, the motion is dominated by higher frequencies for smaller seismic events. The above wave propagation problems are all governed by systems of hyperbolic partial governed by systems of hyperbolic partial differential equations in second order differential form, i.e., they contain second order partial derivatives of the dependent variables. Our general research theme in this project has been to develop numerical methods that directly discretize the wave equations in second order differential form. The obvious advantage of working with hyperbolic systems in second order differential form, as opposed to rewriting them as first order hyperbolic systems, is that the number of differential equations in the second order system is significantly smaller. Another issue with re-writing a second order system into first order form is that compatibility conditions often must be imposed on the first order form. These (Saint-Venant) conditions ensure that the solution of the first order system also satisfies the original second order system. However, such conditions can be difficult to enforce on the discretized equations, without introducing additional modeling errors. This project has previously developed robust and memory efficient algorithms for wave propagation including effects of curved boundaries, heterogeneous isotropic, and viscoelastic materials. Partially supported by internal funding from Lawrence Livermore National Laboratory, many of these methods have been implemented in the open source software WPP, which is geared towards 3-D seismic wave propagation applications. This code has shown excellent scaling on up to 32,768 processors and has enabled seismic wave calculations with up to 26 Billion grid points. TheWPP calculations have resulted in several publications in the field of computational seismology, e.g.. All of our current methods are second order accurate in both space and time. The benefits of higher order accurate schemes for wave propagation have been known for a long time, but have mostly been developed for first order hyperbolic systems. For second order hyperbolic systems, it has not been known how to make finite difference schemes stable with free surface boundary conditions, heterogeneous material properties, and curvilinear coordinates. The importance of higher order accurate methods is not necessarily to make the numerical solution more accurate, but to reduce the computational cost for obtaining a solution within an acceptable error tolerance. This is because the accuracy in the solution can always be improved by reducing the grid size h. However, in practice, the available computational resources might not be large enough to solve the problem with a low order method.
Serpentine: Finite Difference Methods for Wave Propagation in Second Order Formulation
Energy Technology Data Exchange (ETDEWEB)
Petersson, N A; Sjogreen, B
2012-03-26
Wave propagation phenomena are important in many DOE applications such as nuclear explosion monitoring, geophysical exploration, estimating ground motion hazards and damage due to earthquakes, non-destructive testing, underground facilities detection, and acoustic noise propagation. There are also future applications that would benefit from simulating wave propagation, such as geothermal energy applications and monitoring sites for carbon storage via seismic reflection techniques. In acoustics and seismology, it is of great interest to increase the frequency bandwidth in simulations. In seismic exploration, greater frequency resolution enables shorter wave lengths to be included in the simulations, allowing for better resolution in the seismic imaging. In nuclear explosion monitoring, higher frequency seismic waves are essential for accurate discrimination between explosions and earthquakes. When simulating earthquake induced motion of large structures, such as nuclear power plants or dams, increased frequency resolution is essential for realistic damage predictions. Another example is simulations of micro-seismic activity near geothermal energy plants. Here, hydro-fracturing induces many small earthquakes and the time scale of each event is proportional to the square root of the moment magnitude. As a result, the motion is dominated by higher frequencies for smaller seismic events. The above wave propagation problems are all governed by systems of hyperbolic partial differential equations in second order differential form, i.e., they contain second order partial derivatives of the dependent variables. Our general research theme in this project has been to develop numerical methods that directly discretize the wave equations in second order differential form. The obvious advantage of working with hyperbolic systems in second order differential form, as opposed to rewriting them as first order hyperbolic systems, is that the number of differential equations in the second order system is significantly smaller. Another issue with re-writing a second order system into first order form is that compatibility conditions often must be imposed on the first order form. These (Saint-Venant) conditions ensure that the solution of the first order system also satisfies the original second order system. However, such conditions can be difficult to enforce on the discretized equations, without introducing additional modeling errors. This project has previously developed robust and memory efficient algorithms for wave propagation including effects of curved boundaries, heterogeneous isotropic, and viscoelastic materials. Partially supported by internal funding from Lawrence Livermore National Laboratory, many of these methods have been implemented in the open source software WPP, which is geared towards 3-D seismic wave propagation applications. This code has shown excellent scaling on up to 32,768 processors and has enabled seismic wave calculations with up to 26 Billion grid points. TheWPP calculations have resulted in several publications in the field of computational seismology, e.g.. All of our current methods are second order accurate in both space and time. The benefits of higher order accurate schemes for wave propagation have been known for a long time, but have mostly been developed for first order hyperbolic systems. For second order hyperbolic systems, it has not been known how to make finite difference schemes stable with free surface boundary conditions, heterogeneous material properties, and curvilinear coordinates. The importance of higher order accurate methods is not necessarily to make the numerical solution more accurate, but to reduce the computational cost for obtaining a solution within an acceptable error tolerance. This is because the accuracy in the solution can always be improved by reducing the grid size h. However, in practice, the available computational resources might not be large enough to solve the problem with a low order method.
Continuous dependence and differentiation of solutions of finite difference equations
Linda Lee; Johnny Henderson
1991-01-01
Conditions are given for the continuity and differentiability of solutions of initial value problems and boundary value problems for the nth order finite difference equation, u(m+n)=f(m,u(m),u(m+1),Ã¢Â€Â¦,u(m+nÃ¢ÂˆÂ’1)),mÃ¢ÂˆÂˆÃ¢Â„Â¤.
Three dimensional finite difference time domain simulations of photonic crystals
Hermann, Christian
2004-01-01
Die vorliegende Arbeit beschäftigt sich mit der numerischen Analyse grundlegender optischer Eigenschaften von photonischen Kristallen mit Hilfe eines Finite-Difference Time-Domain Algorithmus. Nach einer Diskussion der zugrundeliegenden physikalischen und mathematischen Prinzipien, die zur Ausbildung einer photonischen Bandlücke führen, werden zwei Beispielsysteme eingehender untersucht. Als erstes beschäftigen wir uns mit zweidimensional strukturierten Schichtsystemen. Diese Systeme gelt...
Continuous dependence and differentiation of solutions of finite difference equations
Directory of Open Access Journals (Sweden)
Linda Lee
1991-01-01
Full Text Available Conditions are given for the continuity and differentiability of solutions of initial value problems and boundary value problems for the nth order finite difference equation, u(m+n=f(m,u(m,u(m+1,Ã¢Â€Â¦,u(m+nÃ¢ÂˆÂ’1,mÃ¢ÂˆÂˆÃ¢Â„Â¤.
3D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids
Eymard, Robert; Henry, Gérard; Herbin, Rapahele; Hubert, Florence; Klofkorn, Robert; Manzini, Gianmarco
2011-01-01
We present a number of test cases and meshes that were designed as a benchmark for numerical schemes dedicated to the approximation of three-dimensional anisotropic and heterogeneous diffusion problems. These numerical schemes may be applied to general, possibly non conforming, meshes composed of tetrahedra, hexahedra and quite distorted general polyhedra. A number of methods were tested among which conforming ?nite element methods, discontinuous Galerkin ?nite element methods, cell-centered ...
HEATING-7, Multidimensional Finite-Difference Heat Conduction Analysis
International Nuclear Information System (INIS)
1 - Description of program or function: HEATING 7.2i and 7.3 are the most recent developments in a series of heat-transfer codes and obsolete all previous versions distributed by RSICC as SCA-1/HEATING5 and PSR-199/HEATING 6. Note that Unix and PC versions of HEATING7 are available in the CCC-545/SCALE 4.4 package. HEATING can solve steady-state and/or transient heat conduction problems in one-, two-, or three-dimensional Cartesian, cylindrical, or spherical coordinates. A model may include multiple materials, and the thermal conductivity, density, and specific heat of each material may be both time- and temperature-dependent. The thermal conductivity may also be anisotropic. Materials may undergo change of phase. Thermal properties of materials may be input or may be extracted from a material properties library. Heat- generation rates may be dependent on time, temperature, and position, and boundary temperatures may be time- and position-dependent. The boundary conditions, which may be surface-to-environment or surface-to-surface, may be specified temperatures or any combination of prescribed heat flux, forced convection, natural convection, and radiation. The boundary condition parameters may be time- and/or temperature-dependent. General gray body radiation problems may be modeled with user-defined factors for radiant exchange. The mesh spacing may be variable along each axis. HEATING uses a run-time memory allocation scheme to avoid having to recompile to match mee to avoid having to recompile to match memory requirements for each specific problem. HEATING utilizes free-form input. In June 1997 HEATING 7.3 was added to the HEATING 7.2i packages, and the Unix and PC versions of both 7.2i and 7.3 were merged into one package. HEATING 7.3 is being released as a beta-test version; therefore, it does not entirely replace HEATING 7.2i. There is no published documentation for HEATING 7.3; but a listing of input specifications, which reflects changes for 7.3, is included in the PSR-199 documentation. For 3-D problems, surface fluxes may be plotted with H7TECPLOT which requires the proprietary software TECPLOT. HEATING 7.3 runs under Windows95 and WindowsNT on PC's. No future modifications are planned for HEATING7. See README.1ST for more information. 2 - Method of solution: Three steady-state solution techniques are available: point-successive over-relaxation iterative method with extrapolation, direct-solution (for one-dimensional or two-dimensional problems), and conjugate gradient. Transient problems may be solved using any one of several finite-difference schemes: Crank-Nicolson implicit, Classical Implicit Procedure (CIP), Classical Explicit Procedure (CEP), or Levy explicit method (which for some circumstances allows a time step greater than the CEP stability criterion.) The solution of the system of equations arising from the implicit techniques is accomplished by point-successive over-relaxation iteration and includes procedures to estimate the optimum acceleration parameter. 3 - Restrictions on the complexity of the problem: All surfaces in a model must be parallel to one of the coordinate axes which makes modeling complex geometries difficult. Transient change of phase problems can only be solved with one of the explicit techniques - an implicit change-of-phase capability has not been implemented
An outgoing energy flux boundary condition for finite difference ICRP antenna models
International Nuclear Information System (INIS)
For antennas at the ion cyclotron range of frequencies (ICRF) modeling in vacuum can now be carried out to a high level of detail such that shaping of the current straps, isolating septa, and discrete Faraday shield structures can be included. An efficient approach would be to solve for the fields in the vacuum region near the antenna in three dimensions by finite methods and to match this solution at the plasma-vacuum interface to a solution obtained in the plasma region in one dimension by Fourier methods. This approach has been difficult to carry out because boundary conditions must be imposed at the edge of the finite difference grid on a point-by-point basis, whereas the condition for outgoing energy flux into the plasma is known only in terms of the Fourier transform of the plasma fields. A technique is presented by which a boundary condition can be imposed on the computational grid of a three-dimensional finite difference, or finite element, code by constraining the discrete Fourier transform of the fields at the boundary points to satisfy an outgoing energy flux condition appropriate for the plasma. The boundary condition at a specific grid point appears as a coupling to other grid points on the boundary, with weighting determined by a kemel calctdated from the plasma surface impedance matrix for the various plasma Fourier modes. This boundary condition has been implemented in a finite difference solution of a simple problem in two dimensions, which can also be solved directly by Fourier transformation. Results are presented, and it is shown that the proposed boundary condition does enforce outgoing energy flux and yields the same solution as is obtained by Fourier methods
Staircase-free finite-difference time-domain formulation for general materials in complex geometries
DEFF Research Database (Denmark)
Dridi, Kim; Hesthaven, J.S.
2001-01-01
A stable Cartesian grid staircase-free finite-difference time-domain formulation for arbitrary material distributions in general geometries is introduced. It is shown that the method exhibits higher accuracy than the classical Yee scheme for complex geometries since the computational representation of physical structures is not of a staircased nature, Furthermore, electromagnetic boundary conditions are correctly enforced. The method significantly reduces simulation times as fewer points per wavelength are needed to accurately resolve the wave and the geometry. Both perfect electric conductors and dielectric structures have been investigated, Numerical results are presented and discussed.
Finite difference analysis for Navier-Stokes and energy equations of Couette-Poiseuille flow
International Nuclear Information System (INIS)
Numerical results for the problem of combined convective heat transfer in a vertical annular gap between two concentric Isothermal cylinders, are presented. Emphasis was given to the effects of the inlet temperature, flow direction and the inner cylinder rotation on hydrodynamic and heat transfer characteristics. The boundary layer simplifications of the Navier-Stokes equations and the energy equation were solved by means of an extension of the linearized finite difference scheme used previously by Coney and El-Shaarawi (1975). The results were obtained for Re of 100, 200 and 250, 04 and -2x1044. 14 refs.; 11 figs
Liang, Wen-Quan; Wang, Yan-Fei; Yang, Chang-Chun
2015-02-01
Numerical simulation of the wave equation is widely used to synthesize seismograms theoretically and is also the basis of the reverse time migration and full waveform inversion. For the finite difference methods, grid dispersion often exists because of the discretization of the time and the spatial derivatives in the wave equation. How to suppress the grid dispersion is therefore a key problem for finite difference (FD) approaches. The FD operators for the space derivatives are usually obtained in the space domain. However, the wave equations are discretized in the time and space directions simultaneously. So it would be better to design the FD operators in the time–space domain. We improved the time–space domain method for obtaining the FD operators in an acoustic vertically transversely isotropic (VTI) media so as to cover a much wider range of frequencies. Dispersion analysis and seismic numerical simulation demonstrate the effectiveness of the proposed method.
On the modeling of the compressive behaviour of metal foams: a comparison of discretization schemes.
Czech Academy of Sciences Publication Activity Database
Koudelka_ml., Petr; Zlámal, Petr; Kytý?, Daniel; Doktor, Tomáš; Fíla, Tomáš; Jiroušek, Ond?ej
Kippen : Civil-Comp Press, 2013 - ( Topping , B.; Iványi, P.) ISBN 978-1-905088-57-7. ISSN 1759-3433. - (Civil-Comp Proceedings. 102). [International Conference on Civil, Structural and Environmental Engineering Computing /14./. Cagliari (IT), 03.09.2013-06.09.2013] R&D Projects: GA ?R(CZ) GAP105/12/0824 Institutional support: RVO:68378297 Keywords : aluminium foam * micromechanical properties * discretization * compressive behaviour * closed-cell geometry * microCT Subject RIV: JI - Composite Materials
Mueller-gronbach, Thoms; Ritter, Klaus
2006-01-01
We present an algorithm for solving stochastic heat equations, whose key ingredient is a non-uniform time discretization of the driving Brownian motion $W$. For this algorithm we derive an error bound in terms of its number of evaluations of one-dimensional components of $W$. The rate of convergence depends on the spatial dimension of the heat equation and on the decay of the eigenfunctions of the covariance of $W$. According to known lower bounds, our algorithm is optimal, ...
Chirvasa, Mihaela
2010-01-01
This thesis is concerned with the development of numerical methods using finite difference techniques for the discretization of initial value problems (IVPs) and initial boundary value problems (IBVPs) of certain hyperbolic systems which are first order in time and second order in space. This type of system appears in some formulations of Einstein equations, such as ADM, BSSN, NOR, and the generalized harmonic formulation. For IVP, the stability method proposed in [14] is extended from sec...
Recursive computation of finite difference of associated Legendre functions
Fukushima, Toshio
2012-09-01
The existing methods to compute the definite integral of associated Legendre function (ALF) with respect to the argument suffer from a loss of significant figures independently of the latitude. This is caused by the subtraction of similar quantities in the additional term of their recurrence formulas, especially the finite difference of their values between two endpoints of the integration interval. In order to resolve the problem, we develop a recursive algorithm to compute their finite difference. Also, we modify the algorithm to evaluate their definite integrals assuming that their values at one endpoint are known. We numerically confirm a significant increase in computing precision of the integral by the new method. When the interval is one arc minute, for example, the gain amounts to 2-4 digits for the degree of harmonics in the range 2 ? n ? 2,048. This improvement in precision is achieved at a negligible increase in CPU time, say less than 5%.
Time dependent wave envelope finite difference analysis of sound propagation
Baumeister, K. J.
1984-01-01
A transient finite difference wave envelope formulation is presented for sound propagation, without steady flow. Before the finite difference equations are formulated, the governing wave equation is first transformed to a form whose solution tends not to oscillate along the propagation direction. This transformation reduces the required number of grid points by an order of magnitude. Physically, the transformed pressure represents the amplitude of the conventional sound wave. The derivation for the wave envelope transient wave equation and appropriate boundary conditions are presented as well as the difference equations and stability requirements. To illustrate the method, example solutions are presented for sound propagation in a straight hard wall duct and in a two dimensional straight soft wall duct. The numerical results are in good agreement with exact analytical results.
Variational finite-difference representation of the kinetic energy operator
Maragakis, P.; Soler, Jose. M.; Kaxiras, Efthimios
2001-01-01
A potential disadvantage of real-space-grid electronic structure methods is the lack of a variational principle and the concomitant increase of total energy with grid refinement. We show that the origin of this feature is the systematic underestimation of the kinetic energy by the finite difference representation of the Laplacian operator. We present an alternative representation that provides a rigorous upper bound estimate of the true kinetic energy and we illustrate its p...
Finite difference approach for modeling multispecies transport in porous media
Directory of Open Access Journals (Sweden)
N.Natarajan
2010-08-01
Full Text Available An alternative approach to the decomposition method for solving multispecies transport in porous media, coupled with first-order reactions has been proposed. The numerical solution is based on implicit finite difference method. The task of decoupling the coupled partial differential equations has been overcome in this method. The proposed approach is very much advantageous because of its simplicity and also can be adopted in situations where non linear processes are coupled with multi-species transport problems.
Finite difference approach for modeling multispecies transport in porous media
Natarajan, N.; Suresh Kumar, G.
2010-01-01
An alternative approach to the decomposition method for solving multispecies transport in porous media, coupled with first-order reactions has been proposed. The numerical solution is based on implicit finite difference method. The task of decoupling the coupled partial differential equations has been overcome in this method. The proposed approach is very much advantageous because of its simplicity and also can be adopted in situations where non linear processes are coupled with multi-species...
Optimized Finite-Difference Coefficients for Hydroacoustic Modeling
Preston, L. A.
2014-12-01
Responsible utilization of marine renewable energy sources through the use of current energy converter (CEC) and wave energy converter (WEC) devices requires an understanding of the noise generation and propagation from these systems in the marine environment. Acoustic noise produced by rotating turbines, for example, could adversely affect marine animals and human-related marine activities if not properly understood and mitigated. We are utilizing a 3-D finite-difference acoustic simulation code developed at Sandia that can accurately propagate noise in the complex bathymetry in the near-shore to open ocean environment. As part of our efforts to improve computation efficiency in the large, high-resolution domains required in this project, we investigate the effects of using optimized finite-difference coefficients on the accuracy of the simulations. We compare accuracy and runtime of various finite-difference coefficients optimized via criteria such as maximum numerical phase speed error, maximum numerical group speed error, and L-1 and L-2 norms of weighted numerical group and phase speed errors over a given spectral bandwidth. We find that those coefficients optimized for L-1 and L-2 norms are superior in accuracy to those based on maximal error and can produce runtimes of 10% of the baseline case, which uses Taylor Series finite-difference coefficients at the Courant time step limit. We will present comparisons of the results for the various cases evaluated as well as recommendations for utilization of the cases studied. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
Finite-Difference Frequency-Domain Method in Nanophotonics
Ivinskaya, Aliaksandra; Lavrinenko, Andrei
2011-01-01
Optics and photonics are exciting, rapidly developing fields building their success largely on use of more and more elaborate artificially made, nanostructured materials. To further advance our understanding of light-matter interactions in these complicated artificial media, numerical modeling is often indispensable. This thesis presents the development of rigorous finite-difference method, a very general tool to solve Maxwell’s equations in arbitrary geometries in three dime...
An assessment of semi-discrete central schemes for hyperbolic conservation laws
International Nuclear Information System (INIS)
High-resolution finite volume methods for solving systems of conservation laws have been widely embraced in research areas ranging from astrophysics to geophysics and aero-thermodynamics. These methods are typically at least second-order accurate in space and time, deliver non-oscillatory solutions in the presence of near discontinuities, e.g., shocks, and introduce minimal dispersive and diffusive effects. High-resolution methods promise to provide greatly enhanced solution methods for Sandia's mainstream shock hydrodynamics and compressible flow applications, and they admit the possibility of a generalized framework for treating multi-physics problems such as the coupled hydrodynamics, electro-magnetics and radiative transport found in Z pinch physics. In this work, we describe initial efforts to develop a generalized 'black-box' conservation law framework based on modern high-resolution methods and implemented in an object-oriented software framework. The framework is based on the solution of systems of general non-linear hyperbolic conservation laws using Godunov-type central schemes. In our initial efforts, we have focused on central or central-upwind schemes that can be implemented with only a knowledge of the physical flux function and the minimal/maximal eigenvalues of the Jacobian of the flux functions, i.e., they do not rely on extensive Riemann decompositions. Initial experimentation with high-resolution central schemes suggests that contact discontinuitiesemes suggests that contact discontinuities with the concomitant linearly degenerate eigenvalues of the flux Jacobian do not pose algorithmic difficulties. However, central schemes can produce significant smearing of contact discontinuities and excessive dissipation for rotational flows. Comparisons between 'black-box' central schemes and the piecewise parabolic method (PPM), which relies heavily on a Riemann decomposition, shows that roughly equivalent accuracy can be achieved for the same computational cost with both methods. However, PPM clearly outperforms the central schemes in terms of accuracy at a given grid resolution and the cost of additional complexity in the numerical flux functions. Overall we have observed that the finite volume schemes, implemented within a well-designed framework, are extremely efficient with (potentially) very low memory storage. Finally, we have found by computational experiment that second and third-order strong-stability preserving (SSP) time integration methods with the number of stages greater than the order provide a useful enhanced stability region. However, we observe that non-SSP and non-optimal SSP schemes with SSP factors less than one can still be very useful if used with time-steps below the standard CFL limit. The 'well-designed' integration schemes that we have examined appear to perform well in all instances where the time step is maintained below the standard physical CFL limit
An assessment of semi-discrete central schemes for hyperbolic conservation laws.
Energy Technology Data Exchange (ETDEWEB)
Christon, Mark Allen; Robinson, Allen Conrad; Ketcheson, David Isaac
2003-09-01
High-resolution finite volume methods for solving systems of conservation laws have been widely embraced in research areas ranging from astrophysics to geophysics and aero-thermodynamics. These methods are typically at least second-order accurate in space and time, deliver non-oscillatory solutions in the presence of near discontinuities, e.g., shocks, and introduce minimal dispersive and diffusive effects. High-resolution methods promise to provide greatly enhanced solution methods for Sandia's mainstream shock hydrodynamics and compressible flow applications, and they admit the possibility of a generalized framework for treating multi-physics problems such as the coupled hydrodynamics, electro-magnetics and radiative transport found in Z pinch physics. In this work, we describe initial efforts to develop a generalized 'black-box' conservation law framework based on modern high-resolution methods and implemented in an object-oriented software framework. The framework is based on the solution of systems of general non-linear hyperbolic conservation laws using Godunov-type central schemes. In our initial efforts, we have focused on central or central-upwind schemes that can be implemented with only a knowledge of the physical flux function and the minimal/maximal eigenvalues of the Jacobian of the flux functions, i.e., they do not rely on extensive Riemann decompositions. Initial experimentation with high-resolution central schemes suggests that contact discontinuities with the concomitant linearly degenerate eigenvalues of the flux Jacobian do not pose algorithmic difficulties. However, central schemes can produce significant smearing of contact discontinuities and excessive dissipation for rotational flows. Comparisons between 'black-box' central schemes and the piecewise parabolic method (PPM), which relies heavily on a Riemann decomposition, shows that roughly equivalent accuracy can be achieved for the same computational cost with both methods. However, PPM clearly outperforms the central schemes in terms of accuracy at a given grid resolution and the cost of additional complexity in the numerical flux functions. Overall we have observed that the finite volume schemes, implemented within a well-designed framework, are extremely efficient with (potentially) very low memory storage. Finally, we have found by computational experiment that second and third-order strong-stability preserving (SSP) time integration methods with the number of stages greater than the order provide a useful enhanced stability region. However, we observe that non-SSP and non-optimal SSP schemes with SSP factors less than one can still be very useful if used with time-steps below the standard CFL limit. The 'well-designed' integration schemes that we have examined appear to perform well in all instances where the time step is maintained below the standard physical CFL limit.
Traoré, Philippe; Ahipo, Yves Marcel; Louste, Christophe
2009-08-01
In this paper an improved finite volume scheme to discretize diffusive flux on a non-orthogonal mesh is proposed. This approach, based on an iterative technique initially suggested by Khosla [P.K. Khosla, S.G. Rubin, A diagonally dominant second-order accurate implicit scheme, Computers and Fluids 2 (1974) 207-209] and known as deferred correction, has been intensively utilized by Muzaferija [S. Muzaferija, Adaptative finite volume method for flow prediction using unstructured meshes and multigrid approach, Ph.D. Thesis, Imperial College, 1994] and later Fergizer and Peric [J.H. Fergizer, M. Peric, Computational Methods for Fluid Dynamics, Springer, 2002] to deal with the non-orthogonality of the control volumes. Using a more suitable decomposition of the normal gradient, our scheme gives accurate solutions in geometries where the basic idea of Muzaferija fails. First the performances of both schemes are compared for a Poisson problem solved in quadrangular domains where control volumes are increasingly skewed in order to test their robustness and efficiency. It is shown that convergence properties and the accuracy order of the solution are not degraded even on extremely skewed mesh. Next, the very stable behavior of the method is successfully demonstrated on a randomly distorted grid as well as on an anisotropically distorted one. Finally we compare the solution obtained for quadrilateral control volumes to the ones obtained with a finite element code and with an unstructured version of our finite volume code for triangular control volumes. No differences can be observed between the different solutions, which demonstrates the effectiveness of our approach.
Supervisory control synthesis of discrete-event systems using a coordination scheme.
Czech Academy of Sciences Publication Activity Database
Komenda, Jan; Masopust, Tomáš; van Schuppen, J. H.
2012-01-01
Ro?. 48, ?. 2 (2012), s. 247-254. ISSN 0005-1098 R&D Projects: GA ?R(CZ) GAP103/11/0517; GA ?R GPP202/11/P028 Grant ostatní: European Commission(XE) EU.ICT.DISC 224498 Institutional research plan: CEZ:AV0Z10190503 Keywords : discrete-event systems * supervisory control * distributed control * closed-loop systems * controllability Subject RIV: BA - General Mathematics Impact factor: 2.919, year: 2012 http://www.sciencedirect.com/science/article/pii/S0005109811005395
A finite difference method for the design of gradient coils in MRI--an initial framework.
Zhu, Minhua; Xia, Ling; Liu, Feng; Zhu, Jianfeng; Kang, Liyi; Crozier, Stuart
2012-09-01
This paper proposes a finite-difference (FD)-based method for the design of gradient coils in MRI. The design method first uses the FD approximation to describe the continuous current density of the coil space and then employs the stream function method to extract the coil patterns. During the numerical implementation, a linear equation is constructed and solved using a regularization scheme. The algorithm details have been exemplified through biplanar and cylindrical gradient coil design examples. The design method can be applied to unusual coil designs such as ultrashort or dedicated gradient coils. The proposed gradient coil design scheme can be integrated into a FD-based electromagnetic framework, which can then provide a unified computational framework for gradient and RF design and patient-field interactions. PMID:22353392
Directory of Open Access Journals (Sweden)
Hai Fang
2013-11-01
Full Text Available Watermarking represents a potentially effective tool for the protection and verification of ownership rights in remote sensing images. Multispectral images (MSIs are the main type of images acquired by remote sensing radiometers. In this paper, a robust multispectral image watermarking technique based on the discrete wavelet transform (DWT and the tucker decomposition (TD is proposed. The core idea behind our proposed technique is to apply TD on the DWT coefficients of spectral bands of multispectral images. We use DWT to effectively separate multispectral images into different sub-images and TD to efficiently compact the energy of sub-images. Then watermark is embedded in the elements of the last frontal slices of the core tensor with the smallest absolute value. The core tensor has a good stability and represents the multispectral image properties. The experimental results on LANDSAT images show the proposed approach is robust against various types of attacks such as lossy compression, cropping, addition of noise etc.
Discrete Wavelet Transform Method: A New Optimized Robust Digital Image Watermarking Scheme
Hassan Talebi; Behzad Poursoleyman
2012-01-01
In this paper, a wavelet-based logo watermarking scheme is presented. The logo watermark is embedded into all sub-blocks of the LLn sub-band of the transformed host image, using quantization technique. Extracted logos from all sub-blocks are mixed to make the extracted watermark from distorted watermarked image. Knowing the quantization step-size, dimensions of logo and the level of wavelet transform, the watermark is extracted, without any need to have access to the original image. Robustnes...
Directory of Open Access Journals (Sweden)
Md. Kamal Hossain
2010-10-01
Full Text Available In this paper, the wave propagation in free space and different dielectric material by using Finite Difference Time Domain (FDTD method has been studied. Among various numerical methods Finite Difference Time Domain method is being used to study the time evolution behavior of electromagnetic field by solving the Maxwell’sequation in time domain. In this paper, FDTD method has been employed to study the wave propagation in free space and different dielectric materials. The wave equations are discretized in time and space as required by this FDTD method and leaf-frog algorithm is used to find the solution. We observed wave propagation for one and two dimensional cases. We also observed wave propagation through lossy medium for one dimensional case. For two dimensional cases the patterns of wave incident on rectangular dielectric slab, square metal, RCC pillar were observed. In order to visualize the wave propagation, the evaluation of the excitation at various locations of problem space is monitored. The numerical results agree with the propagation characteristics as expected.
Seismic imaging using finite-differences and parallel computers
Energy Technology Data Exchange (ETDEWEB)
Ober, C.C. [Sandia National Labs., Albuquerque, NM (United States)
1997-12-31
A key to reducing the risks and costs of associated with oil and gas exploration is the fast, accurate imaging of complex geologies, such as salt domes in the Gulf of Mexico and overthrust regions in US onshore regions. Prestack depth migration generally yields the most accurate images, and one approach to this is to solve the scalar wave equation using finite differences. As part of an ongoing ACTI project funded by the US Department of Energy, a finite difference, 3-D prestack, depth migration code has been developed. The goal of this work is to demonstrate that massively parallel computers can be used efficiently for seismic imaging, and that sufficient computing power exists (or soon will exist) to make finite difference, prestack, depth migration practical for oil and gas exploration. Several problems had to be addressed to get an efficient code for the Intel Paragon. These include efficient I/O, efficient parallel tridiagonal solves, and high single-node performance. Furthermore, to provide portable code the author has been restricted to the use of high-level programming languages (C and Fortran) and interprocessor communications using MPI. He has been using the SUNMOS operating system, which has affected many of his programming decisions. He will present images created from two verification datasets (the Marmousi Model and the SEG/EAEG 3D Salt Model). Also, he will show recent images from real datasets, and point out locations of improved imaging. Finally, he will discuss areas of current research which will hopefully improve the image quality and reduce computational costs.
Finite difference evolution equations and quantum dynamical semigroups
International Nuclear Information System (INIS)
We consider the recently proposed [Bonifacio, Lett. Nuovo Cimento, 37, 481 (1983)] coarse grained description of time evolution for the density operator rho(t) through a finite difference equation with steps tau, and we prove that there exists a generator of the quantum dynamical semigroup type yielding an equation giving a continuous evolution coinciding at all time steps with the one induced by the coarse grained description. The map rho(0)?rho(t) derived in this way takes the standard form originally proposed by Lindblad [Comm. Math. Phys., 48, 119 (1976)], even when the map itself (and, therefore, the corresponding generator) is not bounded. (author)
Finite difference program for calculating hydride bed wall temperature profiles
International Nuclear Information System (INIS)
A QuickBASIC finite difference program was written for calculating one dimensional temperature profiles in up to two media with flat, cylindrical, or spherical geometries. The development of the program was motivated by the need to calculate maximum temperature differences across the walls of the Tritium metal hydrides beds for thermal fatigue analysis. The purpose of this report is to document the equations and the computer program used to calculate transient wall temperatures in stainless steel hydride vessels. The development of the computer code was motivated by the need to calculate maximum temperature differences across the walls of the hydrides beds in the Tritium Facility for thermal fatigue analysis
Finite-difference methods in dynamics of continuous media
International Nuclear Information System (INIS)
In this clear and systematic account, the author introduces numerical analysis methods (finite difference approximations to the field equations) to solve problems of a dynamical nature (time-varying). The author includes background in some of the classical methods of analysis which lie at the heart of the physical models. He also addresses the crucial problems of error analysis, convergence, and stability from first principles, without drawing too heavily on any advanced mathematical machinery. Extensive examples illustrate the principles involved and numerous problems give the reader a mastery of the techniques developed in the text
Iwase, Shigeru; Hoshi, Takeo; Ono, Tomoya
2015-01-01
We propose an efficient procedure to obtain Green's functions by combining the shifted conjugate orthogonal conjugate gradient (shifted COCG) method with the nonequilibrium Green's function (NEGF) method based on a real-space finite-difference (RSFD) approach. The bottleneck of the computation in the NEGF scheme is matrix inversion of the Hamiltonian including the self-energy terms of electrodes to obtain perturbed Green's function in the transition region. This procedure fi...
Franzè, Giuseppe; Lucia, Walter; Tedesco, Francesco
2014-12-01
This paper proposes a Model Predictive Control (MPC) strategy to address regulation problems for constrained polytopic Linear Parameter Varying (LPV) systems subject to input and state constraints in which both plant measurements and command signals in the loop are sent through communication channels subject to time-varying delays (Networked Control System (NCS)). The results here proposed represent a significant extension to the LPV framework of a recent Receding Horizon Control (RHC) scheme developed for the so-called robust case. By exploiting the parameter availability, the pre-computed sequences of one- step controllable sets inner approximations are less conservative than the robust counterpart. The resulting framework guarantees asymptotic stability and constraints fulfilment regardless of plant uncertainties and time-delay occurrences. Finally, experimental results on a laboratory two-tank test-bed show the effectiveness of the proposed approach.
A Skin Tone Based Stenographic Scheme using Double Density Discrete Wavelet Transforms.
Directory of Open Access Journals (Sweden)
Varsha Gupta
2013-07-01
Full Text Available Steganography is the art of concealing the existence of data in another transmission medium i.e. image, audio, video files to achieve secret communication. It does not replace cryptography but rather boosts the security using its obscurity features. In the proposed method Biometric feature (Skin tone region is used to implement Steganography[1]. In our proposed method Instead of embedding secret data anywhere in image, it will be embedded in only skin tone region. This skin region provides excellent secure location for data hiding. So, firstly skin detection is performed using, HSV (Hue, Saturation, Value color space in cover images. Thereafter, a region from skin detected area is selected, which is known as the cropped region. In this cropped region secret message is embedded using DD-DWT (Double Density Discrete Wavelet Transform. DD-DWT overcomes the intertwined shortcomings of DWT (like poor directional selectivity, Shift invariance, oscillations and aliasing[2].optimal pixel adjustment process (OPA is used to enhance the image quality of the stego-image. Hence the image obtained after embedding secret message (i.e. Stego image is far more secure and has an acceptable range of PSNR. The proposed method is much better than the previous works both in terms of PSNR and robustness against various attacks (like Gaussian Noise, salt and pepper Noise, Speckle Noise, rotation, JPEG Compression, Cropping, and Contrast Adjustment etc.
Jia, Jinhong; Wang, Hong
2015-07-01
Numerical methods for space-fractional diffusion equations often generate dense or even full stiffness matrices. Traditionally, these methods were solved via Gaussian type direct solvers, which requires O (N3) of computational work per time step and O (N2) of memory to store where N is the number of spatial grid points in the discretization. In this paper we develop a preconditioned fast Krylov subspace iterative method for the efficient and faithful solution of finite difference methods (both steady-state and time-dependent) space-fractional diffusion equations with fractional derivative boundary conditions in one space dimension. The method requires O (N) of memory and O (Nlog ? N) of operations per iteration. Due to the application of effective preconditioners, significantly reduced numbers of iterations were achieved that further reduces the computational cost of the fast method. Numerical results are presented to show the utility of the method.
Finite-difference solution of the space-angle-lethargy-dependent slowing-down transport equation
International Nuclear Information System (INIS)
A procedure has been developed for solving the slowing-down transport equation for a cylindrically symmetric reactor system. The anisotropy of the resonance neutron flux is treated by the spherical harmonics formalism, which reduces the space-angle-Iethargy-dependent transport equation to a matrix integro-differential equation in space and lethargy. Replacing further the lethargy transfer integral by a finite-difference form, a set of matrix ordinary differential equations is obtained, with lethargy-and space dependent coefficients. If the lethargy pivotal points are chosen dense enough so that the difference correction term can be ignored, this set assumes a lower block triangular form and can be solved directly by forward block substitution. As in each step of the finite-difference procedure a boundary value problem has to be solved for a non-homogeneous system of ordinary differential equations with space-dependent coefficients, application of any standard numerical procedure, for example, the finite-difference method or the method of adjoint equations, is too cumbersome and would make the whole procedure practically inapplicable. A simple and efficient approximation is proposed here, allowing analytical solution for the space dependence of the spherical-harmonics flux moments, and hence the derivation of the recurrence relations between the flux moments at successive lethargy pivotal points. According to the procedure indicated above a computer code has been developed for the CDC -3600 computer, which uses the KEDAK nuclear data file. The space and lethargy distribution of the resonance neutrons can be computed in such a detailed fashion as the neutron cross-sections are known for the reactor materials considered. The computing time is relatively short so that the code can be efficiently used, either autonomously, or as part of some complex modular scheme. Typical results will be presented and discussed in order to prove and illustrate the applicability of the method proposed. (author)
A RBF Based Local Gridfree Scheme for Unsteady Convection-Diffusion Problems
Directory of Open Access Journals (Sweden)
Sanyasiraju VSS Yedida
2009-12-01
Full Text Available In this work a Radial Basis Function (RBF based local gridfree scheme has been presented for unsteady convection diffusion equations. Numerical studies have been made using multiquadric (MQ radial function. Euler and a three stage Runge-Kutta schemes have been used for temporal discretization. The developed scheme is compared with the corresponding finite difference (FD counterpart and found that the solutions obtained using the former are more superior. As expected, for a fixed time step and for large nodal densities, thought the Runge-Kutta scheme is able to maintain higher order of accuracy over the Euler method, the temporal discretization is independent of the improvement in the solution which in the developed scheme has been achived by optimizing the shape parameter of the RBF.
Directory of Open Access Journals (Sweden)
Anjana P. Ghorai
2013-03-01
Full Text Available In this present context, mathematical modeling of the propagation of surface waves in a fluid saturated poro-elastic medium under the influence of initial stress has been considered using time dependent higher order finite difference method (FDM. We have proved that the accuracy of this finite-difference scheme is 2M when we use 2nd order time domain finite-difference and 2M-th order space domain finite-difference. It also has been shown that the dispersion curves of Love waves are less dispersed for higher order FDM than of lower order FDM. The effect of initial stress, porosity and anisotropy of the layer in the propagation of Love waves has been studied here. The numerical results have been shown graphically. As a particular case, the phase velocity in a non porous elastic solid layer derived in this paper is in perfect agreement with that of Liu et al. (2009.
Finite difference analysis of curved deep beams on Winkler foundation
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Adel A. Al-Azzawi
2011-03-01
Full Text Available This research deals with the linear elastic behavior of curved deep beams resting on elastic foundations with both compressional and frictional resistances. Timoshenko’s deep beam theory is extended to include the effect of curvature and the externally distributed moments under static conditions. As an application to the distributed moment generations, the problems of deep beams resting on elastic foundations with both compressional and frictional restraints have been investigated in detail. The finite difference method was used to represent curved deep beams and the results were compared with other methods to check the accuracy of the developed analysis. Several important parameters are incorporated in the analysis, namely, the vertical subgrade reaction, horizontal subgrade reaction, beam width, and also the effect of beam thickness to radius ratio on the deflections, bending moments, and shear forces. The computer program (CDBFDA (Curved Deep Beam Finite Difference Analysis Program coded in Fortran-77 for the analysis of curved deep beams on elastic foundations was formed. The results from this method are compared with other methods exact and numerical and check the accuracy of the solutions. Good agreements are found, the average percentages of difference for deflections and moments are 5.3% and 7.3%, respectively, which indicate the efficiency of the adopted method for analysis.
Finite-Difference Frequency-Domain Method in Nanophotonics
DEFF Research Database (Denmark)
Ivinskaya, Aliaksandra
2011-01-01
Optics and photonics are exciting, rapidly developing fields building their success largely on use of more and more elaborate artificially made, nanostructured materials. To further advance our understanding of light-matter interactions in these complicated artificial media, numerical modeling is often indispensable. This thesis presents the development of rigorous finite-difference method, a very general tool to solve Maxwell’s equations in arbitrary geometries in three dimensions, with an emphasis on the frequency-domain formulation. Enhanced performance of the perfectly matched layers is obtained through free space squeezing technique, and nonuniform orthogonal grids are built to greatly improve the accuracy of simulations of highly heterogeneous nanostructures. Examples of the use of the finite-difference frequency-domain method in this thesis range from simulating localized modes in a three-dimensional photonic-crystal membrane-based cavity, a quasi-one-dimensional nanobeam cavity and arrays of side-coupled nanobeam cavities, to modeling light propagation through metal films with single or periodically arranged multiple subwavelength slits.
A coarse-mesh nodal method-diffusive-mesh finite difference method
International Nuclear Information System (INIS)
Modern nodal methods have been successfully used for conventional light water reactor core analyses where the homogenized, node average cross sections (XSs) and the flux discontinuity factors (DFs) based on equivalence theory can reliably predict core behavior. For other types of cores and other geometries characterized by tightly-coupled, heterogeneous core configurations, the intranodal flux shapes obtained from a homogenized nodal problem may not accurately portray steep flux gradients near fuel assembly interfaces or various reactivity control elements. This may require extreme values of DFs (either very large, very small, or even negative) to achieve a desired solution accuracy. Extreme values of DFs, however, can disrupt the convergence of the iterative methods used to solve for the node average fluxes, and can lead to a difficulty in interpolating adjacent DF values. Several attempts to remedy the problem have been made, but nothing has been satisfactory. A new coarse-mesh nodal scheme called the Diffusive-Mesh Finite Difference (DMFD) technique, as contrasted with the coarse-mesh finite difference (CMFD) technique, has been developed to resolve this problem. This new technique and the development of a few-group, multidimensional kinetics computer program are described in this paper
MasQU: Finite Differences on Masked Irregular Stokes Q,U Grids
Bowyer, Jude; Jaffe, Andrew H.; Novikov, Dmitri I.
2011-01-01
The detection of B-mode polarization in the CMB is one of the most important outstanding tests of inflationary cosmology. One of the necessary steps for extracting polarization information in the CMB is reducing contamination from so-called "ambiguous modes" on a masked sky, which contain leakage from the larger E-mode signal. This can be achieved by utilising derivative operators on the real-space Stokes Q and U parameters. This paper presents an algorithm and a software package to perform this procedure on the nearly full sky, i.e., with projects such as the Planck Surveyor and future satellites in mind; in particular, the package can perform finite differences on masked, irregular grids and is applied to a semi-regular spherical pixellization, the HEALPix grid. The formalism reduces to the known finite-difference solutions in the case of a regular grid. We quantify full-sky improvements on the possible bounds on the CMB B-mode signal. We find that in the specific case of E and B-mode separation, there exists a "pole problem" in our formalism which produces signal contamination at very low multipoles l. Several solutions to the "pole problem" are presented; one proposed solution facilitates a calculation of a general Gaussian quadrature scheme, which finds application in calculating accurate harmonic coefficients on the HEALPix sphere. Nevertheless, on a masked sphere the software represents a considerable reduction in B-mode noise from limited sky coverage.
Wang, Wei; Shu, Chi-Wang; Yee, H. C.; Sjögreen, Björn
2012-01-01
A new high order finite-difference method utilizing the idea of Harten ENO subcell resolution method is proposed for chemical reactive flows and combustion. In reaction problems, when the reaction time scale is very small, e.g., orders of magnitude smaller than the fluid dynamics time scales, the governing equations will become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present proposed method is a modified fractional step method which solves the convection step and reaction step separately. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with the computed flow variables in the shock region modified by the Harten subcell resolution idea. For numerical experiments, a fifth-order finite-difference WENO scheme and its anti-diffusion WENO variant are considered. A wide range of 1D and 2D scalar and Euler system test cases are investigated. Studies indicate that for the considered test cases, the new method maintains high order accuracy in space for smooth flows, and for stiff source terms with discontinuities, it can capture the correct propagation speed of discontinuities in very coarse meshes with reasonable CFL numbers.
Directory of Open Access Journals (Sweden)
Vineet K. Srivastava
2014-03-01
Full Text Available In this paper, an implicit logarithmic finite difference method (I-LFDM is implemented for the numerical solution of one dimensional coupled nonlinear Burgers’ equation. The numerical scheme provides a system of nonlinear difference equations which we linearise using Newton's method. The obtained linear system via Newton's method is solved by Gauss elimination with partial pivoting algorithm. To illustrate the accuracy and reliability of the scheme, three numerical examples are described. The obtained numerical solutions are compared well with the exact solutions and those already available.
Parallel finite-difference time-domain method
Yu, Wenhua
2006-01-01
The finite-difference time-domain (FTDT) method has revolutionized antenna design and electromagnetics engineering. This book raises the FDTD method to the next level by empowering it with the vast capabilities of parallel computing. It shows engineers how to exploit the natural parallel properties of FDTD to improve the existing FDTD method and to efficiently solve more complex and large problem sets. Professionals learn how to apply open source software to develop parallel software and hardware to run FDTD in parallel for their projects. The book features hands-on examples that illustrate the power of parallel FDTD and presents practical strategies for carrying out parallel FDTD. This detailed resource provides instructions on downloading, installing, and setting up the required open source software on either Windows or Linux systems, and includes a handy tutorial on parallel programming.
Effects of sources on time-domain finite difference models.
Botts, Jonathan; Savioja, Lauri
2014-07-01
Recent work on excitation mechanisms in acoustic finite difference models focuses primarily on physical interpretations of observed phenomena. This paper offers an alternative view by examining the properties of models from the perspectives of linear algebra and signal processing. Interpretation of a simulation as matrix exponentiation clarifies the separate roles of sources as boundaries and signals. Boundary conditions modify the matrix and thus its modal structure, and initial conditions or source signals shape the solution, but not the modal structure. Low-frequency artifacts are shown to follow from eigenvalues and eigenvectors of the matrix, and previously reported artifacts are predicted from eigenvalue estimates. The role of source signals is also briefly discussed. PMID:24993210
Visualization of elastic wavefields computed with a finite difference code
Energy Technology Data Exchange (ETDEWEB)
Larsen, S. [Lawrence Livermore National Lab., CA (United States); Harris, D.
1994-11-15
The authors have developed a finite difference elastic propagation model to simulate seismic wave propagation through geophysically complex regions. To facilitate debugging and to assist seismologists in interpreting the seismograms generated by the code, they have developed an X Windows interface that permits viewing of successive temporal snapshots of the (2D) wavefield as they are calculated. The authors present a brief video displaying the generation of seismic waves by an explosive source on a continent, which propagate to the edge of the continent then convert to two types of acoustic waves. This sample calculation was part of an effort to study the potential of offshore hydroacoustic systems to monitor seismic events occurring onshore.
Finite-difference modeling of commercial aircraft using TSAR
Energy Technology Data Exchange (ETDEWEB)
Pennock, S.T.; Poggio, A.J.
1994-11-15
Future aircraft may have systems controlled by fiber optic cables, to reduce susceptibility to electromagnetic interference. However, the digital systems associated with the fiber optic network could still experience upset due to powerful radio stations, radars, and other electromagnetic sources, with potentially serious consequences. We are modeling the electromagnetic behavior of commercial transport aircraft in support of the NASA Fly-by-Light/Power-by-Wire program, using the TSAR finite-difference time-domain code initially developed for the military. By comparing results obtained from TSAR with data taken on a Boeing 757 at the Air Force Phillips Lab., we hope to show that FDTD codes can serve as an important tool in the design and certification of U.S. commercial aircraft, helping American companies to produce safe, reliable air transportation.
Computational electrodynamics the finite-difference time-domain method
Taflove, Allen
2005-01-01
This extensively revised and expanded third edition of the Artech House bestseller, Computational Electrodynamics: The Finite-Difference Time-Domain Method, offers engineers the most up-to-date and definitive resource on this critical method for solving Maxwell's equations. The method helps practitioners design antennas, wireless communications devices, high-speed digital and microwave circuits, and integrated optical devices with unsurpassed efficiency. There has been considerable advancement in FDTD computational technology over the past few years, and the third edition brings professionals the very latest details with entirely new chapters on important techniques, major updates on key topics, and new discussions on emerging areas such as nanophotonics. What's more, to supplement the third edition, the authors have created a Web site with solutions to problems, downloadable graphics and videos, and updates, making this new edition the ideal textbook on the subject as well.
Obtaining Potential Field Solution with Spherical Harmonics and Finite Differences
Toth, Gabor; Huang, Zhenguang; 10.1088/0004-637X/732/2/102
2011-01-01
Potential magnetic field solutions can be obtained based on the synoptic magnetograms of the Sun. Traditionally, a spherical harmonics decomposition of the magnetogram is used to construct the current and divergence free magnetic field solution. This method works reasonably well when the order of spherical harmonics is limited to be small relative to the resolution of the magnetogram, although some artifacts, such as ringing, can arise around sharp features. When the number of spherical harmonics is increased, however, using the raw magnetogram data given on a grid that is uniform in the sine of the latitude coordinate can result in inaccurate and unreliable results, especially in the polar regions close to the Sun. We discuss here two approaches that can mitigate or completely avoid these problems: i) Remeshing the magnetogram onto a grid with uniform resolution in latitude, and limiting the highest order of the spherical harmonics to the anti-alias limit; ii) Using an iterative finite difference algorithm t...
Chevrot, Sébastien; Martin, Roland; Komatitsch, Dimitri
2012-12-01
Wavelets are extremely powerful to compress the information contained in finite-frequency sensitivity kernels and tomographic models. This interesting property opens the perspective of reducing the size of global tomographic inverse problems by one to two orders of magnitude. However, introducing wavelets into global tomographic problems raises the problem of computing fast wavelet transforms in spherical geometry. Using a Cartesian cubed sphere mapping, which grids the surface of the sphere with six blocks or 'chunks', we define a new algorithm to implement fast wavelet transforms with the lifting scheme. This algorithm is simple and flexible, and can handle any family of discrete orthogonal or bi-orthogonal wavelets. Since wavelet coefficients are local in space and scale, aliasing effects resulting from a parametrization with global functions such as spherical harmonics are avoided. The sparsity of tomographic models expanded in wavelet bases implies that it is possible to exploit the power of compressed sensing to retrieve Earth's internal structures optimally. This approach involves minimizing a combination of a ?2 norm for data residuals and a ?1 norm for model wavelet coefficients, which can be achieved through relatively minor modifications of the algorithms that are currently used to solve the tomographic inverse problem.
International Nuclear Information System (INIS)
A general adjoint Monte Carlo-forward discrete ordinates radiation transport calculational scheme has been created to study the effects of the radiation environment in Hiroshima and Nagasaki due to the bombing of these two cities. Various such studies for comparison with physical data have progressed since the end of World War II with advancements in computing machinery and computational methods. These efforts have intensified in the last several years with the U.S.-Japan joint reassessment of nuclear weapons dosimetry in Hiroshima and Nagasaki. Three principal areas of investigation are: (1) to determine by experiment and calculation the neutron and gamma-ray energy and angular spectra and total yield of the two weapons; (2) using these weapons descriptions as source terms, to compute radiation effects at several locations in the two cities for comparison with experimental data collected at various times after the bombings and thus validate the source terms; and (3) to compute radiation fields at the known locations of fatalities and surviving individuals at the time of the bombings and thus establish an absolute cause-and-effect relationship between the radiation received and the resulting injuries to these individuals and any of their descendants as indicated by their medical records. It is in connection with the second and third items, the determination of the radiation effects and the dose received by individuals, that the current study is concernede current study is concerned
Chernyshenko, Dmitri
2014-01-01
In the finite difference method which is commonly used in computational micromagnetics, the demagnetizing field is usually computed as a convolution of the magnetization vector field with the demagnetizing tensor that describes the magnetostatic field of a cuboidal cell with constant magnetization. An analytical expression for the demagnetizing tensor is available, however at distances far from the cuboidal cell, the numerical evaluation of the analytical expression can be very inaccurate. Due to this large-distance inaccuracy numerical packages such as OOMMF compute the demagnetizing tensor using the explicit formula at distances close to the originating cell, but at distances far from the originating cell a formula based on an asymptotic expansion has to be used. In this work, we describe a method to calculate the demagnetizing field by numerical evaluation of the multidimensional integral in the demagnetization tensor terms using a sparse grid integration scheme. This method improves the accuracy of comput...
Strong, Stuart L.; Meade, Andrew J., Jr.
1992-01-01
Preliminary results are presented of a finite element/finite difference method (semidiscrete Galerkin method) used to calculate compressible boundary layer flow about airfoils, in which the group finite element scheme is applied to the Dorodnitsyn formulation of the boundary layer equations. The semidiscrete Galerkin (SDG) method promises to be fast, accurate and computationally efficient. The SDG method can also be applied to any smoothly connected airfoil shape without modification and possesses the potential capability of calculating boundary layer solutions beyond flow separation. Results are presented for low speed laminar flow past a circular cylinder and past a NACA 0012 airfoil at zero angle of attack at a Mach number of 0.5. Also shown are results for compressible flow past a flat plate for a Mach number range of 0 to 10 and results for incompressible turbulent flow past a flat plate. All numerical solutions assume an attached boundary layer.
Gupta, A; Scagliarini, A
2014-01-01
We propose numerical simulations of viscoelastic fluids based on a hybrid algorithm combining Lattice-Boltzmann models (LBM) and Finite Differences (FD) schemes, the former used to model the macroscopic hydrodynamic equations, and the latter used to model the polymer dynamics. The kinetics of the polymers is introduced using constitutive equations for viscoelastic fluids with finitely extensible non-linear elastic dumbbells with Peterlin's closure (FENE-P). The numerical model is first benchmarked by characterizing the rheological behaviour of dilute homogeneous solutions in various configurations, including steady shear, elongational flows, transient shear and oscillatory flows. As an upgrade of complexity, we study the model in presence of non-ideal multicomponent interfaces, where immiscibility is introduced in the LBM description using the "Shan-Chen" model. The problem of a confined viscoelastic (Newtonian) droplet in a Newtonian (viscoelastic) matrix under simple shear is investigated and numerical resu...
A full Eulerian finite difference approach for solving fluid-structure coupling problems
Sugiyama, Kazuyasu; Takeuchi, Shintaro; Takagi, Shu; Matsumoto, Yoichiro
2010-01-01
A new simulation method for solving fluid-structure coupling problems has been developed. All the basic equations are numerically solved on a fixed Cartesian grid using a finite difference scheme. A volume-of-fluid formulation (Hirt and Nichols (1981, J. Comput. Phys., 39, 201)), which has been widely used for multiphase flow simulations, is applied to describing the multi-component geometry. The temporal change in the solid deformation is described in the Eulerian frame by updating a left Cauchy-Green deformation tensor, which is used to express constitutive equations for nonlinear Mooney-Rivlin materials. In this paper, various verifications and validations of the present full Eulerian method, which solves the fluid and solid motions on a fixed grid, are demonstrated, and the numerical accuracy involved in the fluid-structure coupling problems is examined.
Martin, R.; Komatitsch, D.; Barucq, H.
2005-12-01
The Perfectly Matched Layer (PML) technique, introduced in 1994 by Berenger for Maxwell's equations, has become classical in the context of numerical simulations in electromagnetics, in particular for 3D finite difference in the time domain (FDTD) calculations. One of the most attractive properties of a PML model is that no reflection occurs at the interface between the physical domain and the absorbing layer before truncation to a finite-size layer and discretization by a numerical scheme. Therefore, the absorbing layer does not send spurious energy back into the medium. This property holds for any frequency and angle of incidence. However, the layer must be truncated in order to be able to perform numerical simulations, and such truncation creates a reflected wave whose amplitude is amplified by the discretization process. In 2001, Collino and Tsogka introduced a PML model for the elastodynamics equation written as a first-order system in velocity and stress with split unknowns, and discretized it based on the standard 2D staggered-grid FD scheme of Virieux (1986). Unfortunately, this standard PML suffers from two drawbacks: the fact that the unknowns are split adds to the memory cost of the simulations; and after numerical discretization, the numerical reflection coefficient between the physical domain and the PML region becomes large at grazing incidence and thus the efficiency of the absorbing layer is poor. In this work, we apply an idea introduced by Roden and Gedney (2000), for solving Maxwell's equations, in order to develop a Convolution-Perfectly Matched Layer (C-PML) formulation for the 3D seismic wave equation based on a velocity-stress staggered-grid FD technique. C-PML is based on the unsplit components of the wave field and optimized for grazing incidence and surface waves using an analytical integration of the convolution term. C-PML formulation allows one to use very thin mesh slices to study a given region of the Earth, thus significantly reducing the cost of 3D simulations, which is of particular interest in the context of inverse problems.
Energy Technology Data Exchange (ETDEWEB)
Ewing, R.E.; Saevareid, O.; Shen, J. [Texas A& M Univ., College Station, TX (United States)
1994-12-31
A multigrid algorithm for the cell-centered finite difference on equilateral triangular grids for solving second-order elliptic problems is proposed. This finite difference is a four-point star stencil in a two-dimensional domain and a five-point star stencil in a three dimensional domain. According to the authors analysis, the advantages of this finite difference are that it is an O(h{sup 2})-order accurate numerical scheme for both the solution and derivatives on equilateral triangular grids, the structure of the scheme is perhaps the simplest, and its corresponding multigrid algorithm is easily constructed with an optimal convergence rate. They are interested in relaxation of the equilateral triangular grid condition to certain general triangular grids and the application of this multigrid algorithm as a numerically reasonable preconditioner for the lowest-order Raviart-Thomas mixed triangular finite element method. Numerical test results are presented to demonstrate their analytical results and to investigate the applications of this multigrid algorithm on general triangular grids.
Farhat, Charbel; Geuzaine, Philippe; Grandmont, Céline
2001-12-01
Discrete geometric conservation laws (DGCLs) govern the geometric parameters of numerical schemes designed for the solution of unsteady flow problems on moving grids. A DGCL requires that these geometric parameters, which include among others grid positions and velocities, be computed so that the corresponding numerical scheme reproduces exactly a constant solution. Sometimes, this requirement affects the intrinsic design of an arbitrary Lagrangian Eulerian (ALE) solution method. In this paper, we show for sample ALE schemes that satisfying the corresponding DGCL is a necessary and sufficient condition for a numerical scheme to preserve the nonlinear stability of its fixed grid counterpart. We also highlight the impact of this theoretical result on practical applications of computational fluid dynamics.
Ramadan, Omar
2014-12-01
Systematic split-step finite difference time domain (SS-FDTD) formulations, based on the general Lie-Trotter-Suzuki product formula, are presented for solving the time-dependent Maxwell equations in double-dispersive electromagnetic materials. The proposed formulations provide a unified tool for constructing a family of unconditionally stable algorithms such as the first order split-step FDTD (SS1-FDTD), the second order split-step FDTD (SS2-FDTD), and the second order alternating direction implicit FDTD (ADI-FDTD) schemes. The theoretical stability of the formulations is included and it has been demonstrated that the formulations are unconditionally stable by construction. Furthermore, the dispersion relation of the formulations is derived and it has been found that the proposed formulations are best suited for those applications where a high space resolution is needed. Two-dimensional (2-D) and 3-D numerical examples are included and it has been observed that the SS1-FDTD scheme is computationally more efficient than the ADI-FDTD counterpart, while maintaining approximately the same numerical accuracy. Moreover, the SS2-FDTD scheme allows using larger time step than the SS1-FDTD or ADI-FDTD and therefore necessitates less CPU time, while giving approximately the same numerical accuracy.
Marcus, Sherman W.
1992-12-01
A standard parabolic equation is used to approximate the Helmholtz equation for electromagnetic propagation in an inhomogeneneous atmosphere. An implicit finite difference (IFD) scheme to solve the SPE is applied between the irregularly shaped ground and an altitude z = z(h) below which all inhomogeneities of the medium are assumed localized. The boundary condition at z = z(h) is obtained by matching the IFD solution to a surface Green's function solution within the uniform region above z = z(h). For ground slopes above about 1 percent the IFD implementation of the impedance boundary condition at the ground is shown to maintain the validity of the approximation only for vertically polarized waves. Predictions using this hybrid finite difference-surface Green's function method agree well with results obtained using other computational methods.
Semi-discrete numeric solution for the non-stationary heat equation using mimetic techniques
International Nuclear Information System (INIS)
It is proposed that the diffusion equation can be solved using second-order mimetic operators for the spatial partial derivatives, in order to obtain a semi-discrete time scheme that is easy to solve with exponential integrators. The scheme is more stable than the traditional method of finite differences (centered on space and forward on time) and easier to implement than implicit methods. Some numerical examples are shown to illustrate the advantages of the proposed method. In addition, routines written in MATLAB were developed for its implementation. (paper)
Finite difference approximation of hedging quantities in the Heston model
in't Hout, Karel
2012-09-01
This note concerns the hedging quantities Delta and Gamma in the Heston model for European-style financial options. A modification of the discretization technique from In 't Hout & Foulon (2010) is proposed, which enables a fast and accurate approximation of these important quantities. Numerical experiments are given that illustrate the performance.
Explicit finite-difference simulations of Project Salt Vault
International Nuclear Information System (INIS)
A series of two-dimensional, plane strain simulations of Project Salt Vault (PSV) were computed in order to demonstrate the applicability of the Lagrange explicit finite-difference (EFD) method to the analysis of the detailed stability response of a radioactive waste repository. The PSV field project was chosen for the simulations because it is a well documented experiment for which some materials testing data are available. The PSV experiment was essentially a feasibility study of radioactive waste disposal in an underground salt formation. It included a large-scale experiment performed in an inactive salt mine at Lyons, Kansas, where a new mining level consisting of five rooms was excavated at about 1000 ft depth and approximately 15 ft above an existing level. Heat sources were arranged and activated so that the imposed heating was also essentially symmetric about a vertical plane. The model for salt creep is a generalization of the work performed by Starfield and McClain, and is a general model for three-dimensional creep response. For the PSV calculations, it relied on the laboratory salt pillar data of Lomenick for its specific constants. The model is stable for discontinuous stress and temperature changes
A finite difference model of the iron ore sinter process
Scientific Electronic Library Online (English)
J., Muller; T.L., de Vries; B.A., Dippenaar; J.C., Vreugdenburg.
2015-05-01
Full Text Available Iron ore fines are agglomerated to produce sinter, which is an important feed material for blast furnaces worldwide. A model of the iron ore sintering process has been developed with the objective of being representative of the sinter pot test, the standard laboratory process in which the behaviour [...] of specific sinter feed mixtures is evaluated. The model aims to predict sinter quality, including chemical quality and physical strength, as well as key sinter process performance parameters such as production rate and fuel consumption rate. The model uses the finite difference method (FDM) to solve heat and mass distributions within the sinter pot over the height and time dimensions. This model can further be used for establishing empirical relationships between modelled parameters and measured sinter properties. Inputs into the model include the feed material physical properties, chemical compositions, and boundary conditions. Submodels describe relationships between applied pressure differential and gas flow rate through the bed of granulated fine ore particles, combustion of carbonaceous material, calcination of fluxes, evaporation and condensation of water, and melting and solidification. The model was applied to typical sinter test conditions to illustrate the results predicted, and to test sensitivities to parameters such as feed void fraction, feed coke percentage, and the fraction of combustion heat transferred to the gas phase. A model validation and improvement study should follow, ensuring sinter test results are free from experimental errors by conducting repeated tests.
A finite difference model for free surface gravity drainage
Energy Technology Data Exchange (ETDEWEB)
Couri, F.R.; Ramey, H.J. Jr.
1993-09-01
The unconfined gravity flow of liquid with a free surface into a well is a classical well test problem which has not been well understood by either hydrologists or petroleum engineers. Paradigms have led many authors to treat an incompressible flow as compressible flow to justify the delayed yield behavior of a time-drawdown test. A finite-difference model has been developed to simulate the free surface gravity flow of an unconfined single phase, infinitely large reservoir into a well. The model was verified with experimental results in sandbox models in the literature and with classical methods applied to observation wells in the Groundwater literature. The simulator response was also compared with analytical Theis (1935) and Ramey et al. (1989) approaches for wellbore pressure at late producing times. The seepage face in the sandface and the delayed yield behavior were reproduced by the model considering a small liquid compressibility and incompressible porous medium. The potential buildup (recovery) simulated by the model evidenced a different- phenomenon from the drawdown, contrary to statements found in the Groundwater literature. Graphs of buildup potential vs time, buildup seepage face length vs time, and free surface head and sand bottom head radial profiles evidenced that the liquid refills the desaturating cone as a flat moving surface. The late time pseudo radial behavior was only approached after exaggerated long times.
Nigro, Alessandra; De Bartolo, Carmine; Bassi, Francesco; Ghidoni, Antonio
2014-11-01
In this paper a high-order implicit multi-step method, known in the literature as Two Implicit Advanced Step-point (TIAS) method, has been implemented in a high-order Discontinuous Galerkin (DG) solver for the unsteady Euler and Navier-Stokes equations. Application of the absolute stability condition to this class of multi-step multi-stage time discretization methods allowed to determine formulae coefficients which ensure A-stability up to order 6. The stability properties of such schemes have been verified by considering linear model problems. The dispersion and dissipation errors introduced by TIAS method have been investigated by looking at the analytical solution of the oscillation equation. The performance of the high-order accurate, both in space and time, TIAS-DG scheme has been evaluated by computing three test cases: an isentropic convecting vortex under two different testing conditions and a laminar vortex shedding behind a circular cylinder. To illustrate the effectiveness and the advantages of the proposed high-order time discretization, the results of the fourth- and sixth-order accurate TIAS schemes have been compared with the results obtained using the standard second-order accurate Backward Differentiation Formula, BDF2, and the five stage fourth-order accurate Strong Stability Preserving Runge-Kutta scheme, SSPRK4.
Directory of Open Access Journals (Sweden)
G. Shapiro
2013-03-01
Full Text Available Results of a sensitivity study are presented from various configurations of the NEMO ocean model in the Black Sea. The standard choices of vertical discretization, viz. z levels, s coordinates and enveloped s coordinates, all show their limitations in the areas of complex topography. Two new hybrid vertical coordinate schemes are presented: the "s-on-top-of-z" and its enveloped version. The hybrid grids use s coordinates or enveloped s coordinates in the upper layer, from the sea surface to the depth of the shelf break, and z-coordinates are set below this level. The study is carried out for a number of idealised and real world settings. The hybrid schemes help reduce errors generated by the standard schemes in the areas of steep topography. Results of sensitivity tests with various horizontal diffusion formulations are used to identify the optimum value of Smagorinsky diffusivity coefficient to best represent the mesoscale activity.
Directory of Open Access Journals (Sweden)
Paulo C. Oliveira
2003-04-01
Full Text Available Este trabalho foi desenvolvido com o objetivo de se apresentar a aplicação de um esquema de discretização mais eficiente para volumes finitos, denominado Flux-Spline utilizando-se, para tal, de dois casos de transporte difusivo de umidade e calor, através de um meio poroso capilar. Os resultados da solução numérica do sistema de equações formado pelas equações de Luikov mostram desempenho adequado do esquema para este tipo de problema, quando comparado ao tradicional esquema de diferença central e ao método da transformada integral.This study was conducted with the objective to present a more efficient discretization scheme to finite volumes method called Flux-Spline, utilising for the purpose two cases of pure diffusion in capillary porous media. The results of numerical simulation of the equations system formed by Luikov equations showed a good performance of the scheme in comparison to the Central Difference Scheme and Generalised Integral Transform Technique method.
Scientific Electronic Library Online (English)
Paulo C., Oliveira; José L., Lima.
2003-04-01
Full Text Available Este trabalho foi desenvolvido com o objetivo de se apresentar a aplicação de um esquema de discretização mais eficiente para volumes finitos, denominado Flux-Spline utilizando-se, para tal, de dois casos de transporte difusivo de umidade e calor, através de um meio poroso capilar. Os resultados da [...] solução numérica do sistema de equações formado pelas equações de Luikov mostram desempenho adequado do esquema para este tipo de problema, quando comparado ao tradicional esquema de diferença central e ao método da transformada integral. Abstract in english This study was conducted with the objective to present a more efficient discretization scheme to finite volumes method called Flux-Spline, utilising for the purpose two cases of pure diffusion in capillary porous media. The results of numerical simulation of the equations system formed by Luikov equ [...] ations showed a good performance of the scheme in comparison to the Central Difference Scheme and Generalised Integral Transform Technique method.
Coarse mesh finite difference formulation for accelerated Monte Carlo eigenvalue calculation
International Nuclear Information System (INIS)
Highlights: • Coarse Mesh Finite Difference (CMFD) formulation is applied to Monte Carlo (MC) calculations. • CMFD leads very rapid convergence of the MC fission source distribution. • The variance bias problem is less significant in three dimensional problems for local tallies. • CMFD-MC enables using substantially many particles without causing waste in inactive cycles. • CMFD-MC is suitable for power reactor calculations requiring many particles per cycle. - Abstract: An efficient Monte Carlo (MC) eigenvalue calculation method for source convergence acceleration and stabilization is developed by employing the Coarse Mesh Finite Difference (CMFD) formulation. The detailed methods for constructing the CMFD system using proper MC tallies are devised such that the coarse mesh homogenization parameters are dynamically produced. These involve the schemes for tally accumulation and periodic reset of the CMFD system. The method for feedback which is to adjust the MC fission source distribution (FSD) using the CMFD global solution is then introduced through a weight adjustment scheme. The CMFD accelerated MC (CMFD-MC) calculation is examined first for a simple one-dimensional multigroup problem to investigate the effectiveness of the accelerated fission source convergence process and also to analyze the sensitivity of the CMFD-MC solutions on the size of coarse meshes and on the number of CMFD energy groups. The performance of CMFD acceleration is then assessed for a set of two-dimensional and three-dimensional multigroup (3D) pressurized water reactor core problems. It is demonstrated that very rapid convergence of the MC FSD is possible with the CMFD formulation in that a sufficiently converged MC FSD can be obtained within 20 cycles even for large three-dimensional problems which would require more than 600 inactive cycles with the standard MC fission source iteration scheme. It is also shown that the optional application of the CMFD formulation in the active cycles can stabilize FSDs such that the real-to-apparent variance ratio of the local tallies can be reduced. However, due to the reduced importance of the variance bias in fine local tallies of 3D MC eigenvalue problems, the effectiveness of CMFD in tally stabilization turns out to be not so great
International Nuclear Information System (INIS)
A full-vector mode solver for optical dielectric waveguide bends by using an improved finite difference method in terms of transverse-electric-field components is developed in a local cylindrical coordinate system. A six-point finite difference scheme is constructed to approximate the cross-coupling terms for improving the convergent behavior, and the perfectly matched layer absorbing boundary conditions via the complex coordinate stretching technique are used for effectively demonstrating the leaky nature of the waveguide bends. The fundamental and high-order leaky modes for a typical bending rib waveguide are computed, which shows the validity and utility of the established method
Finite-difference time-domain simulations of metamaterials
Hao, Zhengwei
Metamaterials are periodic structures created by many identical scattering objects which are stationary and small compared to the wavelength of electromagnetic wave applied to it so that when combined with different elements, these materials have the potential to be coupled to the applied electromagnetic wave without modifying the structure. Due to their unusual properties that are not readily available in nature, metamaterials have been drawing significant attentions in many research areas, including theoretical, experimental as well as numerical investigations. As one of the major computational electromagnetic modeling methods, finite-difference time-domain (FDTD) technique tackles problems by providing a full wave solution. FDTD, which is able to show transient evolution of interactions between electromagnetic wave and physical objects, not only has the advantage in dispersive and nonlinear material simulations, but also has the ability to model circuit elements including semiconductor devices. All these features make FDTD a competitive candidate in numerical methods of metamaterial simulations. This dissertation presents the implementation of FDTD technique to deal with three dimensional (3D) problems characterized with metamaterial structures. We endeavor to make the FDTD engine multi-functional and fast, as depicted in the following three efforts: (1) We incorporated FDTD engine with the stable and highly efficient model for materials with dispersion, nonlinearity and gain properties. (2) We coupled FDTD engine with SPICE, the general-purpose and powerful analog electronic circuit simulator. This makes FDTD ready to simulate complex semiconductor devices and provides a variety of possibilities for novel metamaterials. (3) We investigated the cutting-edge area of Graphics Processing Units (GPU) computing module to speed up the FDTD engine, and implemented subgridding system to target more efficient modeling for metamaterial applications with embedded fine structures. The contribution of this work is toward the development of a powerful FDTD engine for modern metamaterial analysis. Our implementation could be used to improve the analysis of a number of electromagnetic problems.
Hoogland, Jiri; Neumann, Dimitri
2000-01-01
In this article we present a new approach to the numerical valuation of derivative securities. The method is based on our previous work where we formulated the theory of pricing in terms of tradables. The basic idea is to fit a finite difference scheme to exact solutions of the pricing PDE. This can be done in a very elegant way, due to the fact that in our tradable based formulation there appear no drift terms in the PDE. We construct a mixed scheme based on this idea and a...
Wei, Leilei
2012-01-01
In this paper, a fully discrete local discontinuous Galerkin (LDG) finite element method is considered for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation. The scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. We prove that our scheme is unconditional stable and $L^2$ error estimate for the linear case with the convergence rate $O(h^{k+1}+(\\Delta t)^2+(\\Delta t)^\\frac{\\alpha}{2}h^{k+1/2})$. Numerical examples are presented to show the efficiency and accuracy of our scheme.
Li, H.; Zhang, Z.; Chen, X.
2012-12-01
It is widely accepted that they are oversampled in spatial grid spacing and temporal time step in the high speed medium if uniform grids are used for the numerical simulation. This oversampled grid spacing and time step will lower the efficiency of the calculation, especially high velocity contrast exists. Based on the collocated-grid finite-difference method (FDM), we present an algorithm of spatial discontinuous grid, with localized grid blocks and locally varying time steps, which will increase the efficiency of simulation of seismic wave propagation and earthquake strong ground motion. According to the velocity structure, we discretize the model into discontinuous grid blocks, and the time step of each block is determined according to the local stability. The key problem of the discontinuous grid method is the connection between grid blocks with different grid spacing. We use a transitional area overlapped by both of the finer and the coarser grids to deal with the problem. In the transitional area, the values of finer ghost points are obtained by interpolation from the coarser grid in space and time domain, while the values of coarser ghost points are obtained by downsampling from the finer grid. How to deal with coarser ghost points can influent the stability of long time simulation. After testing different downsampling methods and finally we choose the Gaussian filtering. Basically, 4th order Rung-Kutta scheme will be used for the time integral for our numerical method. For our discontinuous grid FDM, discontinuous time steps for the coarser and the finer grids will be used to increase the simulation efficiency. Numerical tests indicate that our method can provide a stable solution even for the long time simulation without any additional filtration for grid spacing ratio n=2. And for larger grid spacing ratio, Gaussian filtration could be used to preserve the stability. With the collocated-grid FDM, which is flexible and accurate in implementation of free surface condition with topography, our method has high advantage in the simulation of strong ground motion of real earthquake.
Bettaibi, Soufiene; Kuznik, Frédéric; Sediki, Ezeddine
2014-06-01
Mixed convection heat transfer in two-dimensional lid-driven rectangular cavity filled with air (Pr=0.71) is studied numerically. A hybrid scheme with multiple relaxation time lattice Boltzmann method (MRT-LBM) is used to obtain the velocity field while the temperature field is deduced from energy balance equation by using the finite difference method (FDM). The main objective of this work is to investigate the model effectiveness for mixed convection flow simulation. Results are presented in terms of streamlines, isotherms and Nusselt numbers. Excellent agreement is obtained between our results and previous works. The different comparisons demonstrate the robustness and the accuracy of our proposed approach.
A convergent explicit finite difference scheme for a mechanical model for tumor growth
Trivisa, Konstantina; Weber, Franziska
2015-01-01
Mechanical models for tumor growth have been used extensively in recent years for the analysis of medical observations and for the prediction of cancer evolution based on imaging analysis. This work deals with the numerical approximation of a mechanical model for tumor growth and the analysis of its dynamics. The system under investigation is given by a multi-phase flow model: The densities of the different cells are governed by a transport equation for the evolution of tumo...
International Nuclear Information System (INIS)
The scattering source iterative (SI) scheme is traditionally applied to converge fine-mesh numerical solutions to fixed-source discrete ordinates (SN) neutron transport problems. The SI scheme is very simple to implement under a computational viewpoint. However, the SI scheme may show very slow convergence rate, mainly for diffusive media (low absorption) with several mean free paths in extent. In this work we describe an acceleration technique based on an improved initial guess for the scattering source distribution within the slab. In other words, we use as initial guess for the fine-mesh scattering source, the coarse-mesh solution of the neutron diffusion equation with special boundary conditions to account for the classical SN prescribed boundary conditions, including vacuum boundary conditions. Therefore, we first implement a spectral nodal method that generates coarse-mesh diffusion solution that is completely free from spatial truncation errors, then we reconstruct this coarse-mesh solution within each spatial cell of the discretization grid, to further yield the initial guess for the fine-mesh scattering source in the first SN transport sweep (?m > 0 and ?m < 0, m = 1:N) across the spatial grid. We consider a number of numerical experiments to illustrate the efficiency of the offered diffusion synthetic acceleration (DSA) technique. (author)
Directory of Open Access Journals (Sweden)
Tsugio Fukuchi
2014-06-01
Full Text Available The finite difference method (FDM based on Cartesian coordinate systems can be applied to numerical analyses over any complex domain. A complex domain is usually taken to mean that the geometry of an immersed body in a fluid is complex; here, it means simply an analytical domain of arbitrary configuration. In such an approach, we do not need to treat the outer and inner boundaries differently in numerical calculations; both are treated in the same way. Using a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. Discretization of the space differential in the FDM is usually derived using the Taylor series expansion; however, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences. In using the polynomial interpolations, we can numerically solve the Poisson equation freely over any complex domain. Only a particular type of partial differential equation, Poisson's equations, is treated; however, the arguments put forward have wider generality in numerical calculations using the FDM.
Finite difference time domain modeling of light matter interaction in light-propelled microtools
DEFF Research Database (Denmark)
Bañas, Andrew Rafael; Palima, Darwin
2013-01-01
Direct laser writing and other recent fabrication techniques offer a wide variety in the design of microdevices. Hence, modeling such devices requires analysis methods capable of handling arbitrary geometries. Recently, we have demonstrated the potential of microtools, optically actuated microstructures with functionalities geared towards biophotonics applications. Compared to dynamic beam shaping alone, microtools allow more complex interactions between the shaped light and the biological samples at the receiving end. For example, strongly focused light coming from a tapered tip of a microtool may trigger highly localized non linear processes in the surface of a cell. Since these functionalities are strongly dependent on design, it is important to use models that can handle complexities and take in little simplifying assumptions about the system. Hence, we use the finite difference time domain (FDTD) method which is a direct discretization of the fundamental Maxwell's equations applicable to many optical systems. Using the FDTD, we investigate light guiding through microstructures as well as the field enhancement as light comes out of our tapered wave guide designs. Such calculations save time as it helps optimize the structures prior to fabrication and experiments. In addition to field distributions, optical forces can also be obtained using the Maxwell stress tensor formulation. By calculating the forces on bent waveguides subjected to tailored static light distributions, we demonstrate novel methods of optical micromanipulation which primarily result from the particle's geometry as opposed to the directly moving the light distributions as in conventional trapping.
Mimetic finite difference method for the stokes problem on polygonal meshes
Energy Technology Data Exchange (ETDEWEB)
Lipnikov, K [Los Alamos National Laboratory; Beirao Da Veiga, L [DIPARTIMENTO DI MATE; Gyrya, V [PENNSYLVANIA STATE UNIV; Manzini, G [ISTIUTO DI MATEMATICA
2009-01-01
Various approaches to extend the finite element methods to non-traditional elements (pyramids, polyhedra, etc.) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. The mimetic finite difference (MFD) method has many similarities with low-order finite element methods. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we develop a MFD method for the Stokes problem on arbitrary polygonal meshes. The method is constructed for tensor coefficients, which will allow to apply it to the linear elasticity problem. The numerical experiments show the second-order convergence for the velocity variable and the first-order for the pressure.
Finite difference modelling of bulk high temperature superconducting cylindrical hysteresis machines
International Nuclear Information System (INIS)
A mathematical model of the critical state based on averaged fluxon motion has been implemented to solve for the current and field distributions inside a high temperature superconducting hysteresis machine. The machine consists of a rotor made from a solid cylindrical single domain HTS placed in a perpendicular rotating field. The solution technique uses the finite difference approximation for a two-dimensional domain, discretized in cylindrical polar co-ordinates. The torque generated or equivalently the hysteresis loss in such a machine has been investigated using the model. It was found that to maximize the efficiency, the field needs to penetrate the rotor such that B0/?0JcR=0.56, where B0 is the applied field amplitude, Jc is the critical current density and R is the rotor radius. This corresponds to a penetration that is 27% greater than that which reaches the centre of the rotor. An examination of the torque density distributions across the rotor reveal that for situations where the field is less than optimal, a significant increase in the performance can be achieved by removing an inner cylinder from the rotor. (author)
New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows.
Li, Zhilin; Lai, Ming-Chih
2011-01-01
In this paper, new finite difference methods based on the augmented immersed interface method (IIM) are proposed for simulating an inextensible moving interface in an incompressible two-dimensional flow. The mathematical models arise from studying the deformation of red blood cells in mathematical biology. The governing equations are incompressible Stokes or Navier-Stokes equations with an unknown surface tension, which should be determined in such a way that the surface divergence of the velocity is zero along the interface. Thus, the area enclosed by the interface and the total length of the interface should be conserved during the evolution process. Because of the nonlinear and coupling nature of the problem, direct discretization by applying the immersed boundary or immersed interface method yields complex nonlinear systems to be solved. In our new methods, we treat the unknown surface tension as an augmented variable so that the augmented IIM can be applied. Since finding the unknown surface tension is essentially an inverse problem that is sensitive to perturbations, our regularization strategy is to introduce a controlled tangential force along the interface, which leads to a least squares problem. For Stokes equations, the forward solver at one time level involves solving three Poisson equations with an interface. For Navier-Stokes equations, we propose a modified projection method that can enforce the pressure jump condition corresponding directly to the unknown surface tension. Several numerical experiments show good agreement with other results in the literature and reveal some interesting phenomena. PMID:23795308
Abgrall, R.; De Santis, D.
2015-02-01
A robust and high order accurate Residual Distribution (RD) scheme for the discretization of the steady Navier-Stokes equations is presented. The proposed method is very flexible: it is formulated for unstructured grids, regardless the shape of the elements and the number of spatial dimensions. A continuous approximation of the solution is adopted and standard Lagrangian shape functions are used to construct the discrete space, as in Finite Element methods. The traditional technique for designing RD schemes is adopted: evaluate, for any element, a total residual, split it into nodal residuals sent to the degrees of freedom of the element, solve the non-linear system that has been assembled and then iterate up to convergence. The main issue addressed by the paper is that the technique relies in depth on the continuity of the normal flux across the element boundaries: this is no longer true since the gradient of the state solution appears in the flux, hence continuity is lost when using standard finite element approximations. Naive solution methods lead to very poor accuracy. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, a continuous approximation of the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution, preserving the optimal accuracy of the method. Linear and non-linear schemes are constructed, and their accuracy is tested with the method of the manufactured solutions. The numerical method is also used for the discretization of smooth and shocked laminar flows in two and three spatial dimensions.
International Nuclear Information System (INIS)
The use of the albedo boundary conditions for multigroup one-dimensional neutron transport eigenvalue problems in the discrete ordinates (SN) formulation is described. The hybrid spectral diamond-spectral Green's function (SD-SGF) nodal method that is completely free from all spatial truncation errors, is used to determine the multigroup albedo operator. In the inner iteration it is used the 'one-node block inversion' (NBI) iterative scheme, which has convergence rate greater than the modified source iteration (SI) scheme. The power method for convergence of the dominant numerical solution is accelerated by the Tchebycheff method. Numerical results are given to illustrate the method's efficiency. (author). 7 refs, 4 figs, 3 tabs
Schwarz, S.; Kempe, T.; Fröhlich, J.
2015-01-01
The paper introduces a time scheme for an immersed boundary method which enables the efficient, phase-resolving simulation of very light particles in viscous flow. A simple modification of the time scheme of the method detailed in Kempe and Fröhlich (2012) [34] is proposed to extend the range of applicability to particle-to-fluid density ratios as they occur with bubbles in liquids. This modification is termed 'virtual mass approach'. It is shown for the generic test case of a sphere moving under Stokes flow conditions that the approach can be used in conjunction with several time integration schemes without altering the order of convergence of the base scheme. The new scheme is rigorously validated for the three-dimensional case of a sphere rising or settling at finite Reynolds number, as well as for the rotation of a sphere in viscous flow.
Rigorous interpolation near tilted interfaces in 3-D finite-difference EM modelling
Shantsev, Daniil V.; Maaø, Frank A.
2015-02-01
We present a rigorous method for interpolation of electric and magnetic fields close to an interface with a conductivity contrast. The method takes into account not only a well-known discontinuity in the normal electric field, but also discontinuity in all the normal derivatives of electric and magnetic tangential fields. The proposed method is applied to marine 3-D controlled-source electromagnetic modelling (CSEM) where sources and receivers are located close to the seafloor separating conductive seawater and resistive formation. For the finite-difference scheme based on the Yee grid, the new interpolation is demonstrated to be much more accurate than alternative methods (interpolation using nodes on one side of the interface or interpolation using nodes on both sides, but ignoring the derivative jumps). The rigorous interpolation can handle arbitrary orientation of interface with respect to the grid, which is demonstrated on a marine CSEM example with a dipping seafloor. The interpolation coefficients are computed by minimizing a misfit between values at the nearest nodes and linear expansions of the continuous field components in the coordinate system aligned with the interface. The proposed interpolation operators can handle either uniform or non-uniform grids and can be applied to interpolation for both sources and receivers.
Gupta, A.; Sbragaglia, M.; Scagliarini, A.
2015-06-01
We propose numerical simulations of viscoelastic fluids based on a hybrid algorithm combining Lattice-Boltzmann models (LBM) and Finite Differences (FD) schemes, the former used to model the macroscopic hydrodynamic equations, and the latter used to model the polymer dynamics. The kinetics of the polymers is introduced using constitutive equations for viscoelastic fluids with finitely extensible non-linear elastic dumbbells with Peterlin's closure (FENE-P). The numerical model is first benchmarked by characterizing the rheological behavior of dilute homogeneous solutions in various configurations, including steady shear, elongational flows, transient shear and oscillatory flows. As an upgrade of complexity, we study the model in presence of non-ideal multicomponent interfaces, where immiscibility is introduced in the LBM description using the "Shan-Chen" interaction model. The problem of a confined viscoelastic (Newtonian) droplet in a Newtonian (viscoelastic) matrix under simple shear is investigated and numerical results are compared with the predictions of various theoretical models. The proposed numerical simulations explore problems where the capabilities of LBM were never quantified before.
International Nuclear Information System (INIS)
An immersed boundary method to achieve the consistency with a desired wall velocity was developed. Existing schemes of immersed boundary methods for incompressible flow violate the wall condition in the discrete equation system during time-advancement. This problem arises from the inconsistency of the pressure with the velocity interpolated to represent the solid wall, which does not coincide with the computational grid. The numerical discrepancy does not become evident in the laminar flow simulation but in the turbulent flow simulation. To eliminate this inconsistency, a modified pressure equation based on the interpolated pressure gradient was derived for the spatial second-order discrete equation system. The conservation of the wall condition, mass, momentum and energy in the present method was theoretically demonstrated. To verify the theory, large eddy simulations for a plane channel, circular pipe and nuclear rod bundle were successfully performed. Both these theoretical and numerical validations improve the reliability and the applicability of the immersed boundary method
International Nuclear Information System (INIS)
A finite-difference scheme and a Galerkin scheme are compared with respect to a very accurate solution describing time-dependent advection and diffusion of air pollutants from a line source in an atmosphere vertically stratified and limited by an inversion layer. The accurate solution was achieved by applying the finite-difference scheme on a very refined grid with a very small time step. The grid size and time step were defined according to stability and accuracy criteria discussed in the text. It is found that for the problem considered the two methods can be considered equally accurate. However, the Galerkin method gives a better approximation in the vicinity of the source. This was assumed to be partly due to the different way the source term is taken into account in the two methods. Improvement of the accuracy of the finite-difference scheme was achieved by approximating, at every step, the contribution of the source term by a Gaussian puff moving and diffusing with the velocity and diffusivity of the source location, instead of utilizing a stepwise function for the numerical approximation of the delta function representing the source term
Hoogland, Jiri Kamiel; Hoogland, Jiri; Neumann, Dimitri
2000-01-01
In this article we present a new approach to the numerical valuation of derivative securities. The method is based on our previous work where we formulated the theory of pricing in terms of tradables. The basic idea is to fit a finite difference scheme to exact solutions of the pricing PDE. This can be done in a very elegant way, due to the fact that in our tradable based formulation there appear no drift terms in the PDE. We construct a mixed scheme based on this idea and apply it to price various types of arithmetic Asian options, as well as plain vanilla options (both european and american style) on stocks paying known cash dividends. We find prices which are accurate to $\\sim 0.1%$ in about 10ms on a Pentium 233MHz computer and to $\\sim 0.001%$ in a second. The scheme can also be used for market conform pricing, by fitting it to observed option prices.
S. S. Das, M. R. Saran, S. Mohanty, R. K. Padhy
2014-01-01
This paper focuses on the unsteady hydromagnetic mixed convective heat and mass transfer boundary layer flow of a viscous incompressible electrically conducting fluid past an accelerated infinite vertical porous flat plate in a porous medium with suction in presence of foreign species such as H2, He, H2O vapour and NH3. The governing equations are solved both analytically and numerically using error function and finite difference scheme. The flow phenomenon has been characterized with the hel...
Directory of Open Access Journals (Sweden)
Georgios S. Stamatakos
2009-10-01
Full Text Available The tremendous rate of accumulation of experimental and clinical knowledge pertaining to cancer dictates the development of a theoretical framework for the meaningful integration of such knowledge at all levels of biocomplexity. In this context our research group has developed and partly validated a number of spatiotemporal simulation models of in vivo tumour growth and in particular tumour response to several therapeutic schemes. Most of the modeling modules have been based on discrete mathematics and therefore have been formulated in terms of rather complex algorithms (e.g. in pseudocode and actual computer code. However, such lengthy algorithmic descriptions, although sufficient from the mathematical point of view, may render it difficult for an interested reader to readily identify the sequence of the very basic simulation operations that lie at the heart of the entire model. In order to both alleviate this problem and at the same time provide a bridge to symbolic mathematics, we propose the introduction of the notion of hypermatrix in conjunction with that of a discrete operator into the already developed models. Using a radiotherapy response simulation example we demonstrate how the entire model can be considered as the sequential application of a number of discrete operators to a hypermatrix corresponding to the dynamics of the anatomic area of interest. Subsequently, we investigate the operators’ commutativity and outline the “summarize and jump” strategy aiming at efficiently and realistically address multilevel biological problems such as cancer. In order to clarify the actual effect of the composite discrete operator we present further simulation results which are in agreement with the outcome of the clinical study RTOG 83–02, thus strengthening the reliability of the model developed.
Popov, Anton; Kaus, Boris
2015-04-01
This software project aims at bringing the 3D lithospheric deformation modeling to a qualitatively different level. Our code LaMEM (Lithosphere and Mantle Evolution Model) is based on the following building blocks: * Massively-parallel data-distributed implementation model based on PETSc library * Light, stable and accurate staggered-grid finite difference spatial discretization * Marker-in-Cell pedictor-corector time discretization with Runge-Kutta 4-th order * Elastic stress rotation algorithm based on the time integration of the vorticity pseudo-vector * Staircase-type internal free surface boundary condition without artificial viscosity contrast * Geodynamically relevant visco-elasto-plastic rheology * Global velocity-pressure-temperature Newton-Raphson nonlinear solver * Local nonlinear solver based on FZERO algorithm * Coupled velocity-pressure geometric multigrid preconditioner with Galerkin coarsening Staggered grid finite difference, being inherently Eulerian and rather complicated discretization method, provides no natural treatment of free surface boundary condition. The solution based on the quasi-viscous sticky-air phase introduces significant viscosity contrasts and spoils the convergence of the iterative solvers. In LaMEM we are currently implementing an approximate stair-case type of the free surface boundary condition which excludes the empty cells and restores the solver convergence. Because of the mutual dependence of the stress and strain-rate tensor components, and their different spatial locations in the grid, there is no straightforward way of implementing the nonlinear rheology. In LaMEM we have developed and implemented an efficient interpolation scheme for the second invariant of the strain-rate tensor, that solves this problem. Scalable efficient linear solvers are the key components of the successful nonlinear problem solution. In LaMEM we have a range of PETSc-based preconditioning techniques that either employ a block factorization of the velocity-pressure matrix, or treat it as a monolithic piece. In particular we have implemented the custom restriction-interpolation operators for the coupled Galerkin multigrid preconditioner. We have found that this type of algorithm is very robust with respect to the high grid resolutions and realistic viscosity variations. The coupled Galerking geometric multigrid implemented with the custom restriction-interpolation operators currently enables LaMEM to run efficiently with the grid sizes up to 1000-cube cells on the IBM Blue Gene/Q machines. This project is funded by ERC Starting Grant 258830 Computer facilities are provided by Jülich supercomputer center (Germany)
Finite-Difference Approach for a 6th-Order Nonlinear Phase Equation with Self-Excitation
Mohammed, Mayada; Strunin, Dmitry
A range of physical systems, particularly of chemical nature involving reactions, perform self-excited oscillations coupled by diffusion. The role of diffusion is not trivial so that initial differences in the phase of the oscillations between different points in space do not necessarily disappear as time goes; they may self-sustain. The dynamics of the phase depend on the values of the controlling parameters of the system. We consider a 6th-order nonlinear partial differential equation resulting insuch dynamics. The equation is solved using central finite-difference discretization in space. The resulting system of ordinary differential equations is integrated in time using a Matlab solver. The numerical code is tested using forced versions of the equation, which admit exact analytical solutions. The comparison of the exact and numerical solutions demonstrates satisfactory agreement.
International Nuclear Information System (INIS)
We introduce a new numerical grid-based method on unstructured grids in the three-dimensional real-space to investigate the electronic structure of polyatomic molecules. The Voronoi-cell finite difference (VFD) method realizes a discrete Laplacian operator based on Voronoi cells and their natural neighbors, featuring high adaptivity and simplicity. To resolve multicenter Coulomb singularity in all-electron calculations of polyatomic molecules, this method utilizes highly adaptive molecular grids which consist of spherical atomic grids. It provides accurate and efficient solutions for the Schroedinger equation and the Poisson equation with the all-electron Coulomb potentials regardless of the coordinate system and the molecular symmetry. For numerical examples, we assess accuracy of the VFD method for electronic structures of one-electron polyatomic systems, and apply the method to the density-functional theory for many-electron polyatomic molecules.
Barucq, Hélène; Calandra, Henri; Diaz, Julien; Ventimiglia, Florent
2013-01-01
Reverse Time Migration (RTM) is one of the most widely used techniques for Seismic Imaging, but it in- duces very high computational cost since it is based on many successive solutions to the full wave equation. High-Order Discontinuous Galerkin Methods (DGM), coupled with High Performance Computing techniques, can be used to solve accurately this equation in complex geophysic media without increasing the computational burden. However, to fully exploit the high-order space discretization, it ...
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G. Shapiro
2012-11-01
Full Text Available Results of a sensitivity study are presented from various configurations of the NEMO ocean model in the Black Sea. The standard choices of vertical discretization, viz. z-levels, s-coordinates and enveloped s-coordinates, all show their limitations in the areas of complex topography. Two new hydrid vertical coordinate schemes are presented: the "s-on-top-of-z" and its enveloped version. The hybrid grids use s-coordinates or enveloped s-coordinates in the upper layer, from the sea surface to the depth of the shelf break, and z-coordinates are set below this level. The study is carried out for a number of idealised and real world settings. The hybrid schemes help reduce errors generated by the standard schemes in the areas of steep topography. Results of sensitivity tests with various horizontal diffusion formulations show that the mesoscale activity is better captured with a significantly smaller value of Smagorinsky viscosity coefficient than it was previously suggested.
Wei, Leilei; He, Yinnian
2012-01-01
In this paper, a fully discrete local discontinuous Galerkin (LDG) finite element method is considered for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation. The scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. We prove that our scheme is unconditional stable and $L^2$ error estimate for the linear case with the convergence rate $O(h^{k+1}+(\\Delta t)^2+(\\Delta t)^\\frac{\\alpha}{2}h^{k+1/2})$. Numerical exam...
Rayleigh Wave Numerical Dispersion in a 3D Finite-Difference Algorithm
Preston, L. A.; Aldridge, D. F.
2010-12-01
A Rayleigh wave propagates laterally without dispersion in the vicinity of the plane stress-free surface of a homogeneous and isotropic elastic halfspace. The phase speed is independent of frequency and depends only on the Poisson ratio of the medium. However, after temporal and spatial discretization, a Rayleigh wave simulated by a 3D staggered-grid finite-difference (FD) seismic wave propagation algorithm suffers from frequency- and direction-dependent numerical dispersion. The magnitude of this dispersion depends critically on FD algorithm implementation details. Nevertheless, proper gridding can control numerical dispersion to within an acceptable level, leading to accurate Rayleigh wave simulations. Many investigators have derived dispersion relations appropriate for body wave propagation by various FD algorithms. However, the situation for surface waves is less well-studied. We have devised a numerical search procedure to estimate Rayleigh phase speed and group speed curves for 3D O(2,2) and O(2,4) staggered-grid FD algorithms. In contrast with the continuous time-space situation (where phase speed is obtained by extracting the appropriate root of the Rayleigh cubic), we cannot develop a closed-form mathematical formula governing the phase speed. Rather, we numerically seek the particular phase speed that leads to a solution of the discrete wave propagation equations, while holding medium properties, frequency, horizontal propagation direction, and gridding intervals fixed. Group speed is then obtained by numerically differentiating the phase speed with respect to frequency. The problem is formulated for an explicit stress-free surface positioned at two different levels within the staggered spatial grid. Additionally, an interesting variant involving zero-valued medium properties above the surface is addressed. We refer to the latter as an implicit free surface. Our preliminary conclusion is that an explicit free surface, implemented with O(4) spatial FD operators and positioned at the level of the compressional stress components, leads to superior numerical dispersion performance. Phase speeds measured from fixed-frequency synthetic seismograms agree very well with the numerical predictions. Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Company, for the US Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
Vibration analysis of rotating turbomachinery blades by an improved finite difference method
Subrahmanyam, K. B.; Kaza, K. R. V.
1985-01-01
The problem of calculating the natural frequencies and mode shapes of rotating blades is solved by an improved finite difference procedure based on second-order central differences. Lead-lag, flapping and coupled bending-torsional vibration cases of untwisted blades are considered. Results obtained by using the present improved theory have been observed to be close lower bound solutions. The convergence has been found to be rapid in comparison with the classical first-order finite difference method. While the computational space and time required by the present approach is observed to be almost the same as that required by the first-order theory for a given mesh size, accuracies of practical interest can be obtained by using the improved finite difference procedure with a relatively smaller matrix size, in contrast to the classical finite difference procedure which requires either a larger matrix or an extrapolation procedure for improvement in accuracy.
The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion
International Nuclear Information System (INIS)
Numerical modeling of seismic wave propagation and earthquake motion is an irreplaceable tool in investigation of the Earth's structure, processes in the Earth, and particularly earthquake phenomena. Among various numerical methods, the finite-difference method is the dominant method in the modeling of earthquake motion. Moreover, it is becoming more important in the seismic exploration and structural modeling. At the same time we are convinced that the best time of the finite-difference method in seismology is in the future. This monograph provides tutorial and detailed introduction to the application of the finite-difference, finite-element, and hybrid finite-difference-finite-element methods to the modeling of seismic wave propagation and earthquake motion. The text does not cover all topics and aspects of the methods. We focus on those to which we have contributed. (Author)
Prokopidis, Konstantinos P
2013-01-01
A novel 3-D higher-order finite-difference time-domain framework with complex frequency-shifted perfectly matched layer for the modeling of wave propagation in cold plasma is presented. Second- and fourth-order spatial approximations are used to discretize Maxwell's curl equations and a uniaxial perfectly matched layer with the complex frequency-shifted equations is introduced to terminate the computational domain. A numerical dispersion study of second- and higher-order techniques is elaborated and their stability criteria are extracted for each scheme. Comparisons with analytical solutions verify the accuracy of the proposed methods and the low dispersion error of the higher-order schemes.
Zhou, Dong; Seibold, Benjamin; Shirokoff, David; Chidyagwai, Prince; Rosales, Rodolfo Ruben
2013-01-01
We demonstrate how meshfree finite difference methods can be applied to solve vector Poisson problems with electric boundary conditions. In these, the tangential velocity and the incompressibility of the vector field are prescribed at the boundary. Even on irregular domains with only convex corners, canonical nodal-based finite elements may converge to the wrong solution due to a version of the Babuska paradox. In turn, straightforward meshfree finite differences converge to...
A fourth order finite difference method for solving elliptic partial differential equations
International Nuclear Information System (INIS)
A fourth order finite difference method for solving second order elliptic partial differential equations with Dirichlet boundary conditions on a rectangle is developed. The simplicity of the standard 5-POINT STAR finite difference method (which is second order accurate) in two dimensions is retained. Four problems with known exact solutions are used to test the method and it is demonstrated that, for smooth problems, the method is indeed fourth order accurate. (author)
Enhancing coronary Wave Intensity Analysis robustness by high order central finite differences
DEFF Research Database (Denmark)
Rivolo, Simone; Asrress, Kaleab N
2014-01-01
BACKGROUND: Coronary Wave Intensity Analysis (cWIA) is a technique capable of separating the effects of proximal arterial haemodynamics from cardiac mechanics. Studies have identified WIA-derived indices that are closely correlated with several disease processes and predictive of functional recovery following myocardial infarction. The cWIA clinical application has, however, been limited by technical challenges including a lack of standardization across different studies and the derived indices' sensitivity to the processing parameters. Specifically, a critical step in WIA is the noise removal for evaluation of derivatives of the acquired signals, typically performed by applying a Savitzky-Golay filter, to reduce the high frequency acquisition noise. METHODS: The impact of the filter parameter selection on cWIA output, and on the derived clinical metrics (integral areas and peaks of the major waves), is first analysed. The sensitivity analysis is performed either by using the filter as a differentiator to calculate the signals' time derivative or by applying the filter to smooth the ensemble-averaged waveforms. Furthermore, the power-spectrum of the ensemble-averaged waveforms contains little high-frequency components, which motivated us to propose an alternative approach to compute the time derivatives of the acquired waveforms using a central finite difference scheme. RESULTS AND CONCLUSION: The cWIA output and consequently the derived clinical metrics are significantly affected by the filter parameters, irrespective of its use as a smoothing filter or a differentiator. The proposed approach is parameter-free and, when applied to the 10 in-vivo human datasets and the 50 in-vivo animal datasets, enhances the cWIA robustness by significantly reducing the outcome variability (by 60%).
Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows
Fu, S. C.; So, R. M. C.; Leung, W. W. F.
2010-08-01
With the advent of state-of-the-art computers and their rapid availability, the time is ripe for the development of efficient uncertainty quantification (UQ) methods to reduce the complexity of numerical models used to simulate complicated systems with incomplete knowledge and data. The spectral stochastic finite element method (SSFEM) which is one of the widely used UQ methods, regards uncertainty as generating a new dimension and the solution as dependent on this dimension. A convergent expansion along the new dimension is then sought in terms of the polynomial chaos system, and the coefficients in this representation are determined through a Galerkin approach. This approach provides an accurate representation even when only a small number of terms are used in the spectral expansion; consequently, saving in computational resource can be realized compared to the Monte Carlo (MC) scheme. Recent development of a finite difference lattice Boltzmann method (FDLBM) that provides a convenient algorithm for setting the boundary condition allows the flow of Newtonian and non-Newtonian fluids, with and without external body forces to be simulated with ease. Also, the inherent compressibility effect in the conventional lattice Boltzmann method, which might produce significant errors in some incompressible flow simulations, is eliminated. As such, the FDLBM together with an efficient UQ method can be used to treat incompressible flows with built in uncertainty, such as blood flow in stenosed arteries. The objective of this paper is to develop a stochastic numerical solver for steady incompressible viscous flows by combining the FDLBM with a SSFEM. Validation against MC solutions of channel/Couette, driven cavity, and sudden expansion flows are carried out.
Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows
International Nuclear Information System (INIS)
With the advent of state-of-the-art computers and their rapid availability, the time is ripe for the development of efficient uncertainty quantification (UQ) methods to reduce the complexity of numerical models used to simulate complicated systems with incomplete knowledge and data. The spectral stochastic finite element method (SSFEM) which is one of the widely used UQ methods, regards uncertainty as generating a new dimension and the solution as dependent on this dimension. A convergent expansion along the new dimension is then sought in terms of the polynomial chaos system, and the coefficients in this representation are determined through a Galerkin approach. This approach provides an accurate representation even when only a small number of terms are used in the spectral expansion; consequently, saving in computational resource can be realized compared to the Monte Carlo (MC) scheme. Recent development of a finite difference lattice Boltzmann method (FDLBM) that provides a convenient algorithm for setting the boundary condition allows the flow of Newtonian and non-Newtonian fluids, with and without external body forces to be simulated with ease. Also, the inherent compressibility effect in the conventional lattice Boltzmann method, which might produce significant errors in some incompressible flow simulations, is eliminated. As such, the FDLBM together with an efficient UQ method can be used to treat incompressible flows with built in uncertainty, such as blood flith built in uncertainty, such as blood flow in stenosed arteries. The objective of this paper is to develop a stochastic numerical solver for steady incompressible viscous flows by combining the FDLBM with a SSFEM. Validation against MC solutions of channel/Couette, driven cavity, and sudden expansion flows are carried out.
Uniformly convergent scheme for Convection-Diffusion problem
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K. Sharath Babu
2012-02-01
Full Text Available In this paper a study of uniformly convergent method proposed by Il’in –Allen-South well scheme was made. A condition was contemplated for uniform convergence in the specified domain. This developed scheme is uniformly convergent for any choice of the diffusion parameter. The search provides a first- order uniformly convergent method with discrete maximum norm. It was observed that the error increases as step size h gets smaller for mid range values of perturbation parameter. Then an analysis carried out by [16] was employed to check the validity of solution with respect to physical aspect and it was in agreement with the analytical solution. The uniformly convergent method gives better results than the finite difference methods. The computed and plotted solutions of this method are in good – agreement with the exact solution.
High resolution schemes for hyperbolic conservation laws
International Nuclear Information System (INIS)
A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an apppropriately modified flux function. The so-derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme. Numerical experiments are presented to demonstrate the performance of these new schemes
Iwase, Shigeru; Ono, Tomoya
2015-01-01
We propose an efficient procedure to obtain Green's functions by combining the shifted conjugate orthogonal conjugate gradient (shifted COCG) method with the nonequilibrium Green's function (NEGF) method based on a real-space finite-difference (RSFD) approach. The bottleneck of the computation in the NEGF scheme is matrix inversion of the Hamiltonian including the self-energy terms of electrodes to obtain perturbed Green's function in the transition region. This procedure first computes unperturbed Green's functions and calculates perturbed Green's functions from the unperturbed ones using a mathematically strict relation. Since the matrices to be inverted to obtain the unperturbed Green's functions are sparse, complex-symmetric and shifted for a given set of sampling energy points, we can use the shifted COCG method, in which once the Green's function for a reference energy point has been calculated, the Green's functions for the other energy points can be obtained with a moderate computational cost. We calc...
A modified symplectic scheme for seismic wave modeling
Liu, Shaolin; Li, Xiaofan; Wang, Wenshuai; Xu, Ling; Li, Bingfei
2015-05-01
Symplectic integrators are well known for their excellent performance in solving partial differential equation of dynamical systems because they are capable of preserving some conservative properties of dynamic equations. However, there are not enough high-order, for example third-order symplectic schemes, which are suitable for seismic wave equations. Here, we propose a strategy to construct a symplectic scheme that is based on a so-called high-order operator modification method. We first employ a conventional two-stage Runge-Kutta-Nyström (RKN) method to solve the ordinary differential equations, which are derived from the spatial discretization of the seismic wave equations. We then add a high-order term to the RKN method. Finally, we obtain a new third-order symplectic scheme with all positive symplectic coefficients, and it is defined based on the order condition, the symplectic condition, the stability condition and the dispersion relation. It is worth noting that the new scheme is independent of the spatial discretization type used, and we simply apply some finite difference operators to approximate the spatial derivatives of the isotropic elastic equations for a straightforward discussion. For the theoretical analysis, we obtain the semi-analytic stability conditions of our scheme with various orders of spatial approximation. The stability and dispersion properties of our scheme are also compared with conventional schemes to illustrate the favorable numerical behaviors of our scheme in terms of precision, stability and dispersion characteristics. Finally, three numerical experiments are employed to further demonstrate the validity of our method. The modified strategy that is proposed in this paper can be used to construct other explicit symplectic schemes.
MacKinnon, Robert J.; Carey, Graham F.
2003-01-01
A new class of positivity-preserving, flux-limited finite-difference and Petrov-Galerkin (PG) finite-element methods are devised for reactive transport problems.The methods are similar to classical TVD flux-limited schemes with the main difference being that the flux-limiter constraint is designed to preserve positivity for problems involving diffusion and reaction. In the finite-element formulation, we also consider the effect of numerical quadrature in the lumped and consistent mass matrix forms on the positivity-preserving property. Analysis of the latter scheme shows that positivity-preserving solutions of the resulting difference equations can only be guaranteed if the flux-limited scheme is both implicit and satisfies an additional lower-bound condition on time-step size. We show that this condition also applies to standard Galerkin linear finite-element approximations to the linear diffusion equation. Numerical experiments are provided to demonstrate the behavior of the methods and confirm the theoretical conditions on time-step size, mesh spacing, and flux limiting for transport problems with and without nonlinear reaction.
Directory of Open Access Journals (Sweden)
Abdur Shahid
2012-09-01
Full Text Available Digital watermarking is the process to hide digital pattern directly into a digital content. Digital watermarking techniques are used to address digital rights management, protect information and conceal secrets. An invisible non-blind watermarking approach for gray scale images is proposed in this paper. The host image is decomposed into 3-levels using Discrete Wavelet Transform. Based on the parent-child relationship between the wavelet coefficients the Set Partitioning in Hierarchical Trees (SPIHT compression algorithm is performed on the LH3, LH2, HL3 and HL2 subbands to find out the significant coefficients. The most significant coefficients of LH2 and HL2 bands are selected to embed a binary watermark image. The selected significant coefficients are modulated using Noise Visibility Function, which is considered as the best strength to ensure better imperceptibility. The approach is tested against various image processing attacks such as addition of noise, filtering, cropping, JPEG compression, histogram equalization and contrast adjustment. The experimental results reveal the high effectiveness of the method.
Iturrarán-Viveros, Ursula; Molero, Miguel
2013-07-01
This paper presents an implementation of a 2.5-D finite-difference (FD) code to model acoustic full waveform monopole logging in cylindrical coordinates accelerated by using the new parallel computing devices (PCDs). For that purpose we use the industry open standard Open Computing Language (OpenCL) and an open-source toolkit called PyOpenCL. The advantage of OpenCL over similar languages is that it allows one to program a CPU (central processing unit) a GPU (graphics processing unit), or multiple GPUs and their interaction among them and with the CPU, or host device. We describe the code and give a performance test in terms of speed using six different computing devices under different operating systems. A maximum speedup factor over 34.2, using the GPU is attained when compared with the execution of the same program in parallel using a CPU quad-core. Furthermore, the results obtained with the finite differences are validated using the discrete wavenumber method (DWN) achieving a good agreement. To provide the Geoscience and the Petroleum Science communities with an open tool for numerical simulation of full waveform sonic logs that runs on the PCDs, the full implementation of the 2.5-D finite difference with PyOpenCL is included.
Deubelbeiss, Y.; Kaus, B. J.
2007-12-01
Numerical modeling of geodynamic processes typically requires the solution of the Stokes equations for creeping, highly viscous flows. Since material properties such as effective viscosity of rocks can vary many orders of magnitudes over small spatial scales, the Stokes solver needs to be robust even in the case of highly variable viscosity. Currently, a number of different techniques (e.g. finite element, finite difference and spectral methods) are in use by different authors. Benchmark studies indicate that the accuracy of the velocity solution is satisfying for most methods. The accuracy of deviatoric stresses and pressures, however, is typically less than that of velocities. In the case of highly varying viscosity, some methods even result in oscillating pressures. Over recent years there has been an increased demand for accurate pressures. E.g. melt migration through compacting, two- phase flow materials requires solving equations for the fluid and the solid matrix. Shear-localization in partially molten rocks couples moving fluid within a deforming solid. Therefore it is necessary to have accurate knowledge of pressures, which feed back to the solution. It becomes increasingly important to understand the accuracy of numerical methods for Stokes flow in the presence of large variations in material properties. The objective of this study is therefore to evaluate the accuracy of the pressure solution for a number of numerical techniques. Thereby, we make use of a 2-D analytical solution for the stress distribution inside and around a viscous inclusion in a matrix of different viscosity subjected to pure-shear boundary conditions. Furthermore, numerical simulations have been compared with the analytical solution for density-driven flow (Rayleigh-Taylor instabilities). Results are presented for a staggered grid velocity- pressure finite difference method, a stream function finite difference method and a rotated staggered grid velocity- pressure finite difference method. The staggered grid and stream function formulations require viscosities to be defined both at center and at corner points of control volumes, while the rotated staggered grid finite difference method only requires viscosity defined at center points. We demonstrate that the manner in which viscosities are defined at these locations is of extreme importance for the accuracy of the overall solution. The problem is investigated by studying a simple physical quasi 1-D model with a contact of two media representing the contact between an inclusion embedded in a matrix (2-D case). Analytically and numerically, it is demonstrated that viscosity interpolation using harmonic averaging yields the best results. 2-D numerical results for the above mentioned setups show that for different interpolation methods the errors can vary one order of magnitude. Accuracy of velocity solutions are more than half an order of magnitude better than pressure solutions. The Rayleigh-Taylor instability test, on the other hand, has a weaker sensitivity to viscosity interpolation methods. Results are mainly dependent on the manner in which density is interpolated, which is the driving force in this system. Differences between the three numerical schemes for both setups are secondary compared to the effect of the viscosity interpolation. The best averaging method, for the setups studied here, is a geometric- harmonic averaging of viscosity and an arithmetic averaging of density.
A Nonstandard Dynamically Consistent Numerical Scheme Applied to Obesity Dynamics
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Gilberto González-Parra
2008-12-01
Full Text Available The obesity epidemic is considered a health concern of paramount importance in modern society. In this work, a nonstandard finite difference scheme has been developed with the aim to solve numerically a mathematical model for obesity population dynamics. This interacting population model represented as a system of coupled nonlinear ordinary differential equations is used to analyze, understand, and predict the dynamics of obesity populations. The construction of the proposed discrete scheme is developed such that it is dynamically consistent with the original differential equations model. Since the total population in this mathematical model is assumed constant, the proposed scheme has been constructed to satisfy the associated conservation law and positivity condition. Numerical comparisons between the competitive nonstandard scheme developed here and Euler's method show the effectiveness of the proposed nonstandard numerical scheme. Numerical examples show that the nonstandard difference scheme methodology is a good option to solve numerically different mathematical models where essential properties of the populations need to be satisfied in order to simulate the real world.
A Second Order Finite Difference Approximation for the Fractional Diffusion Equation
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H. M. Nasir
2013-07-01
Full Text Available We consider an approximation of one-dimensional fractional diffusion equation. We claim and show that the finite difference approximation obtained from the Grünwald-Letnikov formulation, often claimed to be of first order accuracy, is in fact a second order approximation of the fractional derivative at a point away from the grid points. We use this fact to device a second order accurate finite difference approximation for the fractional diffusion equation. The proposed method is also shown to be unconditionally stable. By this approach, we treat three cases of difference approximations in a unified setting. The results obtained are justified by numerical examples.
Flood routing using finite differences and the fourth order Runge-Kutta method
International Nuclear Information System (INIS)
The Saint-Venant continuity and momentum equations are solved numerically by discretising the time variable using finite differences and then the Runge-Kutta method is employed to solve the resulting ODE. A model of the Rufiji river downstream on the proposed Stiegler Gourge Dam is used to provide numerical results for comparison. The present approach is found to be superior to an earlier analysis using finite differences in both space and time. Moreover, the steady and unsteady flow analyses give almost identical predictions for the stage downstream provided that the variations of the discharge and stage upstream are small. (author)
Introduction to the Finite-Difference Time-Domain (FDTD) Method for Electromagnetics
Gedney, Stephen
2011-01-01
Introduction to the Finite-Difference Time-Domain (FDTD) Method for Electromagnetics provides a comprehensive tutorial of the most widely used method for solving Maxwell's equations -- the Finite Difference Time-Domain Method. This book is an essential guide for students, researchers, and professional engineers who want to gain a fundamental knowledge of the FDTD method. It can accompany an undergraduate or entry-level graduate course or be used for self-study. The book provides all the background required to either research or apply the FDTD method for the solution of Maxwell's equations to p
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Vineet K. Srivastava
2013-12-01
Full Text Available This article describes a new implicit finite-difference method: an implicit logarithmic finite-difference method (I-LFDM, for the numerical solution of two dimensional time-dependent coupled viscous Burgers’ equation on the uniform grid points. As the Burgers’ equation is nonlinear, the proposed technique leads to a system of nonlinear systems, which is solved by Newton's iterative method at each time step. Computed solutions are compared with the analytical solutions and those already available in the literature and it is clearly shown that the results obtained using the method is precise and reliable for solving Burgers’ equation.
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Srivastava, Vineet K., E-mail: vineetsriiitm@gmail.com [ISRO Telemetry, Tracking and Command Network (ISTRAC), Bangalore-560058 (India); Awasthi, Mukesh K. [Department of Mathematics, University of Petroleum and Energy Studies, Dehradun-248007 (India); Singh, Sarita [Department of Mathematics, WIT- Uttarakhand Technical University, Dehradun-248007 (India)
2013-12-15
This article describes a new implicit finite-difference method: an implicit logarithmic finite-difference method (I-LFDM), for the numerical solution of two dimensional time-dependent coupled viscous Burgers’ equation on the uniform grid points. As the Burgers’ equation is nonlinear, the proposed technique leads to a system of nonlinear systems, which is solved by Newton's iterative method at each time step. Computed solutions are compared with the analytical solutions and those already available in the literature and it is clearly shown that the results obtained using the method is precise and reliable for solving Burgers’ equation.
Studies on the higher order spatial difference schemes at the interface of two media
International Nuclear Information System (INIS)
Several higher order finite difference schemes have been proposed for the solution of a discrete ordinates transport equation. The performance characteristics of these methods have been studied through numerical and mathematical analyses. In these studies, attention was restricted to a single, homogeneous medium and to uniform meshes only. However, in practice one has to employ nonuniform meshes such as near the interface of any two media. A second criterion that needs examination is the influence of the cross section of the medium on the behavior of these schemes. Finally, the mathematical analysis is, in principle, restricted to a single energy group. Although it is believed that there should be no significant differences in the conclusions with respect to the multigroup problem, it appears that the order of convergence is not as high as estimated when the higher order schemes are applied to a multigroup neutron transport. The results of test cases are presented and discussed, where some of the finite difference schemes, when applied to an interface and multigroup problems, exhibit different behavior than reported earlier
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Lindemberg Lima Fernandes
2009-03-01
Full Text Available Este trabalho tem por objetivo apresentar os resultados da modelagem sísmica em meios com fortes descontinuidades de propriedades físicas, com ênfase na existência de difrações e múltiplas reflexões, tendo a Bacia do Amazonas como referência à modelagem. As condições de estabilidade e de fronteiras utilizadas no cálculo do campo de ondas sísmicas foram analisadas numericamente pelo método das diferenças finitas, visando melhor compreensão e controle da interpretação de dados sísmicos. A geologia da Bacia do Amazonas é constituída por rochas sedimentares depositadas desde o Ordoviciano até o Recente que atingem espessuras da ordem de 5 km. Os corpos de diabásio, presentes entre os sedimentos paleozóicos, estão dispostos na forma de soleiras, alcançam espessuras de centenas de metros e perfazem um volume total de aproximadamente 90000 Km³. A ocorrência de tais estruturas é responsável pela existência de reflexões múltiplas durante a propagação da onda sísmica o que impossibilita melhor interpretação dos horizontes refletores que se encontram abaixo destas soleiras. Para representar situações geológicas desse tipo foram usados um modelo (sintético acústico de velocidades e um código computacional elaborado via método das diferenças finitas com aproximação de quarta ordem no espaço e no tempo da equação da onda. A aplicação dos métodos de diferenças finitas para o estudo de propagação de ondas sísmicas melhorou a compreensão sobre a propagação em meios onde existem heterogeneidades significativas, tendo como resultado boa resolução na interpretação dos eventos de reflexão sísmica em áreas de interesse. Como resultado dos experimentos numéricos realizados em meio de geologia complexa, foi observada a influência significativa das reflexões múltiplas devido à camada de alta velocidade, isto provocou maior perda de energia e dificultou a interpretação dos alvos. Por esta razão recomenda-se a integração de dados de superfície com os de poço, com o objetivo de obter melhor imagem dos alvos abaixo das soleiras de diabásio.This paper discusses the seismic modeling in medium with strong discontinuities in its physical properties. The approach takes in consideration the existences diffractions and multiple reflections in the analyzed medium, which, at that case, is the Amazon Basin. The stability and boundary conditions of modeling were analyzed by the method of the finite differences. Sedimentary rocks deposited since the Ordovician to the present, reaching depth up to 5 Km. The bodies of diabasic between the paleozoic sediments are layers reaching thickness of hundred meters, which add to 90.000 km3, form the geology of the Amazon Basin. The occurrence of these structures is responsible for multiple reflections during the propagation of the seismic waves, which become impossible a better imaging of horizons located bellow the layers. The representation this geological situation was performed an (synthetic acoustic velocity model. The numerical discretization scheme is based in a fourth order approximation of the acoustic wave equation in space and time The understanding of the wave propagation heterogeneous medium has improved for the application of the finite difference method. The method achieves a good resolution in the interpretation of seismic reflection events. The numerical results discusses in this paper have allowed to observed the influence of the multiple reflection in a high velocity layer. It increase a loss of energy and difficult the interpretation of the target. For this reason the integration of surface data with the well data is recommended, with the objective to get one better image of the targets below of the diabasic layer.
Scientific Electronic Library Online (English)
Lindemberg Lima, Fernandes; João Carlos Ribeiro, Cruz; Claudio José Cavalcante, Blanco; Ana Rosa Baganha, Barp.
2009-03-01
Full Text Available Este trabalho tem por objetivo apresentar os resultados da modelagem sísmica em meios com fortes descontinuidades de propriedades físicas, com ênfase na existência de difrações e múltiplas reflexões, tendo a Bacia do Amazonas como referência à modelagem. As condições de estabilidade e de fronteiras [...] utilizadas no cálculo do campo de ondas sísmicas foram analisadas numericamente pelo método das diferenças finitas, visando melhor compreensão e controle da interpretação de dados sísmicos. A geologia da Bacia do Amazonas é constituída por rochas sedimentares depositadas desde o Ordoviciano até o Recente que atingem espessuras da ordem de 5 km. Os corpos de diabásio, presentes entre os sedimentos paleozóicos, estão dispostos na forma de soleiras, alcançam espessuras de centenas de metros e perfazem um volume total de aproximadamente 90000 Km³. A ocorrência de tais estruturas é responsável pela existência de reflexões múltiplas durante a propagação da onda sísmica o que impossibilita melhor interpretação dos horizontes refletores que se encontram abaixo destas soleiras. Para representar situações geológicas desse tipo foram usados um modelo (sintético) acústico de velocidades e um código computacional elaborado via método das diferenças finitas com aproximação de quarta ordem no espaço e no tempo da equação da onda. A aplicação dos métodos de diferenças finitas para o estudo de propagação de ondas sísmicas melhorou a compreensão sobre a propagação em meios onde existem heterogeneidades significativas, tendo como resultado boa resolução na interpretação dos eventos de reflexão sísmica em áreas de interesse. Como resultado dos experimentos numéricos realizados em meio de geologia complexa, foi observada a influência significativa das reflexões múltiplas devido à camada de alta velocidade, isto provocou maior perda de energia e dificultou a interpretação dos alvos. Por esta razão recomenda-se a integração de dados de superfície com os de poço, com o objetivo de obter melhor imagem dos alvos abaixo das soleiras de diabásio. Abstract in english This paper discusses the seismic modeling in medium with strong discontinuities in its physical properties. The approach takes in consideration the existences diffractions and multiple reflections in the analyzed medium, which, at that case, is the Amazon Basin. The stability and boundary conditions [...] of modeling were analyzed by the method of the finite differences. Sedimentary rocks deposited since the Ordovician to the present, reaching depth up to 5 Km. The bodies of diabasic between the paleozoic sediments are layers reaching thickness of hundred meters, which add to 90.000 km3, form the geology of the Amazon Basin. The occurrence of these structures is responsible for multiple reflections during the propagation of the seismic waves, which become impossible a better imaging of horizons located bellow the layers. The representation this geological situation was performed an (synthetic) acoustic velocity model. The numerical discretization scheme is based in a fourth order approximation of the acoustic wave equation in space and time The understanding of the wave propagation heterogeneous medium has improved for the application of the finite difference method. The method achieves a good resolution in the interpretation of seismic reflection events. The numerical results discusses in this paper have allowed to observed the influence of the multiple reflection in a high velocity layer. It increase a loss of energy and difficult the interpretation of the target. For this reason the integration of surface data with the well data is recommended, with the objective to get one better image of the targets below of the diabasic layer.
Iwase, Shigeru; Hoshi, Takeo; Ono, Tomoya
2015-06-01
We propose an efficient procedure to obtain Green's functions by combining the shifted conjugate orthogonal conjugate gradient (shifted COCG) method with the nonequilibrium Green's function (NEGF) method based on a real-space finite-difference (RSFD) approach. The bottleneck of the computation in the NEGF scheme is matrix inversion of the Hamiltonian including the self-energy terms of electrodes to obtain the perturbed Green's function in the transition region. This procedure first computes unperturbed Green's functions and calculates perturbed Green's functions from the unperturbed ones using a mathematically strict relation. Since the matrices to be inverted to obtain the unperturbed Green's functions are sparse, complex-symmetric, and shifted for a given set of sampling energy points, we can use the shifted COCG method, in which once the Green's function for a reference energy point has been calculated the Green's functions for the other energy points can be obtained with a moderate computational cost. We calculate the transport properties of a C60@(10,10) carbon nanotube (CNT) peapod suspended by (10,10)CNTs as an example of a large-scale transport calculation. The proposed scheme opens the possibility of performing large-scale RSFD-NEGF transport calculations using massively parallel computers without the loss of accuracy originating from the incompleteness of the localized basis set.
Li, Hong; Zhang, Wei; Zhang, Zhenguo; Chen, Xiaofei
2015-07-01
A discontinuous grid finite-difference (FD) method with non-uniform time step Runge-Kutta scheme on curvilinear collocated-grid is developed for seismic wave simulation. We introduce two transition zones: a spatial transition zone and a temporal transition zone, to exchange wavefield across the spatial and temporal discontinuous interfaces. A Gaussian filter is applied to suppress artificial numerical noise caused by down-sampling the wavefield from the finer grid to the coarser grid. We adapt the non-uniform time step Runge-Kutta scheme to a discontinuous grid FD method for further increasing the computational efficiency without losing the accuracy of time marching through the whole simulation region. When the topography is included in the modelling, we carry out the discontinuous grid method on a curvilinear collocated-grid to obtain a sufficiently accurate free-surface boundary condition implementation. Numerical tests show that the proposed method can sufficiently accurately simulate the seismic wave propagation on such grids and significantly reduce the computational resources consumption with respect to regular grids.
A finite-difference calculation of point defect migration into a dislocation loop
International Nuclear Information System (INIS)
A finite-difference calculation was carried out of the point defect flux from a surrounding spherical reservoir to a dislocation loop. Computations were done using material parameters pertinent to zirconium. The results were used to comment on the accuracy and range of applicability of some results in the literature. (auth)
Directory of Open Access Journals (Sweden)
Abdollah BORHANIFAR
2013-01-01
Full Text Available In this study fractional Poisson equation is scrutinized through finite difference using shifted Grünwald estimate. A novel method is proposed numerically. The existence and uniqueness of solution for the fractional Poisson equation are proved. Exact and numerical solution are constructed and compared. Then numerical result shows the efficiency of the proposed method.
Detailed balance principle and finite-difference stochastic equation in a field theory
International Nuclear Information System (INIS)
A finite-difference equation, which is a generalization of the Langevin equation in field theory, has been obtained basing upon the principle of detailed balance for the Markov chain. Advantages of the present approach as compared with the conventional Parisi-Wu method are shown for examples of an exactly solvable problem of zero-dimensional quantum theory and a simple numerical simulation
International Nuclear Information System (INIS)
The reactor code PUMA, developed in CNEA, simulates nuclear reactors discretizing space in finite difference elements. Core representation is performed by means a cylindrical mesh, but the reactor channels are arranged in an hexagonal lattice. That is why a mapping using volume intersections must be used. This spatial treatment is the reason of an overestimation of the control rod reactivity values, which must be adjusted modifying the incremental cross sections. Also, a not very good treatment of the continuity conditions between core and reflector leads to an overestimation of channel power of the peripherical fuel elements between 5 to 8 per cent. Another code, DELFIN, developed also in CNEA, treats the spatial discretization using heterogeneous finite elements, allowing a correct treatment of the continuity of fluxes and current among elements and a more realistic representation of the hexagonal lattice of the reactor. A comparison between results obtained using both methods in done in this paper. (author). 4 refs., 3 figs
Energy Technology Data Exchange (ETDEWEB)
Dattoli, Giuseppe; Torre, Amalia [ENEA, Centro Ricerche Frascati, Rome (Italy). Dipt. Innovazione; Ottaviani, Pier Luigi [ENEA, Centro Ricerche Bologna (Italy); Vasquez, Luis [Madris, Univ. Complutense (Spain). Dept. de Matemateca Aplicado
1997-10-01
The finite-difference based integration method for evolution-line equations is discussed in detail and framed within the general context of the evolution operator picture. Exact analytical methods are described to solve evolution-like equations in a quite general physical context. The numerical technique based on the factorization formulae of exponential operator is then illustrated and applied to the evolution-operator in both classical and quantum framework. Finally, the general view to the finite differencing schemes is provided, displaying the wide range of applications from the classical Newton equation of motion to the quantum field theory.
Wind Turbine Micrositing: Comparison of Finite Difference Method and Computational Fluid Dynamics
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Samina Rajper
2012-01-01
Full Text Available For smooth and optimal operation of wind turbines the location of wind turbines in wind farm is critical. Parameters that need to be considered for micrositing of wind turbines are topographic effect and wind effect. The location under consideration for wind farm is Gharo, Sindh, Pakistan. Several techniques are being researched for finding the most optimal location for wind turbines. These techniques are based on linear and nonlinear mathematical models. In this paper wind pressure distribution and its effect on wind turbine on the wind farm are considered. This study is conducted to compare the mathematical model; Finite Difference Method with a computational fluid dynamics software results. Finally the results of two techniques are compared for micrositing of wind turbines and found that finite difference method is not applicable for wind turbine micrositing.
Thermal Analysis of AC Contactor Using Thermal Network Finite Difference Analysis Method
Niu, Chunping; Chen, Degui; Li, Xingwen; Geng, Yingsan
To predict the thermal behavior of switchgear quickly, the Thermal Network Finite Difference Analysis method (TNFDA) is adopted in thermal analysis of AC contactor in the paper. The thermal network model is built with nodes, thermal resistors and heat generators, and it is solved using finite difference method (FDM). The main circuit and the control system are connected by thermal resistors network, which solves the problem of multi-sources interaction in the application of TNFDA. The temperature of conducting wires is calculated according to the heat transfer process and the fundamental equations of thermal conduction. It provides a method to solve the problem of boundary conditions in applying the TNFDA. The comparison between the results of TNFDA and measurements shows the feasibility and practicability of the method.
Zhou, Dong; Shirokoff, David; Chidyagwai, Prince; Rosales, Rodolfo Ruben
2013-01-01
We demonstrate how meshfree finite difference methods can be applied to solve vector Poisson problems with electric boundary conditions. In these, the tangential velocity and the incompressibility of the vector field are prescribed at the boundary. Even on irregular domains with only convex corners, canonical nodal-based finite elements may converge to the wrong solution due to a version of the Babuska paradox. In turn, straightforward meshfree finite differences converge to the true solution, and even high-order accuracy can be achieved in a simple fashion. The methodology is then extended to a specific pressure Poisson equation reformulation of the Navier-Stokes equations that possesses the same type of boundary conditions. The resulting numerical approach is second order accurate and allows for a simple switching between an explicit and implicit treatment of the viscosity terms.
Abert, Claas; Bruckner, Florian; Satz, Armin; Suess, Dieter
2014-01-01
We implement an efficient energy-minimization algorithm for finite-difference micromagnetics that proofs especially usefull for the computation of hysteresis loops. Compared to results obtained by time integration of the Landau-Lifshitz-Gilbert equation, a speedup of up to two orders of magnitude is gained. The method is implemented in a finite-difference code running on CPUs as well as GPUs. This setup enables us to compute accurate hysteresis loops of large systems with a reasonable computational efford. As a benchmark we solve the {\\mu}Mag Standard Problem #1 with a high spatial resolution and compare the results to the solution of the Landau-Lifshitz-Gilbert equation in terms of accuracy and computing time.
Arithmetic Discrete Hyperspheres and Separatingness
Fiorio, Christophe; Toutant, Jean-Luc
2006-01-01
In the framework of the arithmetic discrete geometry, a discrete object is provided with its own analytical definition corresponding to a discretization scheme. It can thus be considered as the equivalent, in a discrete space, of a Euclidean object. Linear objects, namely lines and hyperplanes, have been widely studied under this assumption and are now deeply understood. This is not the case for discrete circles and hyperspheres for which no satisfactory definition exists. In the present pape...
HEATING5-JR: a finite difference computer program for nonlinear heat conduction problems
International Nuclear Information System (INIS)
Computer program HEATING5-JR is a revised version of HEATING5 which is a finite difference computer program used for the solution of multi-dimensional, nonlinear heat conduction problems. Pre- and post-processings for graphical representations of input data and calculation results of HEATING5 are avaiable in HEATING5-JR. The calculation equations, program descriptions and user instructions are presented. Several example problems are described in detail to demonstrate the use of the program. (author)
Finite Difference Time-Domain Modelling of Metamaterials: GPU Implementation of Cylindrical Cloak
Attique Dawood
2013-01-01
Finite difference time-domain (FDTD) technique can be used to model metamaterials by treating them as dispersive material. Drude or Lorentz model can be incorporated into the standard FDTD algorithm for modelling negative permittivity and permeability. FDTD algorithm is readily parallelisable and can take advantage of GPU acceleration to achieve speed-ups of 5x-50x depending on hardware setup. Metamaterial scattering problems are implemented using dispersive FDTD technique on GPU resulting in...
A practical implicit finite-difference method: examples from seismic modelling
International Nuclear Information System (INIS)
We derive explicit and new implicit finite-difference formulae for derivatives of arbitrary order with any order of accuracy by the plane wave theory where the finite-difference coefficients are obtained from the Taylor series expansion. The implicit finite-difference formulae are derived from fractional expansion of derivatives which form tridiagonal matrix equations. Our results demonstrate that the accuracy of a (2N + 2)th-order implicit formula is nearly equivalent to that of a (6N + 2)th-order explicit formula for the first-order derivative, and (2N + 2)th-order implicit formula is nearly equivalent to (4N + 2)th-order explicit formula for the second-order derivative. In general, an implicit method is computationally more expensive than an explicit method, due to the requirement of solving large matrix equations. However, the new implicit method only involves solving tridiagonal matrix equations, which is fairly inexpensive. Furthermore, taking advantage of the fact that many repeated calculations of derivatives are performed by the same difference formula, several parts can be precomputed resulting in a fast algorithm. We further demonstrate that a (2N + 2)th-order implicit formulation requires nearly the same memory and computation as a (2N + 4)th-order explicit formulation but attains the accuracy achieved by a (6N + 2)th-order explicit formulation for the first-order derivative and that of a (4N + 2)th-order explicit method for the second-order derivative whemethod for the second-order derivative when additional cost of visiting arrays is not considered. This means that a high-order explicit method may be replaced by an implicit method of the same order resulting in a much improved performance. Our analysis of efficiency and numerical modelling results for acoustic and elastic wave propagation validates the effectiveness and practicality of the implicit finite-difference method
Development of explicit finite difference-based simulation system for impact studies
Wong, Shaw Voon
2000-01-01
Development of numerical method-based simulation systems is presented. Two types of system development are shown. The former system is developed using conventional structured programming technique. The system incorporates a classical FD hydrocode. The latter system incorporates object-oriented design concept and numerous novel elements are included. Two explicit finite difference models for large deformation of several material characteristics are developed. The models are capable of hand...
Ward, David W.; Nelson, Keith A.
2004-01-01
We describe a simple and intuitive implementation of the method of finite difference time domain simulations for propagating electromagnetic waves using the simplest possible tools available in Microsoft Excel. The method overcomes the usual obstacles of familiarity with programming languages as it relies on little more than the cut and paste features that are standard in Excel. Avenues of exploration by students are proposed and sample graphs are included. The pedagogical e...
RODCON: a finite difference heat conduction computer code in cylindrical coordinates
Energy Technology Data Exchange (ETDEWEB)
Conklin, J.C.
1980-09-16
RODCON, a finite difference computer code, was developed to calculate the internal temperature distribution of the fuel rod simulator (FRS) for the Core Flow Test Loop (CFTL). RODCON solves the implicit, time-dependent forward-differencing heat transfer equation in 2-dimensional (Rtheta) cylindrical coordinates at an axial plane with user specified radial material zones and surface conditions at the FRS periphery. Symmetry of the boundary conditions of coolant bulk temperatures and film coefficients at the FRS periphery is not necessary.
RODCON: a finite difference heat conduction computer code in cylindrical coordinates
International Nuclear Information System (INIS)
RODCON, a finite difference computer code, was developed to calculate the internal temperature distribution of the fuel rod simulator (FRS) for the Core Flow Test Loop (CFTL). RODCON solves the implicit, time-dependent forward-differencing heat transfer equation in 2-dimensional (Rtheta) cylindrical coordinates at an axial plane with user specified radial material zones and surface conditions at the FRS periphery. Symmetry of the boundary conditions of coolant bulk temperatures and film coefficients at the FRS periphery is not necessary
Modeling and Analysis of Printed Antenna Using Finite Difference Time Domain Algorithm
Dr.T.SHANMUGANANTHAM,; Raghavan, Dr S.
2010-01-01
An efficient Finite Difference Time Domain algorithm is developed for printed patch antenna without using commercial software packages like IE3D, HFSS, ADS, and CST. Printed patch antennas which are small andconformity are demanded from the points of carrying and designing. Numerical results of return loss, current distribution, electric field and magnetic field components are plotted. The results presented for the fundamental parameters of the Microstrip patch antenna useful for wireless com...
Simulation of Acoustic Wall Reflections Using the Finite-Difference Time-Domain Method
Haapaniemi, Aki
2012-01-01
In this thesis, the reflection characteristics of layered wall structures are studied using the standard rectilinear (SRL) finite-difference time-domain (FDTD) method for modeling sound wave propagation. The structures consist of a panel with slats combined with a back wall, forming a cavity in between. Outwardly similar structures are known to have resonant properties and are generally known as resonant absorbers or distributed Helmholtz resonators. The structures studied in this thesis, ...
Transport and dispersion of pollutants in surface impoundments: a finite difference model
Energy Technology Data Exchange (ETDEWEB)
Yeh, G.T.
1980-07-01
A surface impoundment model by finite-difference (SIMFD) has been developed. SIMFD computes the flow rate, velocity field, and the concentration distribution of pollutants in surface impoundments with any number of islands located within the region of interest. Theoretical derivations and numerical algorithm are described in detail. Instructions for the application of SIMFD and listings of the FORTRAN IV source program are provided. Two sample problems are given to illustrate the application and validity of the model.
O´reilly, Ossian; Nordstro?m, Jan; Kozdon, Jeremy E.; Dunham, Eric M.
2013-01-01
A numerical method suitable for wave propagation problems in complex geometries is developed for simulating dynamic earthquake ruptures with realistic friction laws. The numerical method couples an unstructured, node-centered finite volume method to a structured, high order finite difference method. In this work we our focus attention on 2-D antiplane shear problems. The finite volume method is used on unstructured triangular meshes to resolve earthquake ruptures propagating along a nonplanar...
Transport and dispersion of pollutants in surface impoundments: a finite difference model
International Nuclear Information System (INIS)
A surface impoundment model by finite-difference (SIMFD) has been developed. SIMFD computes the flow rate, velocity field, and the concentration distribution of pollutants in surface impoundments with any number of islands located within the region of interest. Theoretical derivations and numerical algorithm are described in detail. Instructions for the application of SIMFD and listings of the FORTRAN IV source program are provided. Two sample problems are given to illustrate the application and validity of the model
New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows
Li, Zhilin; Lai, Ming-chih
2011-01-01
In this paper, new finite difference methods based on the augmented immersed interface method (IIM) are proposed for simulating an inextensible moving interface in an incompressible two-dimensional flow. The mathematical models arise from studying the deformation of red blood cells in mathematical biology. The governing equations are incompressible Stokes or Navier-Stokes equations with an unknown surface tension, which should be determined in such a way that the surface divergence of the vel...
Modeling of Nanophotonic Resonators with the Finite-Difference Frequency-Domain Method
Ivinskaya, Aliaksandra; Lavrinenko, Andrei; Shyroki, Dzmitry
2011-01-01
Finite-difference frequency-domain method with perfectly matched layers and free-space squeezing is applied to model open photonic resonators of arbitrary morphology in three dimensions. Treating each spatial dimension independently, nonuniform mesh of continuously varying density can be built easily to better resolve mode features. We explore the convergence of the eigenmode wavelength $lambda $ and quality factor $Q$ of an open dielectric sphere and of a very-high- $Q$ photonic crystal cavi...
Modeling and Simulation of Hamburger Cooking Process Using Finite Difference and CFD Methods
J. Sargolzaei; M. Abarzani; R. Aminzadeh
2011-01-01
Unsteady-state heat transfer in hamburger cooking process was modeled using one dimensional finite difference (FD) and three dimensional computational fluid dynamic (CFD) models. A double-sided cooking system was designed to study the effect of pressure and oven temperature on the cooking process. Three different oven temperatures (114, 152, 204°C) and three different pressures (20, 332, 570 pa) were selected and 9 experiments were performed. Applying pressure to hamburger increases the cont...
Temperature Distribution of Single Slope Solar Still by Finite Difference Method
Kiam Beng Yeo; Kenneth Tze Kin Teo; Cheah Meng Ong
2014-01-01
Single slope solar still utilizes solar distillation technology to clean water from brackish water was investigated. The clean water output of solar still depends on the intensity of sunlight and how well the different mediums in solar still transfer the heat energy around. Thus, the temperature distribution in the single slope solar still was analysed using the explicit finite difference method. Side view of solar still is aligned with a mesh system, which accommodates nodes and specific equ...
An Improvement for the Locally One-Dimensional Finite-Difference Time-Domain Method
Directory of Open Access Journals (Sweden)
Xiuhai Jin
2011-09-01
Full Text Available To reduce the memory usage of computing, the locally one-dimensional reduced finite-difference time-domain method is proposed. It is proven that the divergence relationship of electric-field and magnetic-field is non-zero even in charge-free regions, when the electric-field and magnetic-field are calculated with locally one-dimensional finite-difference time-domain (LOD-FDTD method, and the concrete expression of the divergence relationship is derived. Based on the non-zero divergence relationship, the LOD-FDTD method is combined with the reduced finite-difference time-domain (R-FDTD method. In the proposed method, the memory requirement of LOD-R-FDTD is reduced by1/6 (3D case of the memory requirement of LOD-FDTD averagely. The formulation is presented and the accuracy and efficiency of the proposed method is verified by comparing the results with the conventional results.
International Nuclear Information System (INIS)
Electrical modeling of piezoelectric structronic systems by analog circuits has the disadvantages of huge circuit structure and low precision. However, studies of electrical simulation of segmented distributed piezoelectric structronic plate systems (PSPSs) by using output voltage signals of high-speed digital circuits to evaluate the real-time dynamic displacements are scarce in the literature. Therefore, an equivalent dynamic model based on the finite difference method (FDM) is presented to simulate the actual physical model of the segmented distributed PSPS with simply supported boundary conditions. By means of the FDM, the four-ordered dynamic partial differential equations (PDEs) of the main structure/segmented distributed sensor signals/control moments of the segmented distributed actuator of the PSPS are transformed to finite difference equations. A dynamics matrix model based on the Newmark-? integration method is established. The output voltage signal characteristics of the lower modes (m ? 3, n ? 3) with different finite difference mesh dimensions and different integration time steps are analyzed by digital signal processing (DSP) circuit simulation software. The control effects of segmented distributed actuators with different effective areas are consistent with the results of the analytical model in relevant references. Therefore, the method of digital simulation for vibration analysis of segmented distributed PSPSs presented in this paper can provides presented in this paper can provide a reference for further research into the electrical simulation of PSPSs
Modeling and Simulation of Hamburger Cooking Process Using Finite Difference and CFD Methods
Directory of Open Access Journals (Sweden)
J. Sargolzaei
2011-01-01
Full Text Available Unsteady-state heat transfer in hamburger cooking process was modeled using one dimensional finite difference (FD and three dimensional computational fluid dynamic (CFD models. A double-sided cooking system was designed to study the effect of pressure and oven temperature on the cooking process. Three different oven temperatures (114, 152, 204°C and three different pressures (20, 332, 570 pa were selected and 9 experiments were performed. Applying pressure to hamburger increases the contact area of hamburger with heating plate and hence the heat transfer rate to the hamburger was increased and caused the weight loss due to water evaporation and decreasing cooking time, while increasing oven temperature led to increasing weight loss and decreasing cooking time. CFD predicted results were in good agreement with the experimental results than the finite difference (FD ones. But considering the long time needed for CFD model to simulate the cooking process (about 1 hour, using the finite difference model would be more economic.
Coupled finite-difference/finite-element approach for wing-body aeroelasticity
Guruswamy, Guru P.
1992-01-01
Computational methods using finite-difference approaches for fluids and finite-element approaches for structures have individually advanced to solve almost full-aircraft configurations. However, coupled approaches to solve fluid/structural interaction problems are still in their early stages of development, particularly for complex geometries using complete equations such as the Euler/Navier-Stokes equations. Earlier work demonstrated the success of coupling finite-difference and finite-element methods for simple wing configurations using the Euler/Navier-Stokes equations. In this paper, the same approach is extended for general wing-body configurations. The structural properties are represented by beam-type finite elements. The flow is modeled using the Euler/Navier-Stokes equations. A general procedure to fully couple structural finite-element boundary conditions with fluid finite-difference boundary conditions is developed for wing-body configurations. Computations are made using moving grids that adapt to wing-body structural deformations. Results are illustrated for a typical wing-body configuration.
Lansing, Faiza S.; Rascoe, Daniel L.
1993-01-01
This paper presents a modified Finite-Difference Time-Domain (FDTD) technique using a generalized conformed orthogonal grid. The use of the Conformed Orthogonal Grid, Finite Difference Time Domain (GFDTD) enables the designer to match all the circuit dimensions, hence eliminating a major source o error in the analysis.
International Nuclear Information System (INIS)
We present a novel numerical signal injection technique allowing unidirectional injection of a wave in a wave-guiding structure, applicable to 2D finite-difference time-domain electromagnetic codes, both Maxwell and wave-equation. It is particularly suited to continuous wave radar-like simulations. The scheme gives an unidirectional injection of a signal while being transparent to waves propagating in the opposite direction (directional coupling). The reflected or backscattered waves (returned) are separated from the probing waves allowing direct access to the information on amplitude and phase of the returned wave. It also facilitates the signal processing used to extract the phase derivative (or group delay) when simulating radar systems. Although general, the technique is particularly suited to swept frequency sources (frequency modulated) in the context of reflectometry, a fusion plasma diagnostic. The UTS applications presented here are restricted to fusion plasma reflectometry simulations for different physical situations. This method can, nevertheless, also be used in other dispersive media such as dielectrics, being useful, for example, in the simulation of plasma filled waveguides or directional couplers
Directory of Open Access Journals (Sweden)
V. Ramachandra Prasad
2011-01-01
Full Text Available A numerical solution of the unsteady radiative free convection flow of an incompressible viscous fluid past an impulsively started vertical plate with variable heat and mass flux is presented here. This type of problem finds application in many technological and engineering fields such as rocket propulsion systems, spacecraft re-entry aerothermodynamics, cosmical flight aerodynamics, plasma physics, glass production and furnace engineering. The fluid is gray, absorbing-emitting but non-scattering medium and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The governing non-linear, coupled equations are solved using an implicit finite difference scheme. Numerical results for the velocity, temperature, concentration, the local and average skinfriction, the Nusselt and Sherwood number are shown graphically, for different values of Prandtl number, Schmidt number, thermal Grashof number, mass Grashof number, radiation parameter, heat flux exponent and the mass flux exponent. It is observed that, when the radiation parameter increases, the velocity and temperature decrease in the boundary layer. The local and average skin-friction increases with the increase in radiation parameter. For increasing values of radiation parameter the local as well as average Nusselt number increases.
Sallares, V.; Kormann, J.; Cobo, P.; Biescas, B.; Carbonell, R.
2007-05-01
Holbrook et al. (2003) demonstrated recently the possibility of visualizing fine structures in the water column, like thermohaline intrusion or internal waves, through seismic exploration experiments. Seismic exploration is becoming a popular technique for providing high-lateral resolution images of the explored area, in contrast with the classical oceanography probes, like XBT or XCDT. In this work we present a wave propagation model based upon a high order finite-differences time-domain (FDTD) scheme which includes special absorbing conditions in the boundaries. FDTD algorithms are known for presenting problems with reflections on the computational edges. Classical boundary conditions, like those of Engquist, provide reflection coefficients or the order of 10-2. However, reflection coefficients of fine structures in the water we are trying to model are about 10-4. Thus, the key point of the algorithm we present is in the implementation of Perfectly Matched Layer (PML) boundary conditions. These consist in zones with high absorption (therefore, very low reflection coefficient). The PML implemented in this scheme consists in a second order algorithm in the time domain, to take advantage of its stability and convergence properties. In this work we specify the propagation algorithm, and compare it results with the with Engquist and PML absorbing boundaries conditions. The PML condition affords reflection coefficients in the numerical edges lower than 10-4. Holbrook, W.S., Paramo, P., Pearse, S. and Schmitt, R.W., 2003. Thermohaline fine structure in an oceanographic front from seismic reflection profiling. Science, 301, 821-824.
Li, H.; Zhang, Z.; Chen, X.
2013-12-01
Spatial and temporal oversampling in the high velocity medium lower the efficiency of the calculation, especially high velocity contrast exists if uniform grids are used. In this study, we developed an algorithm of high-order Runge-Kutta method with non-uniform time steps on discontinuous grids, based on the collocated-grid finite-difference method (FDM), to increase the efficiency of simulation of seismic wave propagation. In the numerical simulations, we divided the total computational domain into several blocks with variable spatial and temporal steps, according to the velocity model. The local grid spacing and time step in each block were derived by the target frequency and stability condition, as well as the computing ability. A transitional zone was used to communicate between the coarser grid and the finer grid in the spatial and the temporal domain. To couple the spatial discontinuous grids, a Gaussian filtering was adopted to get wavefield of the coarser grids from that of the finer grids, and the linear interpolation was used to value the wavefield of the finer grids from that of the coarser grids. To exchange wavefields between grids with variable time step, we implemented linear coupling procedures for 4th RK time integeration scheme to achieve the same order of time integration accuracy as normal RK scheme. Then we can save the computational resource required and speed up the calculation both in spatial and temporal domain. Because of the collocated-grid FDM, the grid spacing ratio as well as time ratio in our study could be an arbitrary integer, rather than only odd integer ratio in staggered grid. The stability, accuracy and efficiency of our method would be validated by numerical tests.
O'Reilly, O. J.; Dunham, E. M.; Kozdon, J. E.; Nordström, J.
2012-12-01
We present a 2-D multi-block method for earthquake rupture dynamics in complex geometries using summation-by-parts (SBP) high-order finite differences on structured grids coupled to finite volume methods on unstructured meshes. The node-centered finite volume method is used on unstructured triangular meshes to resolve earthquake ruptures propagating along nonplanar faults with complex geometrical features. The unstructured meshes discretize the fault geometry only in the vicinity of the faults and have collocated nodes on the fault boundaries. Away from faults, where no complex geometry is present, the seismic waves emanating from the earthquake rupture are resolved using the high-order finite difference method on coarsened structured grids, improving the computational efficiency while maintaining the accuracy of the method. In order for the SBP high-order finite difference method to communicate with the node-centered finite volume method in a stable manner, interface conditions are imposed using the simultaneous approximation term (SAT) penalty method. In the SAT method the interface conditions and boundary conditions are imposed in a weak manner. Fault interface conditions (rate-and-state friction) are also imposed in a provably stable manner using the SAT method. Another advantage of the SAT method is the ability to impose multiple boundary conditions at a single boundary node, e.g., at the triple junction of a branching fault. The accuracy and stability of the numerical implementation are verified using the method of manufactured solutions and against other numerical implementations. To demonstrate the potential of the method, we simulate an earthquake rupture propagating in a nonplanar fault geometry resolved with unstructured meshes in the fault zone and structured grids in the far-field.
Directory of Open Access Journals (Sweden)
Paulo Alexandre Costa Rocha
2012-12-01
Full Text Available This work presents a numerical model to simulate the propagation of a liquid front in unsaturated soils. The governing flow equations were discretized using centered finite differences for the space coordinate and backward differences for the time coordinate. The generated scheme is fully implicit, but “lagging the non-linearities” as referred to the determination of the soil characteristic properties as function of the hydraulic head. The soil properties, moisture content and the unsaturated hydraulic conductivity, were curve fitted for two types of soil (Alluvial Eutrophic and Red-Yellow Podsol found in the Northeast region of Brazil. The results show that the Alluvial Eutrophic liquid front diffuses faster than the Red- Yellow Podsol front. The use of a parallel algorithm showed that it can be indicated for bigger problems (2-D, where the processing speed gain can reach values between 2-3 times, against simple problems (1-D. Este trabalho apresenta um modelo numérico para simular a propagação de uma frente de onda líquida em solos insaturados. As equações de governo do fluxo de água foram discretizadas usando o método das diferenças finitas centradas para as coordenadas de espaço e diferenças atrasadas para a coordenada de tempo. O esquema gerado é completamente implícito, mas as “influências das não linearidades” foram referidas na determinação das propriedades características do solo como função do recalque hidráulico. As propriedades do solo, teor de água e condutividade hidráulica insaturada foram representadas por curvas de regressão para dois tipos de solos (Aluvial Eutrófico e Podzólico Vermelho Amarelo normalmente encontrados na região nordeste do Brasil. Os resultados mostrarm que a frente de onda do solo Aluvial Eutrófico difunde-se mais rápido do que o solo Podzólico Vermelho Amarelo. O uso do algoritmo paralelo mostrou-se com grandes problemas na simulação 2-D, onde a velocidade de processamento pode alcançar valores entre 2-3 vezes maiores do que problemas simples (1-D.
A finite difference Hartree-Fock program for atoms and diatomic molecules
Kobus, Jacek
2013-03-01
The newest version of the two-dimensional finite difference Hartree-Fock program for atoms and diatomic molecules is presented. This is an updated and extended version of the program published in this journal in 1996. It can be used to obtain reference, Hartree-Fock limit values of total energies and multipole moments for a wide range of diatomic molecules and their ions in order to calibrate existing and develop new basis sets, calculate (hyper)polarizabilities (?zz, ?zzz, ?zzzz, Az,zz, Bzz,zz) of atoms, homonuclear and heteronuclear diatomic molecules and their ions via the finite field method, perform DFT-type calculations using LDA or B88 exchange functionals and LYP or VWN correlations ones or the self-consistent multiplicative constant method, perform one-particle calculations with (smooth) Coulomb and Krammers-Henneberger potentials and take account of finite nucleus models. The program is easy to install and compile (tarball+configure+make) and can be used to perform calculations within double- or quadruple-precision arithmetic. Catalogue identifier: ADEB_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADEB_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License version 2 No. of lines in distributed program, including test data, etc.: 171196 No. of bytes in distributed program, including test data, etc.: 9481802 Distribution format: tar.gz Programming language: Fortran 77, C. Computer: any 32- or 64-bit platform. Operating system: Unix/Linux. RAM: Case dependent, from few MB to many GB Classification: 16.1. Catalogue identifier of previous version: ADEB_v1_0 Journal reference of previous version: Comput. Phys. Comm. 98(1996)346 Does the new version supersede the previous version?: Yes Nature of problem: The program finds virtually exact solutions of the Hartree-Fock and density functional theory type equations for atoms, diatomic molecules and their ions. The lowest energy eigenstates of a given irreducible representation and spin can be obtained. The program can be used to perform one-particle calculations with (smooth) Coulomb and Krammers-Henneberger potentials and also DFT-type calculations using LDA or B88 exchange functionals and LYP or VWN correlations ones or the self-consistent multiplicative constant method. Solution method: Single-particle two-dimensional numerical functions (orbitals) are used to construct an antisymmetric many-electron wave function of the restricted open-shell Hartree-Fock model. The orbitals are obtained by solving the Hartree-Fock equations as coupled two-dimensional second-order (elliptic) partial differential equations (PDEs). The Coulomb and exchange potentials are obtained as solutions of the corresponding Poisson equations. The PDEs are discretized by the eighth-order central difference stencil on a two-dimensional single grid, and the resulting large and sparse system of linear equations is solved by the (multicolour) successive overrelaxation ((MC)SOR) method. The self-consistent-field iterations are interwoven with the (MC)SOR ones and orbital energies and normalization factors are used to monitor the convergence. The accuracy of solutions depends mainly on the grid and the system under consideration, which means that within double precision arithmetic one can obtain orbitals and energies having up to 12 significant figures. If more accurate results are needed, quadruple-precision floating-point arithmetic can be used. Reasons for new version: Additional features, many modifications and corrections, improved convergence rate, overhauled code and documentation. Summary of revisions: see ChangeLog found in tar.gz archive Restrictions: The present version of the program is restricted to 60 orbitals. The maximum grid size is determined at compilation time. Unusual features: The program uses two C routines for allocating and deallocating memory. Several BLAS (Basic Linear Algebra System) routines are emulated by the program. When possible they should be replaced by their libra
A fully nonlinear, mixed spectral and finite difference model for thermally driven, rotating flows
Miller, Timothy L.; Lu, Huei-Iin; Butler, Karen A.
1992-01-01
Finite difference in time and the meridional plane, in conjunction with a spectral technique in the azimuthal direction, are used to approximate the Navier-Stokes equations in a model that can simulate a variety of thermally driven rotating flows in cylindrical and spherical geometries. Axisymmetric flow, linearized waves relative to a fixed or changing axisymmetric flow, nonlinear waves without wave-wave interaction, and fully nonlinear 3D flow, can in this way be calculated. A reexamination is conducted of the steady baroclinic wave case previously treated by Williams (1971) and Quon (1976).
Finite Difference Time-Domain Modelling of Metamaterials: GPU Implementation of Cylindrical Cloak
Directory of Open Access Journals (Sweden)
Attique Dawood
2013-07-01
Full Text Available Finite difference time-domain (FDTD technique can be used to model metamaterials by treating them as dispersive material. Drude or Lorentz model can be incorporated into the standard FDTD algorithm for modelling negative permittivity and permeability. FDTD algorithm is readily parallelisable and can take advantage of GPU acceleration to achieve speed-ups of 5x-50x depending on hardware setup. Metamaterial scattering problems are implemented using dispersive FDTD technique on GPU resulting in performance gain of 10x-15x compared to conventional CPU implementation.
Calculating modes of quantum wire systems using a finite difference technique
Directory of Open Access Journals (Sweden)
T Mardani
2013-03-01
Full Text Available In this paper, the Schrodinger equation for a quantum wire is solved using a finite difference approach. A new aspect in this work is plotting wave function on cross section of rectangular cross-sectional wire in two dimensions, periodically. It is found that the correct eigen energies occur when wave functions have a complete symmetry. If the value of eigen energy has a small increase or decrease in neighborhood of the correct energy the symmetry will be destroyed and aperturbation value at the first of wave function will be observed. In addition, the demand on computer memory varies linearly with the size of the system under investigation.
WONDY V: a one-dimensional finite-difference wave-propagation code
Energy Technology Data Exchange (ETDEWEB)
Kipp, M.E.; Lawrence, R.J.
1982-06-01
WONDY V solves the finite difference analogs to the Lagrangian equations of motion in one spatial dimension (planar, cylindrical, or spherical). Simulations of explosive detonation, energy deposition, plate impact, and dynamic fracture are possible, using a variety of existing material models. In addition, WONDY has proven to be a powerful tool in the evaluation of new constitutive models. A preprocessor is available to allocate storage arrays commensurate with problem size, and automatic rezoning may be employed to improve resolution. This document provides a description of the equations solved, available material models, operating instructions, and sample problems.
A finite difference approach to find exact solution of differential equations
Zuev, Sergei
2015-04-01
This paper contains the background and samples of an approach to construct exact solutions of a wide range of differential equations (DEs). This approach is based on the finite difference equation which corresponds to the given DE. There are three cases considered: linear partial differential equation (PDE) with constant coefficients and at least one non-zero root of characteristic equation, linear PDE with constant coefficients and completely zero roots of the characteristic equation, and a case of nonlinear autonomous dynamical system of second order. Each of these cases is illustrated by an example.
Drinfeld-Sokolov reduction for finite-difference systems and q-W algebras
Semenov-Tian-Shansky, M A
1997-01-01
We propose a finite-difference version of the Drinfeld-Sokolov reduction. A q-deformation analog of the classical W-algebras is obtained. For the sl(n) case it yields the Poisson structure inherited by the zentrum of the affine quantum group $U_q(\\widehat{{\\frak sl}} (n))$ at the critical level. The nontrivial consistency conditions eliminate the residual freedom in the choice of the classical r-matrix which underlies the entire construction and lead to a new class of elliptic classical r-matrices.
Raj Mittra; Jonathan N. Bringuier
2012-01-01
A rigorous full-wave solution, via the Finite-Difference-Time-Domain (FDTD) method, is performed in an attempt to obtain realistic communication channel models for on-body wireless transmission in Body-Area-Networks (BANs), which are local data networks using the human body as a propagation medium. The problem of modeling the coupling between body mounted antennas is often not amenable to attack by hybrid techniques owing to the complex nature of the human body. For instance, the time-domain ...
DEFF Research Database (Denmark)
Tanev, Stoyan; Sun, Wenbo
2012-01-01
This chapter reviews the fundamental methods and some of the applications of the three-dimensional (3D) finite-difference time-domain (FDTD) technique for the modeling of light scattering by arbitrarily shaped dielectric particles and surfaces. The emphasis is on the details of the FDTD algorithms for particle and surface scattering calculations and the uniaxial perfectly matched layer (UPML) absorbing boundary conditions for truncation of the FDTD grid. We show that the FDTD approach has a significant potential for studying the light scattering by cloud, dust, and biological particles. The applications of the FDTD approach for beam scattering by arbitrarily shaped surfaces are also discussed.
Ward, D W; Cantarella, Jason; Fu, Joseph H.G.; Kusner, Rob; Sullivan, John M.; Wrinkle, Nancy C.; Cantarella, Jason; Fu, Joseph H.G.; Kusner, Rob; Sullivan, John M.; Wrinkle, Nancy C.; Cantarella, Jason; Fu, Joseph H.G.; Kusner, Rob; Sullivan, John M.; Wrinkle, Nancy C.; Cantarella, Jason; Fu, Joseph H.G.; Kusner, Rob; Sullivan, John M.; Wrinkle, Nancy C.; Gelaki, Shlomo; Ward, David W.; Nelson, Keith A.
2004-01-01
We describe a simple and intuitive implementation of the method of finite difference time domain simulations for propagating electromagnetic waves using the simplest possible tools available in Microsoft Excel. The method overcomes the usual obstacles of familiarity with programming languages as it relies on little more than the cut and paste features that are standard in Excel. Avenues of exploration by students are proposed and sample graphs are included. The pedagogical effectiveness of the implementation was tested during an Independent Activities Period class, composed of 80% freshmen, at MIT, and yielded positive results.
Simulation model of a gel shallow solar pond using finite difference method
International Nuclear Information System (INIS)
A solar pond consists of carboxy methyl cellulose as a gel layer and it floats on the convective zone. A simulation model for the gel shallow solar pond is developed by using energy balance equation. The energy balance equation for the convection zone (salt water) is written in the form of partial differential equation. The parameters involved in the energy balance equation are heat loss, heat capacity and solar energy gain though the surface insolation and is solved by finite difference method. The thermal performance of the GSSP model is analyzed by making use of monthly mean hourly radiation for Coimbatore, India (11 degree N latitude). (Author)
Curvilinear vector finite difference approach to the computation of waveguide modes
Directory of Open Access Journals (Sweden)
Alessandro Fanti
2012-05-01
Full Text Available We describe here a Vector Finite Difference approach to the evaluation of waveguide eigenvalues and modes for rectangular, circular and elliptical waveguides. The FD is applied using a 2D cartesian, polar and elliptical grid in the waveguide section. A suitable Taylor expansion of the vector mode function allows to take exactly into account the boundary condition. To prevent the raising of spurious modes, our FD approximation results in a constrained eigenvalue problem, that we solve using a decomposition method. This approach has been evaluated comparing our results to the analytical modes of rectangular and circula rwaveguide, and to known data for the elliptic case.
A computational study of vortex-airfoil interaction using high-order finite difference methods
Svärd, Magnus; Lundberg, Johan; Nordström, Jan
2010-01-01
Simulations of the interaction between a vortex and a NACA0012 airfoil are performed with a stable, high-order accurate (in space and time), multi-block finite difference solver for the compressible Navier–Stokes equations. We begin by computing a benchmark test case to validate the code. Next, the flow with steady inflow conditions are computed on several different grids. The resolution of the boundary layer as well as the amount of the artificial dissipation is studied to establish the ne...
COVE-1: a finite difference creep collapse code for oval fuel pin cladding material
International Nuclear Information System (INIS)
COVE-1 is a time-dependent incremental creep collapse code that estimates the change in ovality of a fuel pin cladding tube. It uses a finite difference method of solving the differential equations which describe the deflection of the tube walls as a function of time. The physical problem is nonlinear, both with respect to geometry and material properties, which requires the use of an incremental, analytical, path-dependent solution. The application of this code is intended primarily for tubes manufactured from Zircaloy. Therefore, provision has been made to include some of the effects of anisotropy in the flow equations for inelastic incremental deformations. 10 references. (U.S.)
International Nuclear Information System (INIS)
An improved solution scheme is developed for the three-dimensional radiative transfer equation (RTE) in inhomogeneous cloudy atmospheres. This solution scheme is deterministic (explicit) and utilizes spherical harmonics series expansion and the finite-volume method for discretization of the RTE. The first-order upwind finite difference is modified to take into account bidirectional flow of radiance in spherical harmonics space, and an iterative solution method is applied. The multigrid method, which is generally employed to achieve rapid convergence in iterative calculation, is incorporated into the solution scheme. The present study suggests that the restriction and prolongation procedure for the multigrid method must be also modified to account for bidirectional flow, and proposes an efficient bidirectional restriction/prolongation procedure that does not increase the computational effort for coarser grids, resulting in a type of wavelet low-pass filter. Several calculation examples for various atmosphere models indicate that the proposed solution scheme is effective for rapid convergence and suitable for obtaining adequate radiation fields in inhomogeneous cloudy atmospheres, although a comparison with the Monte Carlo method suggests that the radiances obtained by this solution scheme at certain angles tends to be smoother. -- Highlights: • We develop a deterministic solution scheme for the 3-D radiative transfer. • The multigrid method is incorporated into an iterative solution scheme. • The multigrid method needs to be modified for the incorporation. • An ingenious procedure for the restriction and prolongation is proposed. • The scheme results in rapid convergence and obtains adequate radiation fields
Okuyama, Yoshifumi
2014-01-01
Discrete Control Systems establishes a basis for the analysis and design of discretized/quantized control systemsfor continuous physical systems. Beginning with the necessary mathematical foundations and system-model descriptions, the text moves on to derive a robust stability condition. To keep a practical perspective on the uncertain physical systems considered, most of the methods treated are carried out in the frequency domain. As part of the design procedure, modified Nyquist–Hall and Nichols diagrams are presented and discretized proportional–integral–derivative control schemes are reconsidered. Schemes for model-reference feedback and discrete-type observers are proposed. Although single-loop feedback systems form the core of the text, some consideration is given to multiple loops and nonlinearities. The robust control performance and stability of interval systems (with multiple uncertainties) are outlined. Finally, the monograph describes the relationship between feedback-control and discrete ev...
Ibral, Asmaa; Zouitine, Asmaa; Assaid, El Mahdi; El Achouby, Hicham; Feddi, El Mustapha; Dujardin, Francis
2015-02-01
Poisson equation is solved analytically in the case of a point charge placed anywhere in a spherical core/shell nanostructure, immersed in aqueous or organic solution or embedded in semiconducting or insulating matrix. Conduction and valence band-edge alignments between core and shell are described by finite height barriers. Influence of polarization charges induced at the surfaces where two adjacent materials meet is taken into account. Original expressions of electrostatic potential created everywhere in the space by a source point charge are derived. Expressions of self-polarization potential describing the interaction of a point charge with its own image-charge are deduced. Contributions of double dielectric constant mismatch to electron and hole ground state energies as well as nanostructure effective gap are calculated via first order perturbation theory and also by finite difference approach. Dependencies of electron, hole and gap energies against core to shell radii ratio are determined in the case of ZnS/CdSe core/shell nanostructure immersed in water or in toluene. It appears that finite difference approach is more efficient than first order perturbation method and that the effect of polarization charge may in no case be neglected as its contribution can reach a significant proportion of the value of nanostructure gap.
International Nuclear Information System (INIS)
A least squares principle is described which uses a penalty function treatment of boundary and interface conditions. Appropriate choices of the trial functions and vectors employed in a dual representation of an approximate solution established complementary principles for the diffusion equation. A geometrical interpretation of the principles provides weighted residual methods for diffusion theory, thus establishing a unification of least squares, variational and weighted residual methods. The complementary principles are used with either a trial function for the flux or a trial vector for the current to establish for regular meshes a connection between finite element, finite difference and nodal methods, which can be exact if the mesh pitches are chosen appropriately. Whereas the coefficients in the usual nodal equations have to be determined iteratively, those derived via the complementary principles are given explicitly in terms of the data. For the further development of the connection between finite element, finite difference and nodal methods, some hybrid variational methods are described which employ both a trial function and a trial vector. (author)
MasQU: Finite Differences on Masked Irregular Stokes Q,U Grids
Bowyer, Jude; Novikov, Dmitri I
2011-01-01
The future detection of B-mode polarization in the CMB is one of the most important outstanding tests of inflationary cosmology. One of the necessary steps for extracting polarization information in the CMB is reducing contamination from so-called 'ambiguous modes' on a masked sky. This can be achieved by utilising derivative operators on the real-space Stokes Q and U parameters. The main result of this paper is the presentation of an algorithm and a software package to perform this procedure on the full sky, i.e. with projects such as the Planck Surveyor and future satellites in mind; in particular, the package can perform finite differences on masked, irregular grids and is applied to a semi-regular spherical pixelisation, the HEALPix grid. The formalism reduces to the known finite difference solutions in the case of a regular grid. We quantify full-sky improvements on the possible bounds of the CMB B-mode signal. We find that in the specific case of E and B-mode separation, there exists a 'pole problem' in...
Baek, Hyoungsu; Karniadakis, George Em
2011-06-01
We present an iterative semi-implicit scheme for the incompressible Navier-Stokes equations, which is stable at CFL numbers well above the nominal limit. We have implemented this scheme in conjunction with spectral discretizations, which suffer from serious time step limitations at very high resolution. However, the approach we present is general and can be adopted with finite element and finite difference discretizations as well. Specifically, at each time level, the nonlinear convective term and the pressure boundary condition - both of which are treated explicitly in time - are updated using fixed-point iteration and Aitken relaxation. Eigenvalue analysis shows that this scheme is unconditionally stable for Stokes flows while numerical results suggest that the same is true for steady Navier-Stokes flows as well. This finding is also supported by error analysis that leads to the proper value of the relaxation parameter as a function of the flow parameters. In unsteady flows, second- and third-order temporal accuracy is obtained for the velocity field at CFL number 5-14 using analytical solutions. Systematic accuracy, stability, and cost comparisons are presented against the standard semi-implicit method and a recently proposed fully-implicit scheme that does not require Newton's iterations. In addition to its enhanced accuracy and stability, the proposed method requires the solution of symmetric only linear systems for which very effective preconditioners exist unlike the fully-implicit schemes.
Simple Numerical Schemes for the Korteweg-deVries Equation
International Nuclear Information System (INIS)
Two numerical schemes, which simulate the propagation of dispersive non-linear waves, are described. The first is a split-step Fourier scheme for the Korteweg-de Vries (KdV) equation. The second is a finite-difference scheme for the modified KdV equation. The stability and accuracy of both schemes are discussed. These simple schemes can be used to study a wide variety of physical processes that involve dispersive nonlinear waves
Luo, Y.; Xia, J.; Xu, Y.; Zeng, C.; Liu, J.
2010-01-01
Love-wave propagation has been a topic of interest to crustal, earthquake, and engineering seismologists for many years because it is independent of Poisson's ratio and more sensitive to shear (S)-wave velocity changes and layer thickness changes than are Rayleigh waves. It is well known that Love-wave generation requires the existence of a low S-wave velocity layer in a multilayered earth model. In order to study numerically the propagation of Love waves in a layered earth model and dispersion characteristics for near-surface applications, we simulate high-frequency (>5 Hz) Love waves by the staggered-grid finite-difference (FD) method. The air-earth boundary (the shear stress above the free surface) is treated using the stress-imaging technique. We use a two-layer model to demonstrate the accuracy of the staggered-grid modeling scheme. We also simulate four-layer models including a low-velocity layer (LVL) or a high-velocity layer (HVL) to analyze dispersive energy characteristics for near-surface applications. Results demonstrate that: (1) the staggered-grid FD code and stress-imaging technique are suitable for treating the free-surface boundary conditions for Love-wave modeling, (2) Love-wave inversion should be treated with extra care when a LVL exists because of a lack of LVL information in dispersions aggravating uncertainties in the inversion procedure, and (3) energy of high modes in a low-frequency range is very weak, so that it is difficult to estimate the cutoff frequency accurately, and "mode-crossing" occurs between the second higher and third higher modes when a HVL exists. ?? 2010 Birkh??user / Springer Basel AG.
Luo, Yinhe; Xia, Jianghai; Xu, Yixian; Zeng, Chong; Liu, Jiangping
2010-12-01
Love-wave propagation has been a topic of interest to crustal, earthquake, and engineering seismologists for many years because it is independent of Poisson's ratio and more sensitive to shear (S)-wave velocity changes and layer thickness changes than are Rayleigh waves. It is well known that Love-wave generation requires the existence of a low S-wave velocity layer in a multilayered earth model. In order to study numerically the propagation of Love waves in a layered earth model and dispersion characteristics for near-surface applications, we simulate high-frequency (>5 Hz) Love waves by the staggered-grid finite-difference (FD) method. The air-earth boundary (the shear stress above the free surface) is treated using the stress-imaging technique. We use a two-layer model to demonstrate the accuracy of the staggered-grid modeling scheme. We also simulate four-layer models including a low-velocity layer (LVL) or a high-velocity layer (HVL) to analyze dispersive energy characteristics for near-surface applications. Results demonstrate that: (1) the staggered-grid FD code and stress-imaging technique are suitable for treating the free-surface boundary conditions for Love-wave modeling, (2) Love-wave inversion should be treated with extra care when a LVL exists because of a lack of LVL information in dispersions aggravating uncertainties in the inversion procedure, and (3) energy of high modes in a low-frequency range is very weak, so that it is difficult to estimate the cutoff frequency accurately, and "mode-crossing" occurs between the second higher and third higher modes when a HVL exists.
An Efficient Finite Difference Method for Parameter Sensitivities of Continuous Time Markov Chains
Anderson, David F
2011-01-01
We present an efficient finite difference method for the computation of parameter sensitivities for a wide class of continuous time Markov chains. The motivating class of models, and the source of our examples, are the stochastic chemical kinetic models commonly used in the biosciences, though other natural application areas include population processes and queuing networks. The method is essentially derived by making effective use of the random time change representation of Kurtz, and is no harder to implement than any standard continuous time Markov chain algorithm, such as "Gillespie's algorithm" or the next reaction method. Further, the method is analytically tractable, and, for a given number of realizations of the stochastic process, produces an estimator with substantially lower variance than that obtained using other common methods. Therefore, the computational complexity required to solve a given problem is lowered greatly. In this work, we present the method together with the theoretical analysis de...
A coupled boundary element-finite difference solution of the elliptic modified mild slope equation
DEFF Research Database (Denmark)
Naserizadeh, R.; Bingham, Harry B.
2011-01-01
The modified mild slope equation of [5] is solved using a combination of the boundary element method (BEM) and the finite difference method (FDM). The exterior domain of constant depth and infinite horizontal extent is solved by a BEM using linear or quadratic elements. The interior domain with variable depth is solved by a flexible order of accuracy FDM in boundary-fitted curvilinear coordinates. The two solutions are matched along the common boundary of two methods (the BEM boundary) to ensure continuity of value and normal flux. Convergence of the individual methods is shown and the combined solution is tested against several test cases. Results for refraction and diffraction of waves from submerged bottom mounted obstacles compare well with experimental measurements and other computed results from the literature.
Finite element-finite difference thermal/structural analysis of large space truss structures
Warren, Andrew H.; Arelt, Joseph E.; Eskew, William F.; Rogers, Karen M.
1992-01-01
A technique of automated and efficient thermal-structural processing of truss structures that interfaces the finite element and finite difference method was developed. The thermal-structural analysis tasks include development of the thermal and structural math models, thermal analysis, development of an interface and data transfer between the models, and finally an evaluation of the thermal stresses and displacements in the structure. Consequently, the objective of the developed technique was to minimize the model development time, in order to assure an automatic transfer of data between the thermal and structural models as well as to minimize the computer resources needed for the analysis itself. The method and techniques described are illustrated on the thermal/structural analysis of the Space Station Freedom main truss.
Numerical finite-difference study of the oscillatory behavior of vertically vented compartments
International Nuclear Information System (INIS)
A numerical finite-difference study has been carried out in two and three dimensions for turbulent buoyant flow in a square compartment with floor and ceiling vents. A volumetric heat source is located either at the center of the floor or at the lower left corner of the compartment. Oscillatory behaviors have been found in the flow and temperature fields for the cases with center heat source, while for cases with heat source at the lower left corner, oscillations only occur during the initial transient period and are thereafter quickly damped out. The physical origin of these oscillations is noted. Results for one of the cases are also shown in terms of the effect of heat source strength on the frequency of oscillations. 6 references
GPU-acceleration of parallel unconditionally stable group explicit finite difference method
Parand, K; Hossayni, Sayyed A
2013-01-01
Graphics Processing Units (GPUs) are high performance co-processors originally intended to improve the use and quality of computer graphics applications. Since researchers and practitioners realized the potential of using GPU for general purpose, their application has been extended to other fields out of computer graphics scope. The main objective of this paper is to evaluate the impact of using GPU in solution of the transient diffusion type equation by parallel and stable group explicit finite difference method. To accomplish that, GPU and CPU-based (multi-core) approaches were developed. Moreover, we proposed an optimal synchronization arrangement for its implementation pseudo-code. Also, the interrelation of GPU parallel programming and initializing the algorithm variables was discussed, using numerical experiences. The GPU-approach results are faster than a much expensive parallel 8-thread CPU-based approach results. The GPU, used in this paper, is an ordinary laptop GPU (GT 335M) and is accessible for e...
A finite difference solution of the polar electrojet current mapping boundary value problem
Energy Technology Data Exchange (ETDEWEB)
Werner, D.H.; Ferraro, A.J. (Pennsylvania State Univ., University Park (USA))
1991-02-01
An investigation is made of how the polar electrojet currents and associated electric fields map down to the ground. The boundary value problem which characterizes the downward mapping of the electrojet current will be formulated using potential theory. The electrojet current is represented by a simple Cowling model in which the geomagnetic field is vertical. A numerical solution to the electrojet mapping boundary value problem is obtained via a finite difference technique. This model is employed to study the downward mapping of the polar electrojet current during intense magnetic storms occuring under sunspot maximum daytime conditions. Results of this analysis suggest that as the current maps down through the D region, from an electrojet source in the E region, it is being attenuated as well as rotated. The rotation, however, is not present at altitudes below the D region. A possible application of electrojet mapping theory to the interpretation of high-latitude ionospheric modification data taken during polar electrojet events is discussed.
Finite-difference modeling of the cryostability of helium II cooled conductor packs
International Nuclear Information System (INIS)
In this chapter, constant heat step inputs are simulated so that the model heat transfer results can be compared with experimental results to understand the accuracy of the analytical simulation for future work. The Convair thermal analyzer finite-difference computer program is used to simulate a typical helium II conductor pack geometry. A transient disturbance in a coil is examined using numeric methods. The conductors, insulation, and helium II are modeled as mass nodes with temperature-dependent densities, heat capacities, and thermal conductivities. The three major heat transfer regimes (Kapitza conductance, constant heat flux, and film boiling) are modeled by defining an effective heat transfer coefficient that simulates the appropriate heat transfer mechanism between the conductor surface and adjacent helium
CASKETSS-HEAT: a finite difference computer program for nonlinear heat conduction problems
International Nuclear Information System (INIS)
A heat conduction program CASKETSS-HEAT has been developed. CASKETSS-HEAT is a finite difference computer program used for the solution of multi-dimensional nonlinear heat conduction problems. Main features of CASKETSS-HEAT are as follows. (1) One, two and three-dimensional geometries for heat conduction calculation are available. (2) Convection and radiation heat transfer of boundry can be specified. (3) Phase change and chemical change can be treated. (4) Finned surface heat transfer can be treated easily. (5) Data memory allocation in the program is variable according to problem size. (6) The program is a compatible heat transfer analysis program to the stress analysis program SAP4 and SAP5. (7) Pre- and post-processing for input data generation and graphic representation of calculation results are available. In the paper, brief illustration of calculation method, input data and sample calculation are presented. (author)
Accelerating a three-dimensional finite-difference wave propagation code using GPU graphics cards
Michéa, David; Komatitsch, Dimitri
2010-07-01
We accelerate a 3-D finite-difference in the time domain wave propagation code by a factor between about 20 and 60 compared to a serial implementation using graphics processing unit computing on NVIDIA graphics cards with the CUDA programming language. We describe the implementation of the code in CUDA to simulate the propagation of seismic waves in a heterogeneous elastic medium. We also implement convolution perfectly matched layers on the graphics cards to efficiently absorb outgoing waves on the fictitious edges of the grid. We show that the code that runs on a graphics card gives the expected results by comparing our results to those obtained by running the same simulation on a classical processor core. The methodology that we present can be used for Maxwell's equations as well because their form is similar to that of the seismic wave equation written in velocity vector and stress tensor.
Study Notes on Numerical Solutions of the Wave Equation with the Finite Difference Method
Adib, A B
2000-01-01
In this introductory work I will present the Finite Difference method for hyperbolic equations, focusing on a method which has second order precision both in time and space (the so-called leap-frog method) and applying it to the case of the 1d and 2d wave equation. A brief derivation of the energy and equation of motion of a wave is done before the numerical part in order to make the transition from the continuum to the lattice clearer. To illustrate the extension of the method to more complex equations, I also add dissipative terms of the kind $-\\eta \\dot{u}$ into the equations. The von Neumann numerical stability analysis and the Courant criterion, two of the most popular in the literature, are briefly discussed. In the end I present some numerical results obtained with the leap-frog algorithm, illustrating the importance of the lattice resolution through energy plots.
Zhu, Lu; Liu, Yuanyuan; Chen, Suhua; Hu, Fei; Chen, Zhizhang (David)
2015-04-01
Synthetic aperture imaging radiometer (SAIR) has the potential to meet the spatial resolution requirement of passive millimeter remote sensing from space. A new two-dimensional (2-D) imaging radiometer at millimeter wave (MMW) band is described in this paper; it uses a one-dimensional (1-D) synthetic aperture digital radiometer (SADR) to obtain an image on one dimension and a rotary platform to provide a scan on the second dimension. Due to the ill-posed inverse problem of SADR, we proposed a new reconstruction algorithm based on Finite Difference (FD) regularization to improve brightness temperature images. Experimental results show that the proposed 2-D MMW radiometer can give the brightness temperature images of natural scenes and the FD regularization reconstruction algorithm is able to improve the quality of brightness temperature images.
Modeling of Nanophotonic Resonators with the Finite-Difference Frequency-Domain Method
DEFF Research Database (Denmark)
Ivinskaya, Aliaksandra; Lavrinenko, Andrei
2011-01-01
Finite-difference frequency-domain method with perfectly matched layers and free-space squeezing is applied to model open photonic resonators of arbitrary morphology in three dimensions. Treating each spatial dimension independently, nonuniform mesh of continuously varying density can be built easily to better resolve mode features. We explore the convergence of the eigenmode wavelength $lambda $ and quality factor $Q$ of an open dielectric sphere and of a very-high- $Q$ photonic crystal cavity calculated with different mesh density distributions. On a grid having, for example, 10 nodes per lattice constant in the region of high field intensity, we are able to find the eigenwavelength $lambda $ with a half-percent precision and the $Q$-factor with an order-of-magnitude accuracy. We also suggest the $lambda /n$ rule (where $n$ is the cavity refractive index) for the optimal cavity-to-PML distance.
Parallel processing algorithms for the finite difference solution to the Navier-Stokes equations
International Nuclear Information System (INIS)
Numerical solutions involving finite difference representations of the equations governing fluid flow, heat conduction, and diffusion processes (including neutron diffusion) usually consist of solving large sparse matrix equations. These matrix equations can be recast into M smaller coupled matrix equations amenable to solution by using M multiple computer processors operating in parallel. A special form of the fluids equations commonly used in nuclear reactor thermal-hydraulic analysis, i.e., one-dimensional flow in closed loop geometry is emphasized. Parallel algorithms for solving these equations are developed and evaluated in terms of computational speed against conventional solutions on a serial machine. Timing studies are performed to assess the efficiency of these methods and to determine the optimum number of parallel processors for these applications
Transient analysis of printed lines using finite-difference time-domain method
Energy Technology Data Exchange (ETDEWEB)
Ahmed, Shahid [JLAB
2013-01-01
Comprehensive studies of ultra-wideband pulses and electromagnetic coupling on printed coupled lines have been performed using full-wave 3D finite-difference time-domain analysis. Effects of unequal phase velocities of coupled modes, coupling between line traces, and the frequency dispersion on the waveform fidelity and crosstalk have been investigated in detail. To discriminate the contributions of different mechanisms into pulse evolution, single and coupled microstrip lines without (?r?=?1) and with (?r?>?1) dielectric substrates have been examined. To consistently compare the performance of the coupled lines with substrates of different permittivities and transients of different characteristic times, a generic metric similar to the electrical wavelength has been introduced. The features of pulse propagation on coupled lines with layered and pedestal substrates and on the irregular traces have been explored. Physical interpretations of the simulation results are discussed in the paper.
An Analysis on 3d Marine Csem Responses Based on a Finite Difference Method
Han, N.; Nam, M.; Kim, H.
2010-12-01
Three-dimensional (3D) marine controlled-source electromagnetic (CSEM) data are analyzed using a modeling algorithm based on a finite difference method. The algorithm employs the secondary-field formulation of a vector Helmholtz equation for electric fields to avoid singularity problems. Primary fields are calculated analytically using a numerical filter for the Hankel transform for a three-layered 1D background model, that consists of air, sea and sub-seafloor; the model includes the air layer to consider air waves. Several numerical filters for the Hankel transform are compared in terms of their accuracy and computation time. Using the analytically-calculated primary fields, we compute secondary fields using a finite difference method with a staggered grid. The grid defines electric fields along cell edges while magnetic fields at cell faces. We verified the developed modeling algorithm using not only 1D analytic solutions but also responses for a 3D model, that are computed by other algorithms. Using disk models, this study analyzes marine CSEM data for horizontal and vertical electric and magnetic dipole sources to determine the most effective source-receiver configuration for the exploration of 3D thin and resistive hydrocarbon targets. Numerical results show that marine CSEM has exciting potential for oilfield characterization. Further, air waves should be properly considered in modeling and interpretation of marine CSEM data because they have great effects on marine CSEM data. For an analysis on bathymetry effects, a stepwise-bathymetry model was constructed. Bathymetry causes significant effects on marine CSEM data because transmitter and receivers are located very far each other. We propose a bathymetry correction method for a proper interpretation of marine CSEM data distorted by bathymetry.
GPU-accelerated 3D neutron diffusion code based on finite difference method
International Nuclear Information System (INIS)
Finite difference method, as a traditional numerical solution to neutron diffusion equation, although considered simpler and more precise than the coarse mesh nodal methods, has a bottle neck to be widely applied caused by the huge memory and unendurable computation time it requires. In recent years, the concept of General-Purpose computation on GPUs has provided us with a powerful computational engine for scientific research. In this study, a GPU-Accelerated multi-group 3D neutron diffusion code based on finite difference method was developed. First, a clean-sheet neutron diffusion code (3DFD-CPU) was written in C++ on the CPU architecture, and later ported to GPUs under NVIDIA's CUDA platform (3DFD-GPU). The IAEA 3D PWR benchmark problem was calculated in the numerical test, where three different codes, including the original CPU-based sequential code, the HYPRE (High Performance Pre-conditioners)-based diffusion code and CITATION, were used as counterpoints to test the efficiency and accuracy of the GPU-based program. The results demonstrate both high efficiency and adequate accuracy of the GPU implementation for neutron diffusion equation. A speedup factor of about 46 times was obtained, using NVIDIA's Geforce GTX470 GPU card against a 2.50 GHz Intel Quad Q9300 CPU processor. Compared with the HYPRE-based code performing in parallel on an 8-core tower server, the speedup of about 2 still could be observed. More encouragingly, without any mathematical acceleration ly, without any mathematical acceleration technology, the GPU implementation ran about 5 times faster than CITATION which was speeded up by using the SOR method and Chebyshev extrapolation technique. (authors)
International Nuclear Information System (INIS)
A code called DIFXYZ has been developed for the finite difference solution of diffusion theory equations in 1-D, 2-D and 3-D Cartesian geometry. It uses central differencing scheme. DIFXYZ can be used for determining the eigenvalue, fluxes, their adjoint and power distribution. Homogeneous as well as inhomogeneous boundary conditions can be applied in all directions. The code is written in variable dimensions so that any combination of number of energy groups and number of spatial meshes can be considered. Diagonal or octant symmetry can be used in radial plane. Acceleration techniques like successive line over relaxation with optimum parameter determined by program, two parameter Chebyschev acceleration for fission source, row or column rebalancing etc. are incorporated in DIFXYZ and facilitate fast convergence of eigenvalue and flux profile. (author)
Scientific Electronic Library Online (English)
Jorge Mauricio, Ruiz Vera; Ignacio, Mantilla Prada.
2013-06-01
Full Text Available La ecuación de Derrida-Lebowitz-Speer-Spohn (DLSS) es una ecuación de evolución no lineal de cuarto orden. Esta aparece en el estudio de las fluctuaciones de interface de sistemas de espín y en la modelación de semicoductores cuánticos. En este artículo, se presenta una discretización por elementos [...] finitos para una formulación exponencial de la ecuación DLSS abordada como un sistema acoplado de ecuaciones. Usando la información disponible acerca del fenómeno físico, se establecen las condiciones de contorno para el sistema acoplado. Se demuestra la existencia de la solución discreta global en el tiempo via un argumento de punto fijo. Los resultados numéricos ilustran el carácter cuántico de la ecuación. Finalmente se presenta un test del orden de convergencia de la discretización porpuesta. Abstract in english The Derrida-Lebowitz-Speer-Spohn (DLSS) equation is a fourth order in space non-linear evolution equation. This equation arises in the study of interface fluctuations in spin systems and quantum semiconductor modelling. In this paper, we present a finite element discretization for a exponential form [...] ulation of a coupled-equation approach to the DLSS equation. Using the available information about the physical phenomena, we are able to set the corresponding boundary conditions for the coupled system. We prove existence of the discrete solution by fixed point argument. Numerical results illustrate the quantum character of the equation. Finally a test of order of convergence of the proposed discretization scheme is presented.
Numerical solution of the 1D kinetics equations using a cubic reduced nodal scheme
International Nuclear Information System (INIS)
In this work a finite differences technique centered in mesh based on a cubic reduced nodal scheme type finite element to solve the equations of the kinetics 1 D that include the equations corresponding to the concentrations of precursors of delayed neutrons is described. The technique of finite elements used is that of Galerkin where so much the neutron flux as the concentrations of precursors its are spatially approached by means of a three grade polynomial. The matrices of rigidity and of mass that arise during this discretization process are numerically evaluated using the open quadrature non standard of Newton-Cotes and that of Radau respectively. The purpose of the application of these quadratures is the one of to eliminate in the global matrices the couplings among the values of the flow in points of the discretization with the consequent advantages as for the reduction of the order of the matrix associated to the discreet problem that is to solve. As for the time dependent part the classical integration scheme known as ? scheme is applied. After carrying out the one reordering of unknown and equations it arrives to a reduced system that it can be solved but quickly. With the McKin compute program developed its were solved three benchmark problems and those results are shown for the relative powers. (Author)
Steinhoff, John; Chitta, Subhashini
2012-08-01
The linear wave equation represents the basis of many linear electromagnetic and acoustic propagation problems. Features that a computational model must have, to capture large scale realistic effects (for over the horizon or "OTH" radar communication, for example), include propagation of short waves with scattering and partial absorption by complex topography. For these reasons, it is not feasible to use Green's Function or any simple integral method, which neglects these intermediate effects and requires a known propagation function between source and observer. In this paper, we describe a new method for propagating such short waves over long distances, including intersecting scattered waves. The new method appears to be much simpler than conventional high frequency schemes: Lagrangian "particle" based approaches, such as "ray tracing" become very complex in 3-D, especially for waves that may be expanding, or even intersecting. The other high frequency scheme in common use, the Eikonal, also has difficulty with intersecting waves. Our approach, based on nonlinear solitary waves concentrated about centroid surfaces of physical wave features, is related to that of Whitham [1], which involves solving wave fronts propagating on characteristics. Then, the evolving electromagnetic (or acoustic) field can be approximated as a collection of propagating co-dimension one surfaces (for example, 2-D surfaces in three dimensions). This approach involves solving propagation equations discretely on an Eulerian grid to approximate the linear wave equation. However, to propagate short waves over long distances, conventional Eulerian numerical methods, which attempt to resolve the structure of each wave, require far too many grid cells and are not feasible on current or foreseeable computers. Instead, we employ an "extended" wave equation that captures the important features of the propagating waves. This method is first formulated at the partial differential equation (PDE) level, as a wave equation with an added "confining" term that involves both a positive and a negative dissipation. Once we have the stable PDE, the discrete formulation is simply a multidimensional PDE with (stable) perturbations caused by the discretization. The resulting discrete solution can then be low order and very simple and yet remain stable over arbitrarily long times. When discretized and solved on an Eulerian grid, this new method allows far coarser grids than required by conventional resolution considerations, while still accounting for the effects of varying atmospheric and topographic features. An important point is that the new method is in the same form as conventional discrete wave equation methods. However, the conventional solution eventually decays, and only the "intermediate asymptotic" solution can be used. Simply by adding an extra term, we show that a nontrivial true asymptotic solution can be obtained. A similar solitary wave based approach has been used successfully in a different problem (involving "Vorticity Confinement"), for a number of years.
Ghosh, Karabi
2008-01-01
A fully implicit finite difference scheme has been developed to solve the hydrodynamic equations coupled with radiation transport. Solution of the time dependent radiation transport equation is obtained using the discrete ordinates method and the energy flow into the Lagrangian meshes as a result of radiation interaction is fully accounted for. A tridiagonal matrix system is solved at each time step to determine the hydrodynamic variables implicitly. The results obtained from this fully implicit radiation hydrodynamics code in the planar geometry agrees well with the scaling law for radiation driven strong shock propagation in aluminium. For the point explosion problem the self similar solutions are compared with results for pure hydrodynamic case in spherical geometry and the effect of radiation energy transfer is determined. Having, thus, benchmarked the code, convergence of the method w.r.t. time step is studied in detail and compared with the results of commonly used semi-implicit method. It is shown that...
Solving the time-dependent Schrödinger equation using finite difference methods
Scientific Electronic Library Online (English)
R, Becerril; F.S, Guzmán; A, Rendón-Romero; S, Valdez-Alvarado.
2008-12-01
Full Text Available Resolvemos la ecuación de Schrödinger dependiente del tiempo en una y dos dimensiones usando diferencias finitas. La evolución se lleva a cabo usando el método de líneas. Los casos ilustrativos incluyen: la partícula en una caja y en un potencial de oscilador armónico en una y dos dimensiones. Como [...] ejemplos poco comunes presentamos la evolución de dos solitones y mostramos la dependencia temporal del comportamiento solitónico en una dimensión y la estabilización de un modelo de gas atómico en dos dimensiones. Los códigos usados para generar los resultados en este manuscrito se encuentran disponibles a la menor petición, y esperamos que este hecho ayude a los estudiantes a adquirir un mejor entendimiento de la solución de ecuaciones diferenciales parciales relacionadas con sistemas dinámicos Abstract in english We solve the time-dependent Schrödinger equation in one and two dimensions using the finite difference approximation. The evolution is carried out using the method of lines. The illustrative cases include: the particle in a box and the harmonic oscillator in one and two dimensions. As non-standard e [...] xamples we evolve two solitons and show the time-dependent solitonic behavior in one dimension and the stabilization of an atomic gas model in two dimensions. The codes used to generate the results in this manuscript are freely available under request, and we expect this material could help students to have a better grasp of the solution of partial differential equations related to dynamical systems
Unsteady streamflow simulation using a linear implicit finite-difference model
Land, Larry F.
1978-01-01
A computer program for simulating one-dimensional subcritical, gradually varied, unsteady flow in a stream has been developed and documented. Given upstream and downstream boundary conditions and channel geometry data, roughness coefficients, stage, and discharge can be calculated anywhere within the reach as a function of time. The program uses a linear implicit finite-difference technique that discritizes the partial differential equations. Then it arranges the coefficients of the continuity and momentum equations into a pentadiagonal matrix for solution. Because it is a reasonable compromise between computational accuracy, speed and ease of use,the technique is one of the most commonly used. The upstream boundary condition is a depth hydrograph. However, options also allow the boundary condition to be discharge or water-surface elevation. The downstream boundary condition is a depth which may be constant, self-setting, or unsteady. The reach may be divided into uneven increments and the cross sections may be nonprismatic and may vary from one to the other. Tributary and lateral inflow may enter the reach. The digital model will simulate such common problems as (1) flood waves, (2) releases from dams, and (3) channels where storage is a consideration. It may also supply the needed flow information for mass-transport simulation. (Woodard-USGS)
International Nuclear Information System (INIS)
Calculations, using the two-dimensional Eulerian finite-difference code CSQ, were performed for the problem of a small spherical high-explosive charge detonated in a closed heavy-walled cylindrical container partially filled with water. Data from corresponding experiments, specifically performed to validate codes used for hypothetical core disruptive accidents of liquid metal fast breeder reactors, are available in the literature. The calculations were performed specifically to test whether Eulerian methods could handle this type of problem, to determine whether water cavitation, which plays a large role in the loadings on the roof of the containment vessel, could be described adequately by an equilibrium liquid-vapor mixed phase model, and to investigate the trade-off between accuracy and cost of the calculations by using different sizes of computational meshes. Comparison of the experimental and computational data shows that the Eulerian method can handle the problem with ease, giving good predictions of wall and floor loadings. While roof loadings are qualitatively correct, peak impulse appears to be affected by numerical resolution and is underestimated somewhat
Kim, Jihye; Lee, Geon Joon; Park, Inkyu; Lee, Young Pak
2012-07-01
The effects of the nanoparticle geometry and the host matrix on the optical properties of silver (Ag) nanocomposites were investigated. The spatial intensity distribution and absorption spectra were obtained by solving Maxwell equations using the finite-difference time-domain method. Local enhancement of the optical field was produced near the surface of the Ag nanoparticle. As the nanoparticle size increased, the plasmon-induced absorption increased and the surface plasmon resonance (SPR) wavelength of the Ag nanocomposite was redshifted. As the nanoparticle geometry was transformed from a sphere to an ellipsoid, two plasmon peaks appeared and their spectral spacing became larger with increasing the aspect ratio. The effects of the nanoparticle size and the anisotropic geometry on the optical properties of the Ag nanocomposites can be described by the Maxwell-Garnett theory and the Drude model. From the absorption spectra of the Ag nanocomposites with five different host matrices (SiO2, Al2O3, ZnO, ZrO2, and TiO2), it was found that the SPR wavelength of the Ag nanocomposite was redshifted with increasing the refractive index of the host matrix. PMID:22966604
A Coupled Finite Difference and Moving Least Squares Simulation of Violent Breaking Wave Impact
DEFF Research Database (Denmark)
Lindberg, Ole; Bingham, Harry B.
2012-01-01
Two model for simulation of free surface flow is presented. The first model is a finite difference based potential flow model with non-linear kinematic and dynamic free surface boundary conditions. The second model is a weighted least squares based incompressible and inviscid flow model. A special feature of this model is a generalized finite point set method which is applied to the solution of the Poisson equation on an unstructured point distribution. The presented finite point set method is generalized to arbitrary order of approximation. The two models are applied to simulation of steep and overturning wave impacts on a vertical breakwater. Wave groups with five different wave heights are propagated from offshore to the vicinity of the breakwater, where the waves are steep, but still smooth and non-overturning. These waves are used as initial condition for the weighted least squares based incompressible and inviscid model and the wave impacts on the vertical breakwater are simulated in this model. The resulting maximum pressures and forces on the breakwater are relatively high when compared with other studies and this is due to the incompressible nature of the present model.
Simulation of acoustic streaming by means of the finite-difference time-domain method
DEFF Research Database (Denmark)
Santillan, Arturo Orozco
2012-01-01
Numerical simulations of acoustic streaming generated by a standing wave in a narrow twodimensional cavity are presented. In this case, acoustic streaming arises from the viscous boundary layers set up at the surfaces of the walls. It is known that streaming vortices inside the boundary layer have directions of rotation that are opposite to those of the outer streaming vortices (Rayleigh streaming). The general objective of the work described in this paper has been to study the extent to which it is possible to simulate both the outer streaming vortices and the inner boundary layer vortices using the finite-difference time-domain method. To simplify the problem, thermal effects are not considered. The motivation of the described investigation has been the possibility of using the numerical method to study acoustic streaming, particularly under non-steady conditions. Results are discussed for channels of different width, which illustrate the applicability of the method. The obtained numerical simulations agree quite will with analytical solutions available in the literature.
Finite difference simulation of biological chromium (VI) reduction in aquifer media columns
Scientific Electronic Library Online (English)
Phalazane J, Mtimunye; Evans MN, Chirwa.
2014-04-03
Full Text Available A mechanistic mathematical model was developed that successfully traced the Cr(VI) concentration profiles inside porous aquifer media columns. The model was thereafter used to calculate Cr(VI) removal rate for a range of Cr(VI) loadings. Internal concentration profiles were modelled against data col [...] lected from intermediate sample ports along the length of the test columns. For the first time, the performance of a simulated barrier was evaluated internally in porous media using a finite difference approach. Parameters in the model were optimised at transient-state and under near steady-state conditions with respect to biomass and effluent Cr(VI) concentration respectively. The best fitting model from this study followed non-competitive inhibition kinetics for Cr(VI) removal with the best fitting steady-state parameters: Cr(VI) reduction rate coefficient, k = 5.2x10(8) l-mg"¹-h"¹; Cr(VI) threshold inhibition concentration, C = 50 mg-l-1; and a semi-empirical reaction order, n = 2. The model results showed that post-barrier infusion of biomass into the clean aquifer downstream of the barrier could be limited by depletion of the substrates within the barrier. The model when fully developed will be used in desktop evaluation of proposed in situ biological barrier systems before implementation in actual aquifer systems.
International Nuclear Information System (INIS)
A calculation program (URA 6.F4) was elaborated on FORTRAN IV language, that through finite differences solves the unidimensional scalar Helmholtz equation, assuming only one energy group, in spherical cylindrical or plane geometry. The purpose is the determination of the flow distribution in a reactor of spherical cylindrical or plane geometry and the critical dimensions. Feeding as entrance datas to the program the geometry, diffusion coefficients and macroscopic transversals cross sections of absorption and fission for each region. The differential diffusion equation is converted with its boundary conditions, to one system of homogeneous algebraic linear equations using the box integration technique. The investigation on criticality is converted then in a succession of eigenvalue problems for the critical eigenvalue. In general, only is necessary to solve the first eigenvalue and its corresponding eigenvector, employing the power method. The obtained results by the program for the critical dimensions of the clean reactors are admissible, the existing error as respect to the analytic is less of 0.5%; by the analysed reactors of three regions, the relative error with respect to the semianalytic result is less of 0.2%. With this program is possible to obtain one quantitative description of one reactor if the transversal sections that appears in the monoenergetic model are adequatedly averaged by the energy group used. (author)
Solving the time-dependent Schrödinger equation using finite difference methods
Directory of Open Access Journals (Sweden)
R Becerril
2008-12-01
Full Text Available We solve the time-dependent Schrödinger equation in one and two dimensions using the finite difference approximation. The evolution is carried out using the method of lines. The illustrative cases include: the particle in a box and the harmonic oscillator in one and two dimensions. As non-standard examples we evolve two solitons and show the time-dependent solitonic behavior in one dimension and the stabilization of an atomic gas model in two dimensions. The codes used to generate the results in this manuscript are freely available under request, and we expect this material could help students to have a better grasp of the solution of partial differential equations related to dynamical systemsResolvemos la ecuación de Schrödinger dependiente del tiempo en una y dos dimensiones usando diferencias finitas. La evolución se lleva a cabo usando el método de líneas. Los casos ilustrativos incluyen: la partícula en una caja y en un potencial de oscilador armónico en una y dos dimensiones. Como ejemplos poco comunes presentamos la evolución de dos solitones y mostramos la dependencia temporal del comportamiento solitónico en una dimensión y la estabilización de un modelo de gas atómico en dos dimensiones. Los códigos usados para generar los resultados en este manuscrito se encuentran disponibles a la menor petición, y esperamos que este hecho ayude a los estudiantes a adquirir un mejor entendimiento de la solución de ecuaciones diferenciales parciales relacionadas con sistemas dinámicos
Directory of Open Access Journals (Sweden)
Raj Mittra
2012-07-01
Full Text Available A rigorous full-wave solution, via the Finite-Difference-Time-Domain (FDTD method, is performed in an attempt to obtain realistic communication channel models for on-body wireless transmission in Body-Area-Networks (BANs, which are local data networks using the human body as a propagation medium. The problem of modeling the coupling between body mounted antennas is often not amenable to attack by hybrid techniques owing to the complex nature of the human body. For instance, the time-domain Green’s function approach becomes more involved when the antennas are not conformal. Furthermore, the human body is irregular in shape and has dispersion properties that are unique. One consequence of this is that we must resort to modeling the antenna network mounted on the body in its entirety, and the number of degrees of freedom (DoFs can be on the order of billions. Even so, this type of problem can still be modeled by employing a parallel version of the FDTD algorithm running on a cluster. Lastly, we note that the results of rigorous simulation of BANs can serve as benchmarks for comparison with the abundance of measurement data.
Zhang, Zhenguo; Zhang, Wei; Li, Hong; Chen, Xiaofei
2013-03-01
Simulating seismic waves with uniform grid in heterogeneous high-velocity contrast media requires small-grid spacing determined by the global minimal velocity, which leads to huge number of grid points and small time step. To reduce the computational cost, discontinuous grids that use a finer grid at the shallow low-velocity region and a coarser grid at high-velocity regions are needed. In this paper, we present a discontinuous grid implementation for the collocated-grid finite-difference (FD) methods to increase the efficiency of seismic wave modelling. The grid spacing ratio n could be an arbitrary integer n ? 2. To downsample the wavefield from the finer grid to the coarser grid, our implementation can simply take the values on the finer grid without employing a downsampling filter for grid spacing ratio n = 2 to achieve stable results for long-time simulation. For grid spacing ratio n ? 3, the Gaussian filter should be used as the downsampling filter to get a stable simulation. To interpolate the wavefield from the coarse grid to the finer grid, the trilinear interpolation is used. Combining the efficiency of discontinuous grid with the flexibility of collocated-grid FD method on curvilinear grids, our method can simulate large-scale high-frequency strong ground motion of real earthquake with consideration of surface topography.
Vapor cooled lead and stacks thermal performance and design analysis by finite difference techniques
International Nuclear Information System (INIS)
Investigation of the combined thermal performance of the stacks and vapor-cooled leads for the Mirror Fusion Test Facility-B (MFTF-B) demonstrates considerable interdependency. For instance, the heat transfer to the vapor-cooled lead (VCL) from warm bus heaters, environmental enclosure, and stack is a significant additional heat load to the joule heating in the leads, proportionately higher for the lower current leads that have fewer current-carrying, counter flow coolant copper tubes. Consequently, the specific coolant flow (G/sec-kA-lead pair) increases as the lead current decreases. The definition of this interdependency and the definition of necessary thermal management has required an integrated thermal model for the entire stack/VCL assemblies. Computer simulations based on finite difference thermal analyses computed all the heat interchanges of the six different stack/VCL configurations. These computer simulations verified that the heat load of the stacks beneficially alters the lead temperature profile to provide added stability against thermal runaway. Significant energy is transferred through low density foam filler in the stack from warm ambient sources to the vapor-cooled leads
Directory of Open Access Journals (Sweden)
P. H. Lauritzen
2013-09-01
Full Text Available Recently, a standard test case suite for 2-D linear transport on the sphere was proposed to assess important aspects of accuracy in geophysical fluid dynamics with a "minimal" set of idealized model configurations/runs/diagnostics. Here we present results from 19 state-of-the-art transport scheme formulations based on finite-difference/finite-volume methods as well as emerging (in the context of atmospheric/oceanographic sciences Galerkin methods. Discretization grids range from traditional regular latitude-longitude grids to more isotropic domain discretizations such as icosahedral and cubed-sphere tessellations of the sphere. The schemes are evaluated using a wide range of diagnostics in idealized flow environments. Accuracy is assessed in single- and two-tracer configurations using conventional error norms as well as novel diagnostics designed for climate and climate-chemistry applications. In addition, algorithmic considerations that may be important for computational efficiency are reported on. The latter is inevitably computing platform dependent, The ensemble of results from a wide variety of schemes presented here helps shed light on the ability of the test case suite diagnostics and flow settings to discriminate between algorithms and provide insights into accuracy in the context of global atmospheric/ocean modeling. A library of benchmark results is provided to facilitate scheme intercomparison and model development. Simple software and data-sets are made available to facilitate the process of model evaluation and scheme intercomparison.
Finite elements schemes for inviscid compressible flows
International Nuclear Information System (INIS)
A finite element scheme is presented for the numerical solution of the two-dimensional, unsteady Euler equations describing high-speed flows of an inviscid and compressible fluid. The proposed scheme represents an extension to the Galerkin finite element method of the well-known Lax-Wendroff finite difference method. The numerical results indicate that the proposed method possesses good shock-capturing properties without requiring the explicit use of artificial viscosity. (orig.)
De Basabe, Jonás D.; Sen, Mrinal K.
2015-01-01
The numerical simulation of wave propagation in media with solid and fluid layers is essential for marine seismic exploration data analysis. The numerical methods for wave propagation that are applicable to this physical settings can be broadly classified as partitioned or monolithic: The partitioned methods use separate simulations in the fluid and solid regions and explicitly satisfy the interface conditions, whereas the monolithic methods use the same method in all the domain without any special treatment of the fluid-solid interface. Despite the accuracy of the partitioned methods, the monolithic methods are more common in practice because of their convenience. In this paper, we analyse the accuracy of several monolithic methods for wave propagation in the presence of a fluid-solid interface. The analysis is based on grid-dispersion criteria and numerical examples. The methods studied here include: the classical finite-difference method (FDM) based on the second-order displacement formulation of the elastic wave equation (DFDM), the staggered-grid finite difference method (SGFDM), the velocity-stress FDM with a standard grid (VSFDM) and the spectral-element method (SEM). We observe that among these, DFDM and the first-order SEM have a large amount of grid dispersion in the fluid region which renders them impractical for this application. On the other hand, SGFDM, VSFDM and SEM of order greater or equal to 2 yield accurate results for the body waves in the fluid and solid regions if a sufficient number of nodes per wavelength is used. All of the considered methods yield limited accuracy for the surface waves because the proper boundary conditions are not incorporated into the numerical scheme. Overall, we demonstrate both by analytic treatment and numerical experiments, that a first-order velocity-stress formulation can, in general, be used in dealing with fluid-solid interfaces without using staggered grids necessarily.
Kaus, B.; Popov, A.
2014-12-01
The complexity of lithospheric rheology and the necessity to resolve the deformation patterns near the free surface (faults and folds) sufficiently well places a great demand on a stable and scalable modeling tool that is capable of efficiently handling nonlinearities. Our code LaMEM (Lithosphere and Mantle Evolution Model) is an attempt to satisfy this demand. The code utilizes a stable and numerically inexpensive finite difference discretization with the spatial staggering of velocity, pressure, and temperature unknowns (a so-called staggered grid). As a time discretization method the forward Euler, or a combination of the predictor-corrector and the fourth-order Runge-Kutta can be chosen. Elastic stresses are rotated on the markers, which are also used to track all relevant material properties and solution history fields. The Newtonian nonlinear iteration, however, is handled at the level of the grid points to avoid spurious averaging between markers and grid. Such an arrangement required us to develop a non-standard discretization of the effective strain-rate second invariant. Important feature of the code is its ability to handle stress-free and open-box boundary conditions, in which empty cells are simply eliminated from the discretization, which also solves the biggest problem of the sticky-air approach - namely large viscosity jumps near the free surface. We currently support an arbitrary combination of linear elastic, nonlinear viscous with multiple creep mechanisms, and plastic rheologies based on either a depth-dependent von Mises or pressure-dependent Drucker-Prager yield criteria.LaMEM is being developed as an inherently parallel code. Structurally all its parts are based on the building blocks provided by PETSc library. These include Jacobian-Free Newton-Krylov nonlinear solvers with convergence globalization techniques (line search), equipped with different linear preconditioners. We have also implemented the coupled velocity-pressure multigrid prolongation and restriction operators specific for staggered grid discretization. The capabilities of the code are demonstrated with a set of geodynamically-relevant benchmarks and example problems on parallel computers.This project is funded by ERC Starting Grant 258830
Pettersson, Per; Doostan, Alireza; Nordström, Jan
2013-01-01
The stochastic Galerkin and collocation methods are used to solve an advection–diffusion equation with uncertain and spatially varying viscosity. We investigate well-posedness, monotonicity and stability for the extended system resulting from the Galerkin projection of the advection–diffusion equation onto the stochastic basis functions. High-order summation-by-parts operators and weak imposition of boundary conditions are used to prove stability of the semi-discrete system. It is essenti...
Hosny, W. M.; Tabakoff, W.
1975-01-01
A two-dimensional finite difference numerical technique is presented to determine the temperature distribution in a solid blade of a radial guide vane. A computer program is written in Fortran IV for IBM 370/165 computer. The computer results obtained from these programs have a similar behavior and trend as those obtained by experimental results.
A finite-difference time-domain technique was used to calculate the specific absorption rate (SAR) at various sites in a heterogeneous block model of man. he block model represented a close approximation to a full-scale heterogeneous phantom model. oth models were comprised of a ...
Mesh-size errors in diffusion-theory calculations using finite-difference and finite-element methods
International Nuclear Information System (INIS)
A study has been performed of mesh-size errors in diffusion-theory calculations using finite-difference and finite-element methods. As the objective was to illuminate the issues, the study was performed for a 1D slab model of a reactor with one neutron-energy group for which analytical solutions were possible. A computer code SLAB was specially written to perform the finite-difference and finite-element calculations and also to obtain the analytical solutions. The standard finite-difference equations were obtained by starting with an expansion of the neutron current in powers of the mesh size, h, and keeping terms as far as h2. It was confirmed that these equations led to the well-known result that the criticality parameter varied with the square of the mesh size. An improved form of the finite-difference equations was obtained by continuing the expansion for the neutron current as far as the term in h4. In this case, the critical parameter varied as the fourth power of the mesh size. The finite-element solutions for 2 and 3 nodes per element revealed that the criticality parameter varied as the square and fourth power of the mesh size, respectively. Numerical results are presented for a bare reactive core of uniform composition with 2 zones of different uniform mesh and for a reactive core with an absorptive reflector. (author)
Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer.
Klose, A D; Hielscher, A H
1999-08-01
We report on the development of an iterative image reconstruction scheme for optical tomography that is based on the equation of radiative transfer. Unlike the commonly applied diffusion approximation, the equation of radiative transfer accurately describes the photon propagation in turbid media without any limiting assumptions regarding the optical properties. The reconstruction scheme consists of three major parts: (1) a forward model that predicts the detector readings based on solutions of the time-independent radiative transfer equation, (2) an objective function that provides a measure of the differences between the detected and the predicted data, and (3) an updating scheme that uses the gradient of the objective function to perform a line minimization to get new guesses of the optical properties. The gradient is obtained by employing an adjoint differentiation scheme, which makes use of the structure of the finite-difference discrete-ordinate formulation of the transport forward model. Based on the new guess of the optical properties a new forward calculation is performed to get new detector predictions. The reconstruction process is completed when the minimum of the objective function is found within a defined error. To illustrate the performance of the code we present initial reconstruction results based on simulated data. PMID:10501069
Implementation of a Courant violating scheme for mixture drift-flux equations
International Nuclear Information System (INIS)
Mixture models are commonly used in the simulation of transient two-phase flows as simplifications of six-equation models, with the drift-flux models as a common way to describe relative phase motion. This is particularly true in simulator and control system modeling where solutions that are faster than real time are necessary, and as a means for incorporating thermal-hydraulic feedback into steady-state and transient neutronics calculations. Variations on semi-implicit finite difference schemes are some of the more commonly used temporal discretization schemes. The maximum time step size associated with these schemes is normally assumed to be limited by stability considerations to the material transport time across any computational cell (Courant limit). In applications requiring solutions that are faster than real time or the calculation of thermal-hydraulic feedback in reactor kinetics codes, time-step sizes that are restricted by the material Courant limit may result in prohibitive run times. A Courant violating scheme is examined for the mixture drift-flux equations, which for rapid transients is at least as fast as classic semi-implicit methods and for slow transients allows time step sizes many times greater than the material Courant limit
Mimetic discretization of two-dimensional magnetic diffusion equations
Lipnikov, Konstantin; Reynolds, James; Nelson, Eric
2013-08-01
In case of non-constant resistivity, cylindrical coordinates, and highly distorted polygonal meshes, a consistent discretization of the magnetic diffusion equations requires new discretization tools based on a discrete vector and tensor calculus. We developed a new discretization method using the mimetic finite difference framework. It is second-order accurate on arbitrary polygonal meshes and a consistent calculation of the Joule heating is intrinsic within it. The second-order convergence rates in L2 and L1 norms were verified with numerical experiments.
New schemes for a two-dimensional inverse problem with temperature overspecification
Directory of Open Access Journals (Sweden)
Dehghan Mehdi
2001-01-01
Full Text Available Two different finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the (3,3 alternating direction implicit (ADI finite difference scheme and the (3,9 alternating direction implicit formula. These schemes are unconditionally stable. The basis of analysis of the finite difference equation considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett [17]. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. These schemes use less central processor times than the fully implicit schemes for two-dimensional diffusion with temperature overspecification. The alternating direction implicit schemes developed in this report use more CPU times than the fully explicit finite difference schemes, but their unconditional stability is significant. The results of numerical experiments are presented, and accuracy and the Central Processor (CPU times needed for each of the methods are discussed. We also give error estimates in the maximum norm for each of these methods.
Directory of Open Access Journals (Sweden)
Berninger Heiko
2013-01-01
Full Text Available In this paper we give an introduction to the Boltzmann equation for neutrino transport used in core collapse supernova models as well as a detailed mathematical description of the Isotropic Diffusion Source Approximation (IDSA established in [6]. Furthermore, we present a numerical treatment of a reduced Boltzmann model problem based on time splitting and finite volumes and revise the discretization of the IDSA in [6] for this problem. Discretization error studies carried out on the reduced Boltzmann model problem and on the IDSA show that the errors are of order one in both cases. By a numerical example, a detailed comparison of the reduced model and the IDSA is carried out and interpreted. For this example the IDSA modeling error with respect to the reduced Boltzmann model is numerically determined and localized. Dans cet article, nous donnons une introduction à l’équation de Boltzmann pour le transport des neutrinos dans les modèles de supernovae à effondrement de cœur ainsi qu’une description détaillée de l’Isotropic Diffusion Source Approximation (IDSA développée dans [6]. De plus, nous présentons le traitement numérique d’un modèle de Boltzmann simplifié basé sur une décomposition en temps de l’opérateur et sur un algorithme de volumes finis ainsi que l’adaptation de la discrétisation de l’IDSA de [6] à notre modèle. Les études de l’erreur de discrétisation faites sur le modèle de Boltzmann simplifié et sur l’IDSA montrent que les erreurs sont d’ordre un dans les deux cas. A l’aide d’un exemple numérique, nous comparons et interprétons en détail les deux modèles. Pour cet exemple, l’erreur de modélisation de l’IDSA par rapport au modèle de Boltzmann simplifié est déterminée numériquement et localisée.
Validation tests of the venture finite-difference diffusion theory neutronics code
International Nuclear Information System (INIS)
The VENTURE code has been programmed for digital computer neutronics calculations. This report discusses the testing done to prove that reliable solutions are obtained. Some of the results produced and discussion about them may be of general interest, especially in application for reactor analysis. The authors believe that the extensive testing done with this code allows a high level of confidence to be placed on results produced if due consideration is given to limits of the model, to discrete mesh requirements, to basic requirements for adequate nuclear properties, and to satisfying adequate convergence criteria for the iterative procedures of solution
3D Finite-Difference Modeling of Acoustic Radiation from Seismic Sources
Chael, E. P.; Aldridge, D. F.; Jensen, R. P.
2013-12-01
Shallow seismic events, earthquakes as well as explosions, often generate acoustic waves in the atmosphere observable at local or even regional distances. Recording both the seismic and acoustic signals can provide additional constraints on source parameters such as epicenter coordinates, depth, origin time, moment, and mechanism. Recent advances in finite-difference (FD) modeling methods enable accurate numerical treatment of wave propagation across the ground surface between the (solid) elastic and (fluid) acoustic domains. Using a fourth-order, staggered-grid, velocity-stress FD algorithm, we are investigating the effects of various source parameters on the acoustic (or infrasound) signals transmitted from the solid earth into the atmosphere. Compressional (P), shear (S), and Rayleigh waves all radiate some acoustic energy into the air at the ground surface. These acoustic wavefronts are typically conical in shape, since their phase velocities along the surface exceed the sound speed in air. Another acoustic arrival with a spherical wavefront can be generated from the vicinity of the epicenter of a shallow event, due to the strong vertical ground motions directly above the buried source. Images of acoustic wavefields just above the surface reveal the radiation patterns and relative amplitudes of the various arrivals. In addition, we compare the relative effectiveness of different seismic source mechanisms for generating acoustic energy. For point sources at a fixed depth, double-couples with almost any orientation produce stronger acoustic signals than isotropic explosions, due to higher-amplitude S and Rayleigh waves. Of course, explosions tend to be shallower than most earthquakes, which can offset the differences due to mechanism. Low-velocity material in the shallow subsurface acts to increase vertical seismic motions there, enhancing the coupling to acoustic waves in air. If either type of source breaks the surface (e.g., an earthquake with surface rupture, or an explosion that causes cratering) then especially strong acoustic signals can emanate from the epicentral zone. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
Finite difference modelling to evaluate seismic P wave and shear wave field data
Burschil, T.; Beilecke, T.; Krawczyk, C. M.
2014-08-01
High-resolution reflection seismic methods are an established non-destructive tool for engineering tasks. In the near surface, shear wave reflection seismic measurements usually offer a higher spatial resolution in the same effective signal frequency spectrum than P wave data, but data quality varies more strongly. To discuss the causes of these differences, we investigated a P wave and a SH wave reflection seismic profile measured at the same location on Föhr island, and applied reflection seismic processing to the field data as well as finite difference modelling of the seismic wavefield (SOFI FD-code). The simulations calculated were adapted to the acquisition field geometry, comprising 2 m receiver distance and 4 m shot distance along the 1.5 km long P wave and 800 m long SH wave profiles. A Ricker-Wavelet and the use of absorbing frames were first order model parameters. The petrophysical parameters to populate the structural models down to 400 m depth are taken from borehole data, VSP measurements and cross-plot relations. The first simulation of the P wave wavefield was based on a simplified hydrogeological model of the survey location containing six lithostratigraphic units. Single shot data were compared and seismic sections created. Major features like direct wave, refracted waves and reflections are imaged, but the reflectors describing a prominent till layer at ca. 80 m depth was missing. Therefore, the P wave input model was refined and 16 units assigned. These define a laterally more variable velocity model (vP = 1600-2300 m s-1) leading to a much better reproduction of the field data. The SH wave model was adapted accordingly but only led to minor correlation with the field data and produced a higher signal-to-noise ratio. Therefore, we suggest to consider for future simulations additional features like intrinsic damping, thin layering, or a near surface weathering layer. These may lead to a better understanding of key parameters determining the data quality of near-surface seismic measurements.
Finite difference frequency domain method for thin, lossy, anisotropic layers in waveguides
Arft, Carl Martin
Thin, lossy layers are present in a variety of optical and microwave waveguiding structures. They may be purposely integrated into a structure, such as in integrated optical devices, optical and microwave sensors, and microwave transmission line devices. Or, they may be incorporated into the structure accidentally via contamination or reactions during the fabrication process. These layers can have a substantial effect upon the properties of the waveguide device; however, incorporation of these layers into a theoretical model is difficult as the layers may be extremely thin. In this work, a method is presented to incorporate thin, lossy, anisotropic layers into a finite difference frequency domain (FDFD) method, without the need for grid points to lie within the layers. This is an advantage over existing FDFD methods, in which the number of grid points quickly becomes unmanageable if the layers are orders of magnitude thinner than the computational domain. In the developed FDFD method, as with most numerical methods used for electromagnetic simulation, the electromagnetic fields often extend outside of the computational domain. In these situations, a transparent boundary condition (TBC) is necessary that absorbs or transmits incident radiation such that the boundaries do not influence the properties of the calculated modes. After reviewing existing TBC's that are applicable to the FDFD method, a new boundary condition, the n-term TBC (nT-TBC) is developed. Numerical examples are presented to demonstrate that the nT-TBC performs very well as compared to other boundary conditions, while requiring less computational overhead. The new FDFD method is then verified by applying it to structures for which analytical solutions are available, and very close correlation is observed. These structures include microwave and optical waveguides, as well as guided surface plasmon modes in very thin metal layers. The method is then used to investigate the effects of thin layers on two different experimental devices. The first is a study of the effects of thin electrode layers present in an asymmetric directional coupler modulator. The second is an investigation of loss arising from contamination incurred during fabrication of silicon oxynitride integrated waveguides.
Finite difference modelling to evaluate seismic P wave and shear wave field data
Directory of Open Access Journals (Sweden)
T. Burschil
2014-08-01
Full Text Available High-resolution reflection seismic methods are an established non-destructive tool for engineering tasks. In the near surface, shear wave reflection seismic measurements usually offer a higher spatial resolution in the same effective signal frequency spectrum than P wave data, but data quality varies more strongly. To discuss the causes of these differences, we investigated a P wave and a SH wave reflection seismic profile measured at the same location on Föhr island, and applied reflection seismic processing to the field data as well as finite difference modelling of the seismic wavefield (SOFI FD-code. The simulations calculated were adapted to the acquisition field geometry, comprising 2 m receiver distance and 4 m shot distance along the 1.5 km long P wave and 800 m long SH wave profiles. A Ricker-Wavelet and the use of absorbing frames were first order model parameters. The petrophysical parameters to populate the structural models down to 400 m depth are taken from borehole data, VSP measurements and cross-plot relations. The first simulation of the P wave wavefield was based on a simplified hydrogeological model of the survey location containing six lithostratigraphic units. Single shot data were compared and seismic sections created. Major features like direct wave, refracted waves and reflections are imaged, but the reflectors describing a prominent till layer at ca. 80 m depth was missing. Therefore, the P wave input model was refined and 16 units assigned. These define a laterally more variable velocity model (vP = 1600–2300 m s?1 leading to a much better reproduction of the field data. The SH wave model was adapted accordingly but only led to minor correlation with the field data and produced a higher signal-to-noise ratio. Therefore, we suggest to consider for future simulations additional features like intrinsic damping, thin layering, or a near surface weathering layer. These may lead to a better understanding of key parameters determining the data quality of near-surface seismic measurements.
A pencil distributed finite difference code for strongly turbulent wall-bounded flows
van der Poel, Erwin P; Donners, John; Verzicco, Roberto
2015-01-01
We present a numerical scheme geared for high performance computation of wall-bounded turbulent flows. The number of all-to-all communications is decreased to only six instances by using a two-dimensional (pencil) domain decomposition and utilizing the favourable scaling of the CFL time-step constraint as compared to the diffusive time-step constraint. As the CFL condition is more restrictive at high driving, implicit time integration of the viscous terms in the wall-parallel directions is no longer required. This avoids the communication of non-local information to a process for the computation of implicit derivatives in these directions. We explain in detail the numerical scheme used for the integration of the equations, and the underlying parallelization. The code is shown to have very good strong and weak scaling to at least 64K cores.
High-order compact schemes for Black-Scholes basket options
Düring, Bertram; Heuer, Christof
2015-01-01
We present a new high-order compact scheme for the multi-dimensional Black-Scholes model with application to European Put options on a basket of two underlying assets. The scheme is second-order accurate in time and fourth-order accurate in space. Numerical examples confirm that a standard second-order finite difference scheme is significantly outperformed.
Spectral Schemes on Triangular Elements
Heinrichs, Wilhelm; Loch, Birgit I.
2001-10-01
The Poisson problem with homogeneous Dirichlet boundary conditions is considered on a triangle. The transformation from square to triangle is realized by mapping an edge of the square onto a corner of the triangle. Then standard Chebyshev collocation techniques can be implemented. Numerical experiments demonstrate the expected high spectral accuracy for smooth solutions. Furthermore, it is shown that finite difference preconditioning can be successfully employed to construct an efficient iterative solver. Then the convection-diffusion equation is considered. Here finite difference preconditioning with central differences leads to instability. However, using the first-order upstream scheme, we obtain a stable method. Finally, a domain decomposition technique is applied to the patching of rectangular and triangular elements.
Staggered Schemes for Fluctuating Hydrodynamics
Balboa, F. B.; Bell, J B; Delgado-Buscalioni, R.; Donev, A.; Fai, T. G.; Griffith, B. E.; Peskin, C. S.
2011-01-01
We develop numerical schemes for solving the isothermal compressible and incompressible equations of fluctuating hydrodynamics on a grid with staggered momenta. We develop a second-order accurate spatial discretization of the diffusive, advective and stochastic fluxes that satisfies a discrete fluctuation-dissipation balance, and construct temporal discretizations that are at least second-order accurate in time deterministically and in a weak sense. Specifically, the methods...
Stability of finite difference approximations of two fluid, two phase flow equations
Energy Technology Data Exchange (ETDEWEB)
Holmes, M.A. [North Carolina State Univ., Raleigh, NC (United States)
1995-12-31
It is well known that the basic single pressure, two fluid model for two phase flow has complex characteristics and is dynamically unstable. Nevertheless, common nuclear reactor thermal-hydraulics codes use variants of this model for reactor safety calculations. In these codes, the non-physical instabilities of the model may be damped by the numerical method and/or additional momentum interchange terms. Both of these effects are investigated using the linearized Von Neumann stability analysis. The stability of the semi-implicit method is of primary concern, because of its computational efficiency and popularity. It is shown that there is likely no completely stable numerical method, including fully implicit methods, for the basic single pressure model. Additionally, the momentum interchange terms commonly added to the basic single pressure model do not result in stable numerical methods for all the physically interesting reference conditions. Although practical stable approximations may be realized on a coarse computational grid, it is concluded that the assumption of instantaneously equilibrated phasic pressures must be relaxed in order to develop a generally stable numerical solution of a two fluid model. The numerical stability of the semi-implicit discretization of the true two pressure models of Ransom and Hicks, and Holm and Kupershmidt is analyzed. The semi-implicit discretization of these models, which possess real characteristics, are found to be numerically stable as long as certain convective limits are satisfied. Based on the form of these models, the general form of a numerically stable, basic two pressure model is proposed. The evolution equation required for closure is a volume fraction transport equation, which may possibly be determined based on void wave propagation considerations. 43 refs., 22 figs., 3 tabs.
DIF3D: a code to solve one-, two-, and three-dimensional finite-difference diffusion theory problems
International Nuclear Information System (INIS)
The mathematical development and numerical solution of the finite-difference equations are summarized. The report provides a guide for user application and details the programming structure of DIF3D. Guidelines are included for implementing the DIF3D export package on several large scale computers. Optimized iteration methods for the solution of large-scale fast-reactor finite-difference diffusion theory calculations are presented, along with their theoretical basis. The computational and data management considerations that went into their formulation are discussed. The methods utilized include a variant of the Chebyshev acceleration technique applied to the outer fission source iterations and an optimized block successive overrelaxation method for the within-group iterations. A nodal solution option intended for analysis of LMFBR designs in two- and three-dimensional hexagonal geometries is incorporated in the DIF3D package and is documented in a companion report, ANL-83-1
Kamalakis, Thomas; Alexandropoulos, Dimitris; Vainos, Nikos
2015-03-01
Polymer photonics have been identified as a strong candidate technology for producing low-cost optical devices. In this paper, we provide a time efficient framework for designing polymer micro-ring devices based on Coupled Mode Theory (CMT) and a Finite Difference Mode Solver. We benchmark two alternative methods for modeling the coupling regions of the micro-ring filter in 3D. We deduce that compared to full blown finite difference time domain simulations, CMT can provide accurate results in just a small fraction of time. The proposed model allows the study of bending losses on the spectral properties of the device, that can be otherwise modelled using time demanding FDTD or less accurate simplified analytical expressions.
Baumeister, K. J.
1977-01-01
Finite difference equations are derived for sound propagation in a two dimensional, straight, soft wall duct with a uniform flow by using the wave envelope concept. This concept reduces the required number of finite difference grid points by one to two orders of magnitude depending on the length of the duct and the frequency of the sound. The governing acoustic difference equations in complex notation are derived. An exit condition is developed that allows a duct of finite length to simulate the wave propagation in an infinitely long duct. Sample calculations presented for a plane wave incident upon the acoustic liner show the numerical theory to be in good agreement with closed form analytical theory. Complete pressure and velocity printouts are given to some sample problems and can be used to debug and check future computer programs.
Energy Technology Data Exchange (ETDEWEB)
Tokuda, Shinji [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment; Watanabe, Tomoko
1996-08-01
The matching problem in resistive MagnetoHydroDynamic stability analysis by the asymptotic matching method has been reformulated as an initial-boundary value problem for the inner-layer equations describing the plasma dynamics in the thin layer around a rational surface. The third boundary conditions at boundaries of a finite interval are imposed on the inner layer equations in the formulation instead of asymptotic conditions at infinities. The finite difference method for this problem has been applied to model equations whose solutions are known in a closed form. It has been shown that the initial value problem and the associated eigenvalue problem for the model equations can be solved by the finite difference method with numerical stability. The formulation presented here enables the asymptotic matching method to be a practical method for the resistive MHD stability analysis. (author)
Argyropoulos, Christos; Zhao, Yan; Hao, Yang
2008-01-01
A radial-dependent dispersive finite-difference time-domain (FDTD) method is proposed to simulate electromagnetic cloaking devices. The Drude dispersion model is applied to model the electromagnetic characteristics of the cloaking medium. Both lossless and lossy cloaking materials are examined and their operating bandwidth is also investigated. It is demonstrated that the perfect "invisibility" from electromagnetic cloaks is only available for lossless metamaterials and with...
International Nuclear Information System (INIS)
The methods and performance of a three-dimensional Sn transport code employing the Discontinuous Finite Element Method (DFEM) and the Coarse Mesh Finite Difference (CMFD) formulation are presented. The mesh generator GMSH and a post processing visualization tool Visit are combined with the code for flexible geometry processing and versatile visualization. The CMFD method for DFEM Sn applications is formulated and the performance of the CMFD acceleration of eigenvalue calculations is demonstrated for a simple set of neutron transport problems. (authors)
Arrigo Calzolari; Marco Buongiorno Nardelli
2013-01-01
Using first principles calculations based on density functional theory and a coupled finite-fields/finite-differences approach, we study the dielectric properties, phonon dispersions and Raman spectra of ZnO, a material whose internal polarization fields require special treatment to correctly reproduce the ground state electronic structure and the coupling with external fields. Our results are in excellent agreement with existing experimental measurements and provide an essential reference fo...
Alemayehu Shiferaw; Ramesh Chand Mittal
2014-01-01
In this work, by extending the method of Hockney into three dimensions, the Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved directly. The Poisson equation is approximated by fourth-order finite differences and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The accuracy of this method is tested for so...
Study of two-dimensional transient cavity fields using the finite-difference time-domain technique
International Nuclear Information System (INIS)
This work is intended to be a study into the application of the finite-difference time-domain, or FD-TD technique, to some of the problems faced by designers of equipment used in modern accelerators. In particular it discusses using the FD-TD algorithm to study the field distribution of a simple two-dimensional cavity in both space and time. 18 refs
International Nuclear Information System (INIS)
Large-area THz antennas are studied by using a finite difference time domain method and a lossy transmission line model. The temporal evolution of an emitted pulse from a specified antenna is experimentally realized, the result is compared with the numerical result and good agreement is observed. The effect of the laser beam spot size, carrier life-time and collision time on temporal evolution of a THz pulse is investigated using this technique
A discourse on sensitivity analysis for discretely-modeled structures
Adelman, Howard M.; Haftka, Raphael T.
1991-01-01
A descriptive review is presented of the most recent methods for performing sensitivity analysis of the structural behavior of discretely-modeled systems. The methods are generally but not exclusively aimed at finite element modeled structures. Topics included are: selections of finite difference step sizes; special consideration for finite difference sensitivity of iteratively-solved response problems; first and second derivatives of static structural response; sensitivity of stresses; nonlinear static response sensitivity; eigenvalue and eigenvector sensitivities for both distinct and repeated eigenvalues; and sensitivity of transient response for both linear and nonlinear structural response.
Fast finite difference methods for solving 1D two-phase flow equations
International Nuclear Information System (INIS)
This paper presents a prospective study of fully implicit numerical methods which are able to compute slow transients fast and with a reasonable accuracy. First a quick survey of recent numerical methods used in two-phase flow computations is given. In a second part, advantages of monotonic schemes and staggered meshes are demonstrated. These two parts introduce the new fully implicit method which is proposed in part three and validated in part four. This numerical method, named FICE (Fully Implicit Compressible Eulerian) by analogy with the I.C.E. method proposed in 1974 by F.H. HARLOW and A.A. AMSDEN, is an improvement of this well known method in the sense that it is fully implicit but reduces automatically to the semi-implicit method if accuracy or convergence criteria lead to time step reduction. The success of this method represents a step toward the final goal of real time reactor transient computation
Numerical Simulation of Electromagnetic Waves Scattering by Discrete Exterior Calculus
International Nuclear Information System (INIS)
We show how to construct discrete Maxwell equations by discrete exterior calculus. The new scheme has many virtues compared to the traditional Yee's scheme: it is a multisymplectic scheme and keeps geometric properties. Moreover, it can be applied on triangular mesh and thus is more adaptive to handle domains with irregular shapes. We have implemented this scheme on a Java platform successfully and our experimental results show that this scheme works well. (fundamental areas of phenomenology (including applications))
Saavedra, Sebastian
2012-07-01
The mathematical model that has been recognized to have the more accurate approximation to the physical laws govern subsurface hydrocarbon flow in reservoirs is the Compositional Model. The features of this model are adequate to describe not only the performance of a multiphase system but also to represent the transport of chemical species in a porous medium. Its importance relies not only on its current relevance to simulate petroleum extraction processes, such as, Primary, Secondary, and Enhanced Oil Recovery Process (EOR) processes but also, in the recent years, carbon dioxide (CO2) sequestration. The purpose of this study is to investigate the subsurface compositional flow under isothermal conditions for several oil well cases. While simultaneously addressing computational implementation finesses to contribute to the efficiency of the algorithm. This study provides the theoretical framework and computational implementation subtleties of an IMplicit Pressure Explicit Composition (IMPEC)-Volume-balance (VB), two-phase, equation-of-state, approach to model isothermal compositional flow based on the finite difference scheme. The developed model neglects capillary effects and diffusion. From the phase equilibrium premise, the model accounts for volumetric performances of the phases, compressibility of the phases, and composition-dependent viscosities. The Equation of State (EoS) employed to approximate the hydrocarbons behaviour is the Peng Robinson Equation of State (PR-EOS). Various numerical examples were simulated. The numerical results captured the complex physics involved, i.e., compositional, gravitational, phase-splitting, viscosity and relative permeability effects. Regarding the numerical scheme, a phase-volumetric-flux estimation eases the calculation of phase velocities by naturally fitting to phase-upstream-upwinding. And contributes to a faster computation and an efficient programming development.
Geometric formulations and variational integrators of discrete autonomous Birkhoff systems
International Nuclear Information System (INIS)
The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of mathematical pendulum shows that the discrete variational method is as effective as symplectic scheme for the autonomous Birkhoff systems. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)
Directory of Open Access Journals (Sweden)
Dora M Ballesteros
2012-04-01
Full Text Available This paper presents FPGA design of ECG compression by using the Discrete Wavelet Transform (DWT and one lossless encoding method. Unlike the classical works based on off-line mode, the current work allows the real-time processing of the ECG signal to reduce the redundant information. A model is developed for a fixed-point convolution scheme which has a good performance in relation to the throughput, the latency, the maximum frequency of operation and the quality of the compressed signal. The quantization of the coefficients of the filters and the selected fixed-threshold give a low error in relation to clinical applications.Este documento presenta el diseño basado en FPGA para la compresión de señales ECG utilizando la Transformada Wavelet Discreta y un método de codificación sin pérdida de información. A diferencia de los trabajos clásicos para modo off-line, el trabajo actual permite la compresión en tiempo real de la señal ECG por medio de la reducción de la información redundante. Se propone un modelo para el esquema de convolución en formato punto fijo, el cual tiene buen desempeño en relación a la tasa de salida, la latencia del sistema, la máxima frecuencia de operación y la calidad de la señal comprimida. La arquitectura propuesta, la cuantización utilizada y el método de codificación proporcionan un PRD que es apto para el análisis clínico.
Scientific Electronic Library Online (English)
Dora M, Ballesteros; Diana Marcela, Moreno; Andrés E, Gaona.
2012-04-01
Full Text Available Este documento presenta el diseño basado en FPGA para la compresión de señales ECG utilizando la Transformada Wavelet Discreta y un método de codificación sin pérdida de información. A diferencia de los trabajos clásicos para modo off-line, el trabajo actual permite la compresión en tiempo real de l [...] a señal ECG por medio de la reducción de la información redundante. Se propone un modelo para el esquema de convolución en formato punto fijo, el cual tiene buen desempeño en relación a la tasa de salida, la latencia del sistema, la máxima frecuencia de operación y la calidad de la señal comprimida. La arquitectura propuesta, la cuantización utilizada y el método de codificación proporcionan un PRD que es apto para el análisis clínico. Abstract in english This paper presents FPGA design of ECG compression by using the Discrete Wavelet Transform (DWT) and one lossless encoding method. Unlike the classical works based on off-line mode, the current work allows the real-time processing of the ECG signal to reduce the redundant information. A model is dev [...] eloped for a fixed-point convolution scheme which has a good performance in relation to the throughput, the latency, the maximum frequency of operation and the quality of the compressed signal. The quantization of the coefficients of the filters and the selected fixed-threshold give a low error in relation to clinical applications.
A hybrid absorbing boundary condition for frequency-domain finite-difference modelling
International Nuclear Information System (INIS)
Liu and Sen (2010 Geophysics 75 A1–6; 2012 Geophys. Prospect. 60 1114–32) proposed an efficient hybrid scheme to significantly absorb boundary reflections for acoustic and elastic wave modelling in the time domain. In this paper, we extend the hybrid absorbing boundary condition (ABC) into the frequency domain and develop specific strategies for regular-grid and staggered-grid modelling, respectively. Numerical modelling tests of acoustic, visco-acoustic, elastic and vertically transversely isotropic (VTI) equations show significant absorptions for frequency-domain modelling. The modelling results of the Marmousi model and the salt model also demonstrate the effectiveness of the hybrid ABC. For elastic modelling, the hybrid Higdon ABC and the hybrid Clayton and Engquist (CE) ABC are implemented, respectively. Numerical simulations show that the hybrid Higdon ABC gets better absorption than the hybrid CE ABC, especially for S-waves. We further compare the hybrid ABC with the classical perfectly matched layer (PML). Results show that the two ABCs cost the same computation time and memory space for the same absorption width. However, the hybrid ABC is more effective than the PML for the same small absorption width and the absorption effects of the two ABCs gradually become similar when the absorption width is increased. (paper)
Gradient Schemes for Stokes problem
Eymard, Robert; Féron, Pierre
2014-01-01
We provide a framework which encompasses a large family of conforming and nonconforming numerical schemes, for the approximation of the steady state incompressible Stokes equations with homogeneous Dirichlet's boundary conditions. Three examples (Taylor-Hood, extended MAC and Crouzeix-Raviart schemes) are shown to enter into this framework. The convergence of the scheme is proved by compactness arguments, thanks to estimates on the discrete solution that allow to prove the weak convergence to...
Computational evaluation of convection schemes in fluid dynamics problems
Ferreira, Valdemir Garcia; Corrêa, Laís; Candezano, Miguel Antonio Caro; Cirilo, Eliandro Rodrigues; Natti, Paulo Laerte; Romeiro, Neyva Maria Lopes; 10.5433/1679-0375.2012v33n2p107
2013-01-01
This article provides a computational evaluation of the popular high resolution upwind WACEB, CUBISTA and ADBQUICKEST schemes for solving non-linear fluid dynamics problems. By using the finite difference methodology, the schemes are analyzed and implemented in the context of normalized variables of Leonard. In order to access the performance of the schemes, Riemann problems for 1D Burgers, Euler and shallow water equations are considered. From the numerical results, the schemes are ranked according to their performance in solving these non-linear equations. The best scheme is then applied in the numerical simulation of tridimensional incompressible moving free surface flows.
Zhang, Jitao
2014-01-01
We numerically describe the physical mechanism underlying the terahertz photoconductive antenna (PCA) by the finite-difference time-domain method in three-dimension. The feature of our approach is that the multi-physical phenomena happening in the PCA, such as light-matter interaction, photo-excited carrier dynamics and full-wave propagation of the THz radiation, are considered and embodied in the simulation. The method has been verified by comparing with existing commercial softwares. In addition, we use this simulation tool to characterize the parameter-dependent performance of a PCA,thereby the design of novel PCA with enhanced optics-to-THz efficiency can be inspired.
Gray, Stephen K.; Kupka, Teobald
2003-07-01
Finite-difference time-domain studies of a variety of silver cylinder arrays with nanometer-scale diameters (nanowires) interacting with light are presented. We show how reasonable estimates of scattering and absorption cross sections for metallic nanoscale objects can be obtained from such calculations. We then study the explicit time-domain behavior of both simple linear chains of such cylinders, as well as more elaborate arrays. A funnel-like configuration is found to be one interesting possibility for achieving propagation of light through features confined in one dimension to less than 100 nm.
International Nuclear Information System (INIS)
A finite-difference time-domain method is developed to treat a nearly-free-electron metal by rewriting the time-domain Maxwell's equations into three coupled partial differential equations for three vector functions. The method is used to calculate the band structures of two-dimensional metallic photonic crystals, and the numerical results are compared with those obtained by other methods. The new time-stepping formulae are also used to investigate the defect modes by calculating the transmission spectrum. (author)
Development of the software Conden 1.0 in finite differences to model electrostatics problems 2D
Directory of Open Access Journals (Sweden)
Wilson Rodríguez Calderón
2012-05-01
Full Text Available The present work consists on the development and implementation of the finite differences method for over-relaxation adapted to irregular meshes to determine the influence of the air frontiers on the potencial values and field electricians, calculated inside a badges parallel condenser, using GID like a pre/post-process platform and Fortran like a programming language of the calculation motor of differences Conden 1.0. The problem domain is constituted by two rectangles that represent the condenser and the air layer that covers it, divided in rectangular meshes no standardize.
International Nuclear Information System (INIS)
With the increment of seismic exploration precision requirement, it is significant to develop the anisotropic migration methods. Pre-stack reverse-time migration (RTM) is performed based on acoustic vertical transversely isotropic (VTI) wave equations, and the accuracy and efficiency of RTM strongly depend on the algorithms used for wave equation numerical solution. Finite-difference (FD) methods have been widely used in numerical solution of wave equations. The conventional FD method derives spatial FD coefficients from the space domain dispersion relation, and it is difficult to satisfy the time–space domain dispersion relation of the wave equation exactly. In this paper, we adopt a time–space domain FD method to solve acoustic VTI wave equations. Dispersion analysis and numerical modelling results demonstrate that the time–space domain FD method has greater accuracy than the conventional FD method under the same discretizations. The time–space domain high-order FD method is also applied in the wavefield extrapolation of acoustic VTI pre-stack RTM. The model tests demonstrate that the acoustic VTI pre-stack RTM based on the time–space domain FD method can obtain better images than that based on the conventional FD method, and the processing results show that the imaging quality of the acoustic VTI RTM is clearer and more correct than that of acoustic isotropic RTM. Meanwhile, in the process of wavefield forward and backward extrapolation, we employ adaptivextrapolation, we employ adaptive variable-length spatial operators to compute spatial derivatives to improve the computational efficiency effectively almost without reducing the imaging accuracy. (paper)
A PRACTICAL PROXY SIGNATURE SCHEME
Directory of Open Access Journals (Sweden)
Sattar Aboud
2012-01-01
Full Text Available A proxy signature scheme is a variation of the ordinary digital signature scheme which enables a proxy signer to generate signatures on behalf of an original signer. In this paper, we present two efficient types of proxy signature scheme. The first one is the proxy signature for warrant partial delegation combines an advantage of two well known warrant partial delegation schemes. This proposed proxy signature scheme is based on the difficulty of solving the discrete logarithm problem. The second proposed scheme is based on threshold delegation the proxy signer power to sign the message is share. We claim that the proposed proxy signature schemes meet the security requirements and more practical than the existing proxy signature schemes.
Expert System to Create Building Design Scheme
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Žilvinas Steckevi?ius
2014-02-01
Full Text Available In this paper the technology for creation of optimal design scheme for building is presented. Optimization of designscheme is based on genetic algorithm. Rectangular perimeter of single-storey structure with a linear load in all its locations isinvestigated. In such case, finite element, finite difference and other methods are not necessary – in order to evaluate the stressof design elements it is sufficient to use formulas of material strength.
International Nuclear Information System (INIS)
A high-order particle-source-in-cell (PSIC) algorithm is presented for the computation of the interaction between shocks, small scale structures, and liquid and/or solid particles in high-speed engineering applications. The improved high-order finite difference weighted essentially non-oscillatory (WENO-Z) method for solution of the hyperbolic conservation laws that govern the shocked carrier gas flow, lies at the heart of the algorithm. Finite sized particles are modeled as points and are traced in the Lagrangian frame. The physical coupling of particles in the Lagrangian frame and the gas in the Eulerian frame through momentum and energy exchange, is numerically treated through high-order interpolation and weighing. The centered high-order interpolation of the fluid properties to the particle location is shown to lead to numerical instability in shocked flow. An essentially non-oscillatory interpolation (ENO) scheme is devised for the coupling that improves stability. The ENO based algorithm is shown to be numerically stable and to accurately capture shocks, small flow features and particle dispersion. Both the carrier gas and the particles are updated in time without splitting with a third-order Runge-Kutta TVD method. One and two-dimensional computations of a shock moving into a particle cloud demonstrates the characteristics of the WENO-Z based PSIC method (PSIC/WENO-Z). The PSIC/WENO-Z computations are not only in excellent agreement with the numerical simulatioent agreement with the numerical simulations with a third-order Rusanov based PSIC and physical experiments in [V. Boiko, V.P. Kiselev, S.P. Kiselev, A. Papyrin, S. Poplavsky, V. Fomin, Shock wave interaction with a cloud of particles, Shock Waves, 7 (1997) 275-285], but also show a significant improvement in the resolution of small scale structures. In two-dimensional simulations of the Mach 3 shock moving into forty thousand bronze particles arranged in the shape of a rectangle, the long time accuracy of the high-order method is demonstrated. The fifth-order PSIC/WENO-Z method with the fifth-order ENO interpolation scheme improves the small scale structure resolution over the third-order PSIC/WENO-Z method with a second-order central interpolation scheme. Preliminary analysis of the particle interaction with the flow structures shows that sharp particle material arms form on the side of the rectangular shape. The arms initially shield the particles from the accelerated flow behind the shock. A reflected compression wave, however, reshocks the particle arm from the shielded area and mixes the particles
The Korteweg-de Vries equation and its symmetry-preserving discretization
Bihlo, Alexander; Coiteux-Roy, Xavier; Winternitz, Pavel
2014-01-01
The Korteweg-de Vries equation is one of the most important nonlinear evolution equations in the mathematical sciences. In this article invariant discretization schemes are constructed for this equation both in the Lagrangian and in the Eulerian form. We also propose invariant schemes that preserve the momentum. Numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes. We find that the invariant discretization schemes...
International Nuclear Information System (INIS)
Highlights: ? Magnesium alloy AE42 was friction stir processed under different cooling conditions. ? Heat flow model was developed using finite difference heat equations. ? Generalized MATLAB code was developed for solving heat flow model. ? Regression equation for estimation of grain size was developed. - Abstract: The present investigation is aimed at developing a heat flow model to simulate temperature history during friction stir processing (FSP). A new approach of developing implicit form of finite difference heat equations solved using MATLAB code was used. A magnesium based alloy AE42 was friction stir processed (FSPed) at different FSP parameters and cooling conditions. Temperature history was continuously recorded in the nugget zone during FSP using data acquisition system and k type thermocouples. The developed code was validated at different FSP parameters and cooling conditions during FSP experimentation. The temperature history at different locations in the nugget zone at different instants of time was further utilized for the estimation of grain growth rate and final average grain size of the FSPed specimen. A regression equation relating the final grain size, maximum temperature during FSP and the cooling rate was developed. The metallurgical characterization was done using optical microscopy, SEM, and FIB-SIM analysis. The simulated temperature profiles and final average grain size were found to be in good agreement with the experimental resultement with the experimental results. The presence of fine precipitate particles generated in situ in the investigated magnesium alloy also contributed in the evolution of fine grain structure through Zener pining effect at the grain boundaries.
A three-point backward finite-difference method has been derived for a system of mixed hyperbolic¯¯parabolic (convection¯¯diffusion) partial differential equations (mixed PDEs). The method resorts to the three-point backward differenci...
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H. Ahmadi
2012-11-01
Full Text Available Solving Laplace equation ?2T = 0 using analytical methods is difficult, so numerical methods are used. One of the numerical methods for solving Laplace equation is finite difference method. We know that knotting and writing finite difference method for a specific body, eventually will give rise to linear algebraic equations. In this study, a new algorithm use for develop finite difference method for solving Laplace equation. In this algorithm, the temperature of the nodes of a specific figure quickly will be evaluated using finite difference method and the number of equations would be reducing significantly. By this method, a new formula for solving Laplace equation for a plate with four different constant temperature boundary conditions (Dirichlet condition derived.
Topalovi? Dušan B.; Pavlovi? Stefan; ?ukari? Nemanja A.; Z?, Tadic? Milan
2014-01-01
The finite-difference and finite-element methods are employed to solve the one-dimensional single-band Schrödinger equation in the planar and cylindrical geometries. The analyzed geometries correspond to semiconductor quantum wells and cylindrical quantum wires. As a typical example, the GaAs/AlGaAs system is considered. The approximation of the lowest order is employed in the finite-difference method and linear shape functions are employed in the finite-e...
Directory of Open Access Journals (Sweden)
S. S. Das, M. R. Saran, S. Mohanty, R. K. Padhy
2014-01-01
Full Text Available This paper focuses on the unsteady hydromagnetic mixed convective heat and mass transfer boundary layer flow of a viscous incompressible electrically conducting fluid past an accelerated infinite vertical porous flat plate in a porous medium with suction in presence of foreign species such as H2, He, H2O vapour and NH3. The governing equations are solved both analytically and numerically using error function and finite difference scheme. The flow phenomenon has been characterized with the help of flow parameters such as magnetic parameter (M, suction parameter (a, permeability parameter (Kp, Grashof number for heat and mass transfer (Gr, Gc, Schmidt number (Sc and Prandtl number (Pr. The effects of the above parameters on the fluid velocity, temperature, concentration distribution, skin friction and heat flux have been analyzed and the results are presented graphically and discussed quantitatively for Grashof number (Gr greater than 0 corresponding to cooling of the plate. It is observed that a growing magnetic parameter (M retards the velocity of the flow field at all points and a greater suction leads to a faster reduction in the velocity of the flow field. Further, as we increase the permeability parameter (Kp and the Grashof numbers for heat and mass transfer (Gr, Gc the velocity of the flow field enhances at all points, while a greater suction/Prandtl number leads to a faster cooling of the plate. It is also observed that a more diffusive species has a significant decrease in the concentration boundary layer of the flow field and a growing suction parameter enhances both skin friction (T’ and heat flux (Nu at the wall corresponding to cooling of the plate (Gr greater than 0.
Energy Technology Data Exchange (ETDEWEB)
Das, S.S. [Department of Physics, K.B.D.A.V. College, Nirakarpur, Khordha-752 019 (Odisha) (India); Saran, M.R. [Department of Physics, Maharishi College of Natural Law, Sahid Nagar, Bhubaneswar-751 007 (Odisha) (India); Mohanty, S. [Department of Chemistry, Christ College, Mission Road, Cuttack-753 001 (Odisha) (India); Padhy, R.K. [Department of Physics, ODM Public School, Shishu Vihar, Patia, Bhubaneswar-751 024 (Odisha) (India)
2013-07-01
This paper focuses on the unsteady hydromagnetic mixed convective heat and mass transfer boundary layer flow of a viscous incompressible electrically conducting fluid past an accelerated infinite vertical porous flat plate in a porous medium with suction in presence of foreign species such as H2, He, H2O vapour and NH3. The governing equations are solved both analytically and numerically using error function and finite difference scheme. The flow phenomenon has been characterized with the help of flow parameters such as magnetic parameter (M), suction parameter (a), permeability parameter (Kp), Grashof number for heat and mass transfer (Gr, Gc), Schmidt number (Sc) and Prandtl number (Pr). The effects of the above parameters on the fluid velocity, temperature, concentration distribution, skin friction and heat flux have been analyzed and the results are presented graphically and discussed quantitatively for Grashof number Gr>0 corresponding to cooling of the plate. It is observed that a growing magnetic parameter (M) retards the velocity of the flow field at all points and a greater suction leads to a faster reduction in the velocity of the flow field. Further, as we increase the permeability parameter (Kp) and the Grashof numbers for heat and mass transfer (Gr, Gc) the velocity of the flow field enhances at all points, while a greater suction/Prandtl number leads to a faster cooling of the plate. It is also observed that a more diffusive species has a significant decrease in the concentration boundary layer of the flow field and a growing suction parameter enhances both skin friction (T') and heat flux (Nu) at the wall corresponding to cooling of the plate (Gr>0).
Scalable Storage Scheme from Forward Key Rotation
Directory of Open Access Journals (Sweden)
Chunbo Ma
2008-01-01
Full Text Available The encryption scheme based on Forward Key Rotation is such a scheme that only the authorized person is allowed access to the designated files and the previous versions. In this study, we present a Forward Key Rotation storage scheme based on discrete logarithm and prove its security under random oracle model. Moreover, we propose another improved Forward Key storage scheme from pairing on elliptic curves. Compared to the scheme presented by Kallahalla et al. our scheme uses relatively short keys to provide equivalent security. In addition, the re-generated keys can be verified to ensure that the keys are valid in the improved scheme.
L1-stability of stationary discrete shocks
International Nuclear Information System (INIS)
The nonlinear stability in the Lp-norm, p ? 1, of stationary weak discrete shocks for the Lax-Friedrichs scheme approximating general m x m systems of nonlinear hyperbolic conservation laws is proved, provided that the summations of the initial perturbations equal zero. The result is proved by using a both weighted estimate and characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme. 13 refs
The multisymplectic diamond scheme
Mclachlan, R. I.; Wilkins, M. C.
2014-01-01
We introduce a class of general purpose linear multisymplectic integrators for Hamiltonian wave equations based on a diamond-shaped mesh. On each diamond, the PDE is discretized by a symplectic Runge--Kutta method. The scheme advances in time by filling in each diamond locally, leading to greater efficiency and parallelization and easier treatment of boundary conditions compared to methods based on rectangular meshes.
Energy Technology Data Exchange (ETDEWEB)
Aldridge, David Franklin; Collier, Sandra L. (U.S. Army Research Laboratory); Marlin, David H. (U.S. Army Research Laboratory); Ostashev, Vladimir E. (NOAA/Environmental Technology Laboratory); Symons, Neill Phillip; Wilson, D. Keith (U.S. Army Cold Regions Research Engineering Lab.)
2005-05-01
This document is intended to serve as a users guide for the time-domain atmospheric acoustic propagation suite (TDAAPS) program developed as part of the Department of Defense High-Performance Modernization Office (HPCMP) Common High-Performance Computing Scalable Software Initiative (CHSSI). TDAAPS performs staggered-grid finite-difference modeling of the acoustic velocity-pressure system with the incorporation of spatially inhomogeneous winds. Wherever practical the control structure of the codes are written in C++ using an object oriented design. Sections of code where a large number of calculations are required are written in C or F77 in order to enable better compiler optimization of these sections. The TDAAPS program conforms to a UNIX style calling interface. Most of the actions of the codes are controlled by adding flags to the invoking command line. This document presents a large number of examples and provides new users with the necessary background to perform acoustic modeling with TDAAPS.
Froese, Brittany D
2010-01-01
The elliptic Monge-Amp\\`ere equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. Novel solution methods are required for stability and convergence to weak solutions. In this article we build a monotone finite difference solver for the \\MA equation, which we prove converges to the weak (viscosity) solution. The resulting nonlinear equations are then solved by a damped Newton's method. We prove convergence and provide a close initial value for Newton's method. Computational results are presented in two and three dimensions, comparing solution time and accuracy to previous solvers using exact solutions which range in regularity from smooth to non-differentiable.
Zhang, Jitao
2014-01-01
The emission properties of terahertz(THz) photoconductive antenna (PCA) have been numerically studied by three-dimensional finite-difference-time-domain method based on the full-wave model. The dependence of the THz radiation on various parameters, such as laser power, bias voltage, substrate's material, pulse duration of the laser, beam spot's size, dimension of the antenna, were comprehensively simulated and analyzed. This work, on one hand, reveals the internal relationship between the THz radiation of a PCA and the involved parameters, so that one can have a better understanding of the PCA. On the other hand, it can inspire new PCA's design that aims at improved performance, such as high radiation power, enhanced optics-to-THz conversion efficiency, and broadband spectrum.
Potter, Michael E.; Okoniewski, Michal; Stuchly, Maria A.
2000-07-01
The implementation of a low frequency line source as a source function in the finite difference time domain (FDTD) method is presented. The total-scattered field formulation is employed, along with a recently developed quasi-static formulation of the FDTD. Line-source modeling is important in the utility industry, where a more accurate prediction of the fields induced in workers in close proximity to power lines is required. The line-source representation is verified, and excellent agreement with analytic solutions is found for two object problems. A practical example of the electric fields and current densities induced in a human body in close proximity to a 60-Hz transmission line is evaluated. The results for predicted organ dosimetry for such a configuration are compared with predictions for the uniform electric field and demonstrate the induced fields and current densities can be significantly higher than originally predicted for the uniform electric field exposure on a ground plane.
Marsden, O; Bogey, C; Bailly, C
2014-03-01
The feasibility of using numerical simulation of fluid dynamics equations for the detailed description of long-range infrasound propagation in the atmosphere is investigated. The two dimensional (2D) Navier Stokes equations are solved via high fidelity spatial finite differences and Runge-Kutta time integration, coupled with a shock-capturing filter procedure allowing large amplitudes to be studied. The accuracy of acoustic prediction over long distances with this approach is first assessed in the linear regime thanks to two test cases featuring an acoustic source placed above a reflective ground in a homogeneous and weakly inhomogeneous medium, solved for a range of grid resolutions. An atmospheric model which can account for realistic features affecting acoustic propagation is then described. A 2D study of the effect of source amplitude on signals recorded at ground level at varying distances from the source is carried out. Modifications both in terms of waveforms and arrival times are described. PMID:24606252
DEFF Research Database (Denmark)
Escolano-Carrasco, José; Jacobsen, Finn
2008-01-01
The finite-difference time-domain (FDTD) method provides a simple and accurate way of solving initial boundary value problems. However, most acoustic problems involve frequency dependent boundary conditions, and it is not easy to include such boundary conditions in an FDTD model. Although solutions to this problem exist, most of them have high computational costs, and stability cannot always be ensured. In this work, a solution is proposed based on "mixing modelling strategies"; this involves separating the FDTD mesh and the boundary conditions (a digital filter representation of the impedance) and combining them into a global solution. This solution is based on an interaction model that involves wave digital filters. The proposed method is validated with several test cases.
International Nuclear Information System (INIS)
The EU will supply the plasma position reflectometer for ITER. The system will have channels located at different poloidal positions, some of them obliquely viewing a plasma which has a poloidal density divergence and curvature, both adverse conditions for profile measurements. To understand the impact of such topology in the reconstruction of density profiles a full-wave two-dimensional finite-difference time domain O-mode code with the capability for frequency sweep was used. Simulations show that the reconstructed density profiles still meet the ITER radial accuracy specifications for plasma position (1 cm), except for the highest densities. Other adverse effects such as multireflections induced by the blanket, density fluctuations, and MHD activity were considered and a first understanding on their impact obtained.
Li, Xihao; Triverio, Piero
2014-01-01
The timestep of the Finite-Difference Time-Domain method (FDTD) is constrained by the stability limit known as the Courant-Friedrichs-Lewy (CFL) condition. This limit can make FDTD simulations quite time consuming for structures containing small geometrical details. Several methods have been proposed in the literature to extend the CFL limit, including implicit FDTD methods and filtering techniques. In this paper, we propose a novel approach which combines model order reduction and a perturbation algorithm to accelerate FDTD simulations beyond the CFL barrier. We compare the proposed algorithm against existing implicit and explicit CFL extension techniques, demonstrating increased accuracy and performance on a large number of test cases, including resonant cavities, a waveguide structure, a focusing metascreen and a microstrip filter.
Huang, Shi-Hao; Wang, Shiang-Jiu; Tseng, Snow H.
2015-03-01
Optical coherence tomography (OCT) provides high resolution, cross-sectional image of internal microstructure of biological tissue. We use the Finite-Difference Time-Domain method (FDTD) to analyze the data acquired by OCT, which can help us reconstruct the refractive index of the biological tissue. We calculate the refractive index tomography and try to match the simulation with the data acquired by OCT. Specifically, we try to reconstruct the structure of melanin, which has complex refractive indices and is the key component of human pigment system. The results indicate that better reconstruction can be achieved for homogenous sample, whereas the reconstruction is degraded for samples with fine structure or with complex interface. Simulation reconstruction shows structures of the Melanin that may be useful for biomedical optics applications.
da Silva, F; Heuraux, S
2008-10-01
The EU will supply the plasma position reflectometer for ITER. The system will have channels located at different poloidal positions, some of them obliquely viewing a plasma which has a poloidal density divergence and curvature, both adverse conditions for profile measurements. To understand the impact of such topology in the reconstruction of density profiles a full-wave two-dimensional finite-difference time domain O-mode code with the capability for frequency sweep was used. Simulations show that the reconstructed density profiles still meet the ITER radial accuracy specifications for plasma position (1 cm), except for the highest densities. Other adverse effects such as multireflections induced by the blanket, density fluctuations, and MHD activity were considered and a first understanding on their impact obtained. PMID:19044591
Singh, Paramjeet
2010-01-01
Explicit numerical methods based on Lax-Friedrichs and Leap-Frog finite difference approximations are constructed to find the numerical solution of the first-order hyperbolic partial differential equation with point-wise delay or advance, i.e., shift in space. The differential equation involving point-wise delay and advance models the distribution of the time intervals between successive neuronal firings. We construct higher order numerical approximations and discuss their consistency, stability and convergence. The numerical approximations constructed in this paper are consistent, stable under CFL condition, and convergent. We also extend our methods to the higher space dimensions. Some test examples are included to illustrate our approach. These examples verify the theoretical estimates and shows the effect of point-wise delay on the solution.
Directory of Open Access Journals (Sweden)
Gabbasov Radek Fatykhovich
Full Text Available Bending plate is widely used in the construction of large-span structures. Its advantage is light weight, industrial production, low cost and easy installation. Implementing the algorithm for calculating bending plates in engineering practice is an important issue of the construction science. The generalized equations of finite difference method is a new trend in the calculation of building construction. FDM with generalized equation provides additional options for an engineer along with other methods (FEM. In the article the algorithm for dynamic calculation of thin bending plates basing on FDM was developed. The computer programs for dynamic calculation were created on the basis of the algorithm. The authors come to the conclusion that the more simple equations of FDM can be used in case of solving the impulse load problems in dynamic load calculation of thin bending plate.
Energy Technology Data Exchange (ETDEWEB)
Arora, H.S. [School of Mechanical, Material and Energy Engineering, Indian Institute of Technology Ropar, Rupnagar, Punjab 140001 (India); Singh, H., E-mail: harpreetsingh@iitrpr.ac.in [School of Mechanical, Material and Energy Engineering, Indian Institute of Technology Ropar, Rupnagar, Punjab 140001 (India); Dhindaw, B.K. [School of Materials and Mineral Resources, Universiti Sains Malaysia, Engineering Campus, Nibong Tebal, Pulau Penang 14300 (Malaysia)
2012-05-01
Highlights: Black-Right-Pointing-Pointer Magnesium alloy AE42 was friction stir processed under different cooling conditions. Black-Right-Pointing-Pointer Heat flow model was developed using finite difference heat equations. Black-Right-Pointing-Pointer Generalized MATLAB code was developed for solving heat flow model. Black-Right-Pointing-Pointer Regression equation for estimation of grain size was developed. - Abstract: The present investigation is aimed at developing a heat flow model to simulate temperature history during friction stir processing (FSP). A new approach of developing implicit form of finite difference heat equations solved using MATLAB code was used. A magnesium based alloy AE42 was friction stir processed (FSPed) at different FSP parameters and cooling conditions. Temperature history was continuously recorded in the nugget zone during FSP using data acquisition system and k type thermocouples. The developed code was validated at different FSP parameters and cooling conditions during FSP experimentation. The temperature history at different locations in the nugget zone at different instants of time was further utilized for the estimation of grain growth rate and final average grain size of the FSPed specimen. A regression equation relating the final grain size, maximum temperature during FSP and the cooling rate was developed. The metallurgical characterization was done using optical microscopy, SEM, and FIB-SIM analysis. The simulated temperature profiles and final average grain size were found to be in good agreement with the experimental results. The presence of fine precipitate particles generated in situ in the investigated magnesium alloy also contributed in the evolution of fine grain structure through Zener pining effect at the grain boundaries.
Energy Technology Data Exchange (ETDEWEB)
Hickman, R.B.
1978-05-01
A diffusion equation modeling the flow of radiation was added to the hydrodynamic equations of two coupled Eulerian and Lagrangian finite-difference computer codes. This addition permits the extension of the range of problems to which these codes may be applied to include those in which temperatures on the order of a thousand electron volts are attained. The coupled codes are first-order-accurate shock hydrodynamics programs designed to calculate transient effects resulting from concentrations of high energy density. Such phenomena occur when a projectile impacts a target or when a high explosive is detonated. When the energy density is very high, as when a nuclear explosive is fired or a laser fusion pellet is imploded, radiation energy becomes a significant portion of the total energy and account must be taken of it. The diffusion approximation has proven to be a useful means of incorporating radiation physics in codes of this type. The three principal problems associated with the finite difference solution of the diffusion equation are the conservation of energy, the spatial differencing on grids that are becoming distorted with the passage of time, and the coupling of calculations done on the separate regional grids that together constitute the geometry of the problem. The difference techniques described are applied to the calculation of the prompt effects of an explosive detonated at the earth's surface. The explosive and the region of the earth more than 3 m from the explosive were zoned with Lagrangian coordinates. The air, the earth directly under the explosive, and a distant sink region were zoned in Eulerian coordinates. The calculation was carried out until most of the energy of the explosive was converted into kinetic energy and thermal energy in the air and earth.
Bases for a discrete special relativity
International Nuclear Information System (INIS)
Following recent developments in the hypothesis of a discrete space-time lattice, some assumptions are postulated that seem necessary to work out this model in the theory of special relativity. In particular, the assumption of space-time coordinates with integer values requires the translation of relativistic mechanics and electrodynamics into the language of finite difference equations. A special study of the covariance of these equations under the inhomogeneous Lorentz group is carried out. Finally, a stronger assumption is postulated, by which the physical magnitudes derived from the space-time coordinates should take rational values. (author)
Discrete spectrum analyses for various mixed discretizations of the Stokes eigenproblem
Evans, John A.; Hughes, Thomas J. R.
2012-12-01
We conduct discrete spectrum analyses for a selection of mixed discretization schemes for the Stokes eigenproblem. In particular, we consider the MINI element, the Crouzeix-Raviart element, the Marker-and-Cell scheme, the Taylor-Hood element, the {{Q}k/P_{k-1}} element, the divergence-conforming discontinuous Galerkin method, and divergence-conforming B-splines. For each of these schemes, we compare the spectrum for the continuous Stokes problem with the spectrum for the discrete Stokes problem, and we discuss the relationship of eigenvalue errors with solution errors associated with unsteady viscous flow problems.
Discrete exterior calculus for numerical simulation of meteor head-echo radar reflections
Räbinä, J.; Mönkölä, S.; Rossi, T.; Penttilä, A.; Markkanen, J.; Muinonen, K.
2014-07-01
The meteor head-echo feature has been studied by high-power large-aperture (HPLA) radars since 1960's (see Evans 1965). Based on the observations conducted by the different radar systems and post-processing techniques, there exist several models for the meteor head-echo simulations. One reason for this is the characteristics of the radar system, e.g., in terms of frequency and antenna geometry (see Kero et al. 2012). It is also worth mentioning that there are significant differences in the meteor sizes. According to the observations reported by, e.g., Vertatschitsch et al. (2011) and Wannberg et al. (2011), the head echo can be modeled as overdense scatter from a plasma layer, surrounding the meteor, with a certain density distribution. In these models, the plasmatic object is assumed to be a conducting spherical object, and the electromagnetic phenomenon can be presented by partial differential equations coupling the electric and magnetic fields. The traditional way of solving electromagnetic problems presented in space-time-domain as partial differential equations is to use the finite-difference time-domain method (FDTD; see Dyrud et al. 2008). In this study, we use more generalized finite differences by applying the discrete exterior calculus (DEC) to the numerical simulation of meteor head-echo radar reflections. The properties and calculus of differential forms is provided in a natural way at the discretization stage, and we associate the degrees of freedom of the electric and magnetic fields to the primal and dual mesh structures, respectively. The connection between the primal and dual forms is obtained by the discrete Hodge operator, the quality of which depends on the mesh construction. Our generalized formulation of the DEC for the Maxwell equations (see Pauly and Rossi 2011) works basically on unstructured grids, and it covers both the classical Yee's FDTD scheme and the Bossavit-Kettunen approach (Bossavit and Kettunen 1999). The method has been shown to give promising results with both time-dependent (Räbinä et al. 2014a) and time-harmonic problems (Räbinä et al. 2014b).
Central schemes for porous media flows
Abreu, E.; Pereira, F.; Ribeiro, S.
2009-01-01
We are concerned with central differencing schemes for solving scalar hyperbolic conservation laws arising in the simulation of multiphase flows in heterogeneous porous media. We compare the Kurganov-Tadmor (KT) [3] semi-discrete central scheme with the Nessyahu-Tadmor (NT) [27] central scheme. The KT scheme uses more precise information about the local speeds of propagation together with integration over nonuniform control volumes, which contain the Riemann fans. These methods can accurately...
Energy Technology Data Exchange (ETDEWEB)
Gomez T, A.M.; Valle G, E. del [IPN-ESFM, 07738 Mexico D.F. (Mexico); Delfin L, A.; Alonso V, G. [ININ, 52045 Ocoyoacac, Estado de Mexico (Mexico)] e-mail: armagotorres@aol.com
2003-07-01
In this work a finite differences technique centered in mesh based on a cubic reduced nodal scheme type finite element to solve the equations of the kinetics 1 D that include the equations corresponding to the concentrations of precursors of delayed neutrons is described. The technique of finite elements used is that of Galerkin where so much the neutron flux as the concentrations of precursors its are spatially approached by means of a three grade polynomial. The matrices of rigidity and of mass that arise during this discretization process are numerically evaluated using the open quadrature non standard of Newton-Cotes and that of Radau respectively. The purpose of the application of these quadratures is the one of to eliminate in the global matrices the couplings among the values of the flow in points of the discretization with the consequent advantages as for the reduction of the order of the matrix associated to the discreet problem that is to solve. As for the time dependent part the classical integration scheme known as {theta} scheme is applied. After carrying out the one reordering of unknown and equations it arrives to a reduced system that it can be solved but quickly. With the McKin compute program developed its were solved three benchmark problems and those results are shown for the relative powers. (Author)
Gridsize induced error in the discretization of exchange processes at the tropopause
Scientific Electronic Library Online (English)
M., FIEBIG-WITTMAACK.
2005-07-01
Full Text Available Estudiamos el error introducido por el método de las diferencias finitas en la discretización de un modelo global simplificado 2-D de transporte de gases traza, para casos en que los coeficientes de difusión, que relacionan el flujo con el gradiente de la razón de mezcla, tienen discontinuidades de [...] salto en la tropopausa. Analizamos el método convencional de celdas tanto para el caso de un flujo ascendente típico, como también para el caso de un flujo descendente típico con reacciones químicas, comparando las aproximaciones de las soluciones correspondientes para diferentes tamaños de paso de discretización. Para el flujo descendente típico resulta que si la rejilla no es suficientemente fina se pueden generar grandes errores; estos se propagan fundamentalmente en la troposfera. En cambio, el flujo típicamente ascendente resulta ser relativamente insensible al tamaño de paso de la discretización. Abstract in english We study the accuracy of the finite differences discretization scheme for a 2-D simplified model of global tracer transport, in the case that the diffusion coefficients relating flux to the gradient of the mixing ratio have discontinuity jumps at the tropopause. We analyze the conventional box metho [...] d for a typical downward flow with chemical reaction and for a typical upward flow, comparing the approximations of the solutions, for different discretization gridsizes. It turns out that the jumps may introduce remarkable errors in the discrete solutions, in the case of a typical downward flow; these errors propagate mainly into the troposphere. A noticeable improvement is achieved by reducing the gridsize. However, a typical upward flow is rather insensitive to the chosen gridsizes.
Scientific Electronic Library Online (English)
Carlos, Chávez; Carlos, Fuentes; Manuel, Zavala; Felipe, Zataráin.
2011-12-01
Full Text Available El drenaje subterráneo es utilizado para eliminar excedentes de agua en la zona radical y suelos salinos para lixiviar las sales. La dinámica del agua es estudiada con la ecuación de Boussinesq, sus soluciones analíticas son obtenidas asumiendo que la transmisibilidad del acuífero y la porosidad dre [...] nable son constantes y que la superficie libre se abate de manera instantánea sobre los drenes. La solución en el caso general requiere de soluciones numéricas. Se ha mostrado que la condición de frontera en los drenes es una condición de radiación fractal y la porosidad drenable es variable y relacionada con la curva de retención de humedad, y ha sido resuelta con el método del elemento finito, que en un esquema unidimensional puede hacerse equivalente al método de diferencias finitas. Aquí se propone una solución en diferencias finitas de la ecuación diferencial considerando la porosidad drenable variable y la condición de radiación fractal. El esquema en diferencias finitas propuesto ha resultado en dos formulaciones: en una aparecen de manera explícita la carga y la porosidad drenable, variables ligadas con una relación funcional, que se ha denominado esquema mixto; en la otra aparece sólo la carga hidráulica, denominada esquema en carga. Los dos esquemas coinciden cuando la porosidad drenable es independiente de la carga. Los esquemas han sido validados con una solución analítica lineal, y para la no linealidad se ha mostrado que la convergencia numérica es estable y concisa. La solución numérica es útil para la caracterización hidrodinámica del suelo a través de una modelación inversa, y para un mejor diseño de los sistemas de drenaje agrícola subterráneo ya que las hipótesis consideradas en las soluciones clásicas han sido eliminadas. Abstract in english The underground drainage is used to remove excess water in the root zone and in saline soils to leach salts. The dynamics of water is studied with the Boussinesq equation; its analytical solutions are obtained assuming that the aquifer transmissivity and drainable porosity are constants and that the [...] free surface instantly lowers on the drains. The solution in the general case requires numerical solution. It has been shown that the boundary condition in the drains is a fractal radiation condition and the drainable porosity is a variable and is related to the moisture retention curve, and has been solved with the finite-element method, which in one-dimensional scheme can become equivalent to the finite-difference method. It is proposed here a finite difference solution of the differential equation considering the variable drainable porosity and fractal radiation condition. The proposed finite difference scheme has resulted in two formulations: in one the head and drainable porosity explicitly appear, variables linked to a functional relationship, which has been called mixed scheme; in the other only the hydraulic head appears, called head scheme. The two schemes coincide when the drainable porosity is independent of the head. The schemes have been validated with a linear analytical solution; for the nonlinearity has been shown that the numerical convergence is stable and concise. The numerical solutions is useful for the hydrodynamic characterization of the soil through an inverse modeling, and for a better design of the agricultural underground drainage systems as the assumptions used in the classical solutions have been eliminated.
International Nuclear Information System (INIS)
This paper analyzes the convergence of the rebalance methods (e.g., Coarse-Mesh Rebalance (CMR), Coarse-Mesh Finite Difference (CMFD), and p-CMFD for accelerating the power iteration method of the discrete ordinates transport equation in the eigenvalue problem. The convergence analysis is done with the well-known Fourier analysis method through a linearization both for spatially continuous and discretized forms of one and two energy group transport equations in an infinite medium
S?onka, ?ukasz
2015-04-01
The Szamotu?y salt diapir, located in NW Poland, was formed within the Mid-Polish Trough - Permo-Mesozoic sedimentary basin that was inverted during the Late Cretaceous to Paleogene. The Szamotu?y diapir was sourced from the Upper Permian (Zechstein) evaporites. Tectonic evolution during extension and compression resulted in a very complex geometry of the salt body and surrounding Mesozoic sedimentary sequences. This, together with significant velocity contrasts between the Zechstein evaporites and the Mesozoic siliciclastic - carbonate sequences resulted in complicated seismic ray paths that hampered seismic imaging of the salt diapir and supra-salt and sub-salt sedimentary strata. The aim of this work was to apply and test selected seismic imaging methods (pre- and post-stack time migrations) for improved imaging of the salt diapir, including its steep and overhang walls, salt wings etc. The study was carried out by using theoretical waveform which was calculated using finite-difference method. The synthetic data were based on the geological depth model which was constructed using results of interpretation of the regional 2-D seismic profile crossing Szamotu?y diapir zone. Additional information was provided by well logs from several wells drilled in the study area. The seismic modelling was completed using Gedco's Omni software. Before performing the full waveform modelling, the offset modelling based on seismic ray theory (seismic ray tracing) was carried out. The results of the offset modelling were used to design the survey methodology of the synthetic seismic profile and to plan the most effective processing workflow. Then, synthetic common-shot gathers were generated using a finite-difference solution of the 2-D wave equation in acoustic variant. Finite-difference modelling results were subsequently used during the next processing stage of that study which was done using Gedco's Vista Seismic Processing software. Important part of this work was multiples attenuation using predictive deconvolution in tau-p domain and iterative seismic velocity analysis before the PreSTM. Tested time migration techniques included Kirchhoff pre- and post-stack migration and post-stack FD migration. Finally, the comparison of the migration results was carried out, especially the analysis of imaging quality of the steep salt diapir flanks, salt overhangs and wings, and Lower Permian (Rotliegendes) and older sub-salt sedimentary sequences. The final results proved that the quality of the time imaging in that area strongly depends on the acquisition parameters, particularly offsets distribution and migration aperture distance. The elimination of the selected multiple waves at the processing stage (especially attenuation of intra-bed multiples) may be a strong challenge for the further studies which may improve the quality of time imaging using real data, and then depth imaging and more advanced seismic interpretation of the Szamotu?y salt diapir.
DEFF Research Database (Denmark)
SØrensen, John Aasted
The objectives of Discrete Mathematics (IDISM2) are: The introduction of the mathematics needed for analysis, design and verification of discrete systems, including the application within programming languages for computer systems. Having passed the IDISM2 course, the student will be able to accomplish the following: -Understand and apply formal representations in discrete mathematics. -Understand and apply formal representations in problems within discrete mathematics. -Understand methods for solving problems in discrete mathematics. -Apply methods for solving problems in discrete mathematics. Having completed this the student is able to carry out the following: Expressions and sets: Define a set; define a logic expression; negate a logic expression; combine logic expressions; construct a truth table for a logic expression; apply reduction rules for logic expressions. Apply these concepts to new problems. Relations and functions: Define a product set; define and apply equivalence relations; construct and apply functions. Apply these concepts to new problems. Natural numbers and induction: Define the natural numbers; apply the principle of induction to verify a selection of properties of natural numbers. Apply these concepts to new problems. Division and factorizing: Define a prime number and apply Euclid´s algorithm for factorizing an integer. Regular languages: Define a language from the elements of a set; define a regular language; form strings from a regular language; construct examples on regular languages. Apply these concepts to new problems. Finite state machines: Define a finite state machine as a 6-tuble; describe simple finite state machines by tables and graphs; pattern recognition by finite state machines; minimizing the number of states in a finite state machine; construct a finite state machine for a given application. Apply these concepts to new problems. The teaching in Discrete Mathematics is a combination of sessions with lectures and students solving problems, either manually or by using Matlab. Furthermore a selection of projects must be solved and handed in during the course. Semester: F2011 Extent: 5 ects
Stabilizability of discrete chaotic systems via unified impulsive control
Xu, Honglei; Teo, Kok Lay
2009-12-01
This Letter is concerned with the asymptotical stabilization problem of discrete chaotic systems by using a novel unified impulsive control scheme. Sufficient conditions for asymptotical stability of the impulsive controlled discrete systems are obtained by means of the Lyapunov stability theory and algebraic inequality techniques. Finally, numerical simulations on the Hénon and Ushio discrete chaotic systems are presented to illustrate the effectiveness and usefulness of the unified impulsive control scheme.
International Nuclear Information System (INIS)
This paper deals with the functional performance of optical surface texture measuring instruments on the market. It is well known that their height response curves against certain referential geometry are not always identical to each other. So, a more precise study on the optical instrument's characteristics is greatly needed. Firstly, we developed a new simulation tool using a finite-difference time-domain technique, which enables the prediction of the height response curve against the fundamental surface geometry in the case of the confocal laser scanning microscope. Secondly, by utilizing this new simulation tool, measurement results, including outliers, were compared with the analytical simulation results. The comparison showed the consistency, which indicates that necessary conditions of surface measurement standards for verifying the instrument performance can be established. Consequently, we suggest that the maximum measurable slope angle must be added to evaluation subjects as significant metrological characteristics of measuring instruments, along with the lateral period limit. Finally, we propose a procedure to determine the lateral period limit in an ISO standard. (paper)
Bohling, G.C.; Butler, J.J., Jr.
2001-01-01
We have developed a program for inverse analysis of two-dimensional linear or radial groundwater flow problems. The program, 1r2dinv, uses standard finite difference techniques to solve the groundwater flow equation for a horizontal or vertical plane with heterogeneous properties. In radial mode, the program simulates flow to a well in a vertical plane, transforming the radial flow equation into an equivalent problem in Cartesian coordinates. The physical parameters in the model are horizontal or x-direction hydraulic conductivity, anisotropy ratio (vertical to horizontal conductivity in a vertical model, y-direction to x-direction in a horizontal model), and specific storage. The program allows the user to specify arbitrary and independent zonations of these three parameters and also to specify which zonal parameter values are known and which are unknown. The Levenberg-Marquardt algorithm is used to estimate parameters from observed head values. Particularly powerful features of the program are the ability to perform simultaneous analysis of heads from different tests and the inclusion of the wellbore in the radial mode. These capabilities allow the program to be used for analysis of suites of well tests, such as multilevel slug tests or pumping tests in a tomographic format. The combination of information from tests stressing different vertical levels in an aquifer provides the means for accurately estimating vertical variations in conductivity, a factor profoundly influencing contaminant transport in the subsurface. ?? 2001 Elsevier Science Ltd. All rights reserved.
Mori, Kazuyoshi; Miyazaki, Ayano; Ogasawara, Hanako; Nakamura, Toshiaki; Takeuchi, Yasuhito
2007-07-01
A convex lens using room temperature vulcanization (RTV) silicone rubber, whose acoustic impedance matches well with that of water, is a typical acoustic lens. However, some considerations are required to reduce the thickness of the lens because attenuation is very large in the RTV silicone rubber. Therefore, we have proposed a phase continuous Fresnel lens, which has some devices to keep the phase continuous in the entire lens aperture without an unequal phase on the exit side. This lens is fabricated by removing its thickness in a staircase shape in which the difference between each step is an integer multiple of wavelength. In this study, the sound fields focused by the phase continuous Fresnel lens are analyzed using a finite difference time domain (FDTD) method. Using a two-dimensional (2-D) FDTD method, we surveyed the sound pressure field of the focal region by changing the burst pulse length, angle of incidence, and frequency. Results show that the lens gain of the phase continuous Fresnel lens is greater than that of the convex lens, that focusing characteristics depend on the burst pulse length of sound source signal, and also that focal points strongly depend on frequency. In another analysis using a three-dimensional FDTD method, we found that the main lobe is the same as that indicated by 2-D analysis results and that the level outer of the main lobe is lower than that in the 2-D analysis results.
International Nuclear Information System (INIS)
A thermionic energy converter (TEC) is a static device that converts heat directly into electricity by boiling electrons off a hot emitter surface across a small inter-electrode gap to a cooler collector surface. The main challenge in TECs is overcoming the space charge limit, which limits the current transmitted across a gap of a given voltage and width. We have verified the feasibility of studying and developing a TEC using a bounded finite-difference time-domain particle-in-cell plasma simulation code, OOPD1, developed by Plasma Theory and Simulation Group, formerly at UC Berkeley and now at Michigan State University. In this preliminary work, a TEC has been modeled kinetically using OOPD1, and the accuracy has been verified by comparing with an analytically solvable case, giving good agreement. With further improvement of the code, one will be able to quickly and cheaply analyze space charge effects, and seek designs that mitigate the space charge effect, allowing TECs to become more efficient and cost-effective
Berezkin, Anatoly V; Kudryavtsev, Yaroslav V
2013-10-21
A novel hybrid approach combining dissipative particle dynamics (DPD) and finite difference (FD) solution of partial differential equations is proposed to simulate complex reaction-diffusion phenomena in heterogeneous systems. DPD is used for the detailed molecular modeling of mass transfer, chemical reactions, and phase separation near the liquid?liquid interface, while FD approach is applied to describe the large-scale diffusion of reactants outside the reaction zone. A smooth, self-consistent procedure of matching the solute concentration is performed in the buffer region between the DPD and FD domains. The new model is tested on a simple model system admitting an analytical solution for the diffusion controlled regime and then applied to simulate practically important heterogeneous processes of (i) reactive coupling between immiscible end-functionalized polymers and (ii) interfacial polymerization of two monomers dissolved in immiscible solvents. The results obtained due to extending the space and time scales accessible to modeling provide new insights into the kinetics and mechanism of those processes and demonstrate high robustness and accuracy of the novel technique. PMID:24160495
Berezkin, Anatoly V.; Kudryavtsev, Yaroslav V.
2013-10-01
A novel hybrid approach combining dissipative particle dynamics (DPD) and finite difference (FD) solution of partial differential equations is proposed to simulate complex reaction-diffusion phenomena in heterogeneous systems. DPD is used for the detailed molecular modeling of mass transfer, chemical reactions, and phase separation near the liquid/liquid interface, while FD approach is applied to describe the large-scale diffusion of reactants outside the reaction zone. A smooth, self-consistent procedure of matching the solute concentration is performed in the buffer region between the DPD and FD domains. The new model is tested on a simple model system admitting an analytical solution for the diffusion controlled regime and then applied to simulate practically important heterogeneous processes of (i) reactive coupling between immiscible end-functionalized polymers and (ii) interfacial polymerization of two monomers dissolved in immiscible solvents. The results obtained due to extending the space and time scales accessible to modeling provide new insights into the kinetics and mechanism of those processes and demonstrate high robustness and accuracy of the novel technique.
Ostashev, Vladimir E; Wilson, D Keith; Liu, Lanbo; Aldridge, David F; Symons, Neill P; Marlin, David
2005-02-01
Finite-difference, time-domain (FDTD) calculations are typically performed with partial differential equations that are first order in time. Equation sets appropriate for FDTD calculations in a moving inhomogeneous medium (with an emphasis on the atmosphere) are derived and discussed in this paper. Two candidate equation sets, both derived from linearized equations of fluid dynamics, are proposed. The first, which contains three coupled equations for the sound pressure, vector acoustic velocity, and acoustic density, is obtained without any approximations. The second, which contains two coupled equations for the sound pressure and vector acoustic velocity, is derived by ignoring terms proportional to the divergence of the medium velocity and the gradient of the ambient pressure. It is shown that the second set has the same or a wider range of applicability than equations for the sound pressure that have been previously used for analytical and numerical studies of sound propagation in a moving atmosphere. Practical FDTD implementation of the second set of equations is discussed. Results show good agreement with theoretical predictions of the sound pressure due to a point monochromatic source in a uniform, high Mach number flow and with Fast Field Program calculations of sound propagation in a stratified moving atmosphere. PMID:15759672
Notaros, Jelena; Popovi?, Miloš A
2015-03-15
We demonstrate a finite-difference approach to complex-wavevector band structure simulation and its use as a tool for the analysis and design of periodic leaky-wave photonic devices. With the (usually real) operating frequency and unit-cell refractive index distribution as inputs, the eigenvalue problem yields the complex-wavevector eigenvalues and Bloch modes of the simulated structure. In a two-dimensional implementation for transverse-electric fields with radiation accounted for by perfectly matched layer boundaries, we validate the method and demonstrate its use in simulating the complex-wavevector band structures and modal properties of a silicon photonic crystal waveguide, an array-antenna-inspired grating coupler with unidirectional radiation, and a recently demonstrated low-loss Bloch-mode-based waveguide crossing array. Additionally, we show the first direct solution of the recently proposed open-system low-loss Bloch modes. We expect this method to be a valuable tool in photonics design, enabling the rigorous analysis and synthesis of advanced periodic and quasi-periodic photonic devices. PMID:25768180