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.
International Nuclear Information System (INIS)
Most of thermal hydraulic processes in nuclear engineering can be described by general convection-diffusion equations that are often can be simulated numerically with finite-difference method (FDM). An effective scheme for finite-difference discretization of such equations is presented in this report. The derivation of this scheme is based on analytical solutions of a simplified one-dimensional equation written for every control volume of the finite-difference mesh. These analytical solutions are constructed using linearized representations of both diffusion coefficient and source term. As a result, the Efficient Finite-Differencing (EFD) scheme makes it possible to significantly improve the accuracy of numerical method even using mesh systems with fewer grid nodes that, in turn, allows to speed-up numerical simulation. EFD has been carefully verified on the series of sample problems for which either analytical or very precise numerical solutions can be found. EFD has been compared with other popular FDM schemes including novel, accurate (as well as sophisticated) methods. Among the methods compared were well-known central difference scheme, upwind scheme, exponential differencing and hybrid schemes of Spalding. Also, newly developed finite-difference schemes, such as the the quadratic upstream (QUICK) scheme of Leonard, the locally analytic differencing (LOAD) scheme of Wong and Raithby, the flux-spline scheme proposed by Varejago and Patankar as well as the latest LENS discretization of Sakai have been compared. Detailed results of this comparison are given in this report. These tests have shown a high efficiency of the EFD scheme. For most of sample problems considered EFD has demonstrated the numerical error that appeared to be in orders of magnitude lower than that of other discretization methods. Or, in other words, EFD has predicted numerical solution with the same given numerical error but using much fewer grid nodes. In this report, the detailed description of EFD is given. It includes basic assumptions, the detailed derivation, the verification procedure, as well verification and comparisons. Conclusion summarizes results and highlights the problems to be solved. (author)
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
A free surface capturing discretization for the staggered grid finite difference scheme
Duretz, T.; May, D. A.; Yamato, P.
2016-03-01
The coupling that exists between surface processes and deformation within both the shallow crust and the deeper mantle-lithosphere has stimulated the development of computational geodynamic models that incorporate a free surface boundary condition. We introduce a treatment of this boundary condition that is suitable for staggered grid, finite difference schemes employing a structured Eulerian mesh. Our interface capturing treatment discretizes the free surface boundary condition via an interface that conforms with the edges of control volumes (e.g. a `staircase' representation) and requires only local stencil modifications to be performed. Comparisons with analytic solutions verify that the method is first-order accurate. Additional intermodel comparisons are performed between known reference models to further validate our free surface approximation. Lastly, we demonstrate the applicability of a multigrid solver to our free surface methodology and demonstrate that the local stencil modifications do not strongly influence the convergence of the iterative solver.
Nonstandard finite difference schemes
Mickens, Ronald E.
1995-01-01
The major research activities of this proposal center on the construction and analysis of nonstandard finite-difference schemes for ordinary and partial differential equations. In particular, we investigate schemes that either have zero truncation errors (exact schemes) or possess other significant features of importance for numerical integration. Our eventual goal is to bring these methods to bear on problems that arise in the modeling of various physical, engineering, and technological systems. At present, these efforts are extended in the direction of understanding the exact nature of these nonstandard procedures and extending their use to more complicated model equations. Our presentation will give a listing (obtained to date) of the nonstandard rules, their application to a number of linear and nonlinear, ordinary and partial differential equations. In certain cases, numerical results will be presented.
Upwind Compact Finite Difference Schemes
Christie, I.
1985-07-01
It was shown by Ciment, Leventhal, and Weinberg ( J. Comput. Phys.28 (1978), 135) that the standard compact finite difference scheme may break down in convection dominated problems. An upwinding of the method, which maintains the fourth order accuracy, is suggested and favorable numerical results are found for a number of test problems.
Conservative properties of finite difference schemes for incompressible flow
Morinishi, Youhei
1995-01-01
The purpose of this research is to construct accurate finite difference schemes for incompressible unsteady flow simulations such as LES (large-eddy simulation) or DNS (direct numerical simulation). In this report, conservation properties of the continuity, momentum, and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discretized equations. Existing finite difference schemes in staggered grid systems are checked for satisfaction of the requirements. Proper higher order accurate finite difference schemes in a staggered grid system are then proposed. Plane channel flow is simulated using the proposed fourth order accurate finite difference scheme and the results compared with those of the second order accurate Harlow and Welch algorithm.
TVD finite difference schemes and artificial viscosity
Davis, S. F.
1984-01-01
The total variation diminishing (TVD) finite difference scheme can be interpreted as a Lax-Wendroff scheme plus an upwind weighted artificial dissipation term. If a particular flux limiter is chosen and the requirement for upwind weighting is removed, an artificial dissipation term which is based on the theory of TVD schemes is obtained which does not contain any problem dependent parameters and which can be added to existing MacCormack method codes. Numerical experiments to examine the performance of this new method are discussed.
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.
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
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.
International Nuclear Information System (INIS)
A new finite difference scheme on a non-uniform staggered grid in cylindrical coordinates is proposed for incompressible flow. The scheme conserves both momentum and kinetic energy for inviscid flow with the exception of the time marching error, provided that the discrete continuity equation is satisfied. A novel pole treatment is also introduced, where a discrete radial momentum equation with the fully conservative convection scheme is introduced at the pole. The pole singularity is removed properly using analytical and numerical techniques. The kinetic energy conservation property is tested for the inviscid concentric annular flow for the proposed and existing staggered finite difference schemes in cylindrical coordinates. The pole treatment is verified for inviscid pipe flow. Mixed second and high order finite difference scheme is also proposed and the effect of the order of accuracy is demonstrated for the large eddy simulation of turbulent pipe flow
Compact finite difference schemes with spectral-like resolution
Lele, Sanjiva K.
1992-01-01
The present finite-difference schemes for the evaluation of first-order, second-order, and higher-order derivatives yield improved representation of a range of scales and may be used on nonuniform meshes. Various boundary conditions may be invoked, and both accurate interpolation and spectral-like filtering can be accomplished by means of schemes for derivatives at mid-cell locations. This family of schemes reduces to the Pade schemes when the maximal formal accuracy constraint is imposed with a specific computational stencil. Attention is given to illustrative applications of these schemes in fluid dynamics.
Fulai Chen; Li Ren
2014-01-01
A new finite difference scheme, the development of the finite difference heterogeneous multiscale method (FDHMM), is constructed for simulating saturated water flow in random porous media. In the discretization framework of FDHMM, we follow some ideas from the multiscale finite element method and construct basic microscopic elliptic models. Tests on a variety of numerical experiments show that, in the case that only about a half of the information of the whole microstructure is used, the cons...
Finite element, discontinuous Galerkin, and finite difference evolution schemes in spacetime
International Nuclear Information System (INIS)
Numerical schemes for Einstein's vacuum equation are developed. Einstein's equation in harmonic gauge is second-order symmetric hyperbolic. It is discretized in four-dimensional spacetime by finite differences, finite elements and interior penalty discontinuous Galerkin methods, the latter being related to Regge calculus. The schemes are split into space and time and new time-stepping schemes for wave equations are derived. The methods are evaluated for linear and nonlinear test problems of the Apples-with-Apples collection.
To the convergence ov finite difference schemes on the generalized solutions of the Poisson equation
International Nuclear Information System (INIS)
The rate of convergence of the finite difference schemes on the generalized solutions of Poisson equation is studied by the energy inequality method. The truncation error is analyzed using Bramle-Hilbert lemma. It is proved that commonly used life-point difference scheme for the Direchlet boundary value problem converges with the rate of O(hsup(1+s)) in a discrete Ws21 norm, and with the rate of O(hsup(1+s)|lnh|sup(1/2)) in discrete C-norm, if the solution is from Wsub(2)sup(2+s), s=0.1
Generalized finite-difference time-domain schemes for solving nonlinear Schrodinger equations
Moxley, Frederick Ira, III
The nonlinear Schrodinger equation (NLSE) is one of the most widely applicable equations in physical science, and characterizes nonlinear dispersive waves, optics, water waves, and the dynamics of molecules. The NLSE satisfies many mathematical conservation laws. Moreover, due to the nonlinearity, the NLSE often requires a numerical solution, which also satisfies the conservation laws. Some of the more popular numerical methods for solving the NLSE include the finite difference, finite element, and spectral methods such as the pseudospectral, split-step with Fourier transform, and integrating factor coupled with a Fourier transform. With regard to the finite difference and finite element methods, higher-order accurate and stable schemes are often required to solve a large-scale linear system. Conversely, spectral methods via Fourier transforms for space discretization coupled with Runge-Kutta methods for time stepping become too complex when applied to multidimensional problems. One of the most prevalent challenges in developing these numerical schemes is that they satisfy the conservation laws. The objective of this dissertation was to develop a higher-order accurate and simple finite difference scheme for solving the NLSE. First, the wave function was split into real and imaginary components and then substituted into the NLSE to obtain coupled equations. These components were then approximated using higher-order Taylor series expansions in time, where the derivatives in time were replaced by the derivatives in space via the coupled equations. Finally, the derivatives in space were approximated using higher-order accurate finite difference approximations. As such, an explicit and higher order accurate finite difference scheme for solving the NLSE was obtained. This scheme is called the explicit generalized finite-difference time-domain (explicit G-FDTD). For purposes of completeness, an implicit G-FDTD scheme for solving the NLSE was also developed. In this dissertation, the discrete energy method is employed to prove that both the explicit and implicit G-FDTD scheme satisfy the discrete analogue form of the first conservation law. To verify the accuracy of the numerical solution and the applicability of the schemes, both schemes were tested by simulating bright and dark soliton propagation and collision in one and two dimensions. Compared with other popular existing methods (e.g., pseudospectral, split-step, integrating factor), numerical results showed that the G-FDTD method provides a more accurate solution, particularly when the time step is large. This solution is particularly important during the long-time period simulations. The explicit G-FDTD method proved to be advantageous in that it was simple and fast in computation. Furthermore, the G-FDTD showed that the solution propagates through the boundary with analytical solution continuation.
A Pseudo-compact Conservative Average Finite Difference Scheme for Dissipation SRLW Eqation
Directory of Open Access Journals (Sweden)
ZHENG Mao-bo
2014-01-01
Full Text Available We study the initial-boundary problem of the dissipative SRLWE by finite difference method. Using pseudo-compact difference scheme constructed thinking; a new three-level conservative average finite difference scheme containing the pseudo-com-pact items * is designed. Then we analyze the discrete conservation properties for the new scheme and simulate two con-Served properties of the problem well. The scheme is linearized and does not require iteration, so it is expected to be more efficient. And the prior estimate of the solution is obtained. It is shown that the finite difference scheme is second-order convergence and un-Conditionally stable. Finally, the results of numerical experiments comparing with existing scheme show that the new scheme will Not only maintain the characteristics of a small amount of calculation and also make calculations with higher precision. At the same time the second-order convergence and conservation properties of the scheme is verified.(* represents formula
The geometry of finite difference discretizations of semilinear elliptic operators
International Nuclear Information System (INIS)
Discretizations by finite differences of some semilinear elliptic equations lead to maps F(u) = Au - f(u), u ? Rn, given by nonlinear convex diagonal perturbations of symmetric matrices A. For natural nonlinearity classes, we consider the equation F(u) = y ? tp, where t is a large positive number and p is a vector with negative coordinates. As the range of the derivative f'i of the coordinates of f encloses more eigenvalues of A, the number of solutions increases geometrically, eventually reaching 2n. This phenomenon, somewhat in contrast with behaviour associated with the LazerMcKenna conjecture, has a very simple geometric explanation: a perturbation of a multiple fold gives rise to a function which sends connected components of its critical set to hypersurfaces with large rotation numbers with respect to vectors with very negative coordinates. Strictly speaking, the results have nothing to do with elliptic equations: they are properties of the interaction of a (self-adjoint) linear map with increasingly stronger nonlinear convex diagonal interactions
Discretizing delta functions via finite differences and gradient normalization
Towers, John D.
2009-06-01
In [J.D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys. 220 (2007) 915-931] the author presented two closely related finite difference methods (referred to here as FDM1 and FDM2) for discretizing a delta function supported on a manifold of codimension one defined by the zero level set of a smooth mapping u :Rn ↦ R . These methods were shown to be consistent (meaning that they converge to the true solution as the mesh size h → 0) in the codimension one setting. In this paper, we concentrate on n ⩽ 3 , but generalize our methods to codimensions other than one - now the level set function is generally a vector valued mapping u → :Rn ↦Rm, 1 ⩽ m ⩽ n ⩽ 3 . Seemingly reasonable algorithms based on simple products of approximate delta functions are not generally consistent when applied to these problems. Motivated by this, we instead use the wedge product formalism to generalize our FDM algorithms, and this approach results in accurate, often consistent approximations. With the goal of ensuring consistency in general, we propose a new gradient normalization process that is applied before our FDM algorithms. These combined algorithms seem to be consistent in all reasonable situations, with numerical experiments indicating O (h2) convergence for our new gradient-normalized FDM2 algorithm. In the full codimension setting (m = n) , our gradient normalization processing also improves accuracy when using more standard approximate delta functions. This combination also yields approximations that appear to be consistent.
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.
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...
Approximation of systems of partial differential equations by finite difference schemes
International Nuclear Information System (INIS)
The approximation of Friedrichs' symmetric systems by a finite difference scheme with second order accuracy with respect to the step of discretization is studied. Unconditional stability of such a scheme is proved by the method of energy increase. This implicit scheme is then solved by three iterative methods: the first one, of the gradient type, converges slowly, the second one, of the Gauss-Seidel type, converges only if the system has been regularized to the first order with respect to the step of discretization by an elliptic operator, the last one, of the under-relaxation type, converges rapidly to a second order accurate solution. Explicit schemes for the integration of linear hyperbolic systems of evolution are considered. Conditional stability is proved for different schemes: Crank Nicolson, Leap-frog, Explicit, Predictor-corrector. Results relative to the explicit scheme are generalized to a quasi-linear, monotone system. Finally, stability and convergence in the solution of a finite difference scheme approximating an elliptic-parabolic equation, and an iterative method of relaxation for solving this scheme are studied. (author)
Computational Aero-Acoustic Using High-order Finite-Difference Schemes
International Nuclear Information System (INIS)
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 equations for time discretization
Hong, Qianying; Lai, Ming-Jun; Wang, Jingyue
2013-01-01
We present a convergence analysis of a finite difference scheme for the time dependent partial different equation called gradient flow associated with the Rudin-Osher-Fatemi model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme. An application for image denoising is given.
Wu, Kailiang; Tang, Huazhong
2015-10-01
The paper develops high-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax-Friedrichs splitting, the WENO reconstruction, the physical-constraints-preserving flux limiter, and the high-order strong stability preserving time discretization. They are extensions of the positivity-preserving finite difference WENO schemes for the non-relativistic Euler equations [20]. However, developing physical-constraints-preserving methods for the RHD system becomes much more difficult than the non-relativistic case because of the strongly coupling between the RHD equations, no explicit formulas of the primitive variables and the flux vectors with respect to the conservative vector, and one more physical constraint for the fluid velocity in addition to the positivity of the rest-mass density and the pressure. The key is to prove the convexity and other properties of the admissible state set and discover a concave function with respect to the conservative vector instead of the pressure which is an important ingredient to enforce the positivity-preserving property for the non-relativistic case. Several one- and two-dimensional numerical examples are used to demonstrate accuracy, robustness, and effectiveness of the proposed physical-constraints-preserving schemes in solving RHD problems with large Lorentz factor, or strong discontinuities, or low rest-mass density or pressure etc.
A fifth-order finite difference scheme for hyperbolic equations on block-adaptive curvilinear grids
Chen, Yuxi; Tóth, Gábor; Gombosi, Tamas I.
2016-01-01
We present a new fifth-order accurate finite difference method for hyperbolic equations on block-adaptive curvilinear grids. The scheme employs the 5th order accurate monotonicity preserving limiter MP5 to construct high order accurate face fluxes. The fifth-order accuracy of the spatial derivatives is ensured by a flux correction step. The method is generalized to curvilinear grids with a free-stream preserving discretization. It is also extended to block-adaptive grids using carefully designed ghost cell interpolation algorithms. Only three layers of ghost cells are required, and the grid blocks can be as small as 6 × 6 × 6 cells. Dynamic grid refinement and coarsening are also fifth-order accurate. All interpolation algorithms employ a general limiter based on the principles of the MP5 limiter. The finite difference scheme is fully conservative on static uniform grids. Conservation is only maintained at the truncation error level at grid resolution changes and during grid adaptation, but our numerical tests indicate that the results are still very accurate. We demonstrate the capabilities of the new method on a number of numerical tests, including smooth but non-linear problems as well as simulations involving discontinuities.
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)
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.
Abramopoulos, Frank
1988-01-01
The conditions under which finite difference schemes for the shallow water equations can conserve both total energy and potential enstrophy are considered. A method of deriving such schemes using operator formalism is developed. Several such schemes are derived for the A-, B- and C-grids. The derived schemes include second-order schemes and pseudo-fourth-order schemes. The simplest B-grid pseudo-fourth-order schemes are presented.
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.
A compact finite difference scheme for div(Rho grad u) - q2u = 0
Phillips, T. N.; Rose, M. E.
1983-01-01
A representative class of elliptic equations is treated by a dissipative compact finite difference scheme and a general solution technique by relaxation methods is discussed in detail for the Laplace equation.
Exact finite difference schemes for the non-linear unidirectional wave equation
Mickens, R. E.
1985-01-01
Attention is given to the construction of exact finite difference schemes for the nonlinear unidirectional wave equation that describes the nonlinear propagation of a wave motion in the positive x-direction. The schemes constructed for these equations are compared with those obtained by using the usual procedures of numerical analysis. It is noted that the order of the exact finite difference models is equal to the order of the differential equation.
Lipnikov, Konstantin; Manzini, Gianmarco; Moulton, J. David; Shashkov, Mikhail
2016-01-01
Numerical schemes for nonlinear parabolic equations based on the harmonic averaging of cell-centered diffusion coefficients break down when some of these coefficients go to zero or their ratio grows. To tackle this problem, we propose new mimetic finite difference schemes that use a staggered discretization of the diffusion coefficient. The primary mimetic operator approximates div (k ṡ); the derived (dual) mimetic operator approximates - ∇ (ṡ). The new mimetic schemes preserve symmetry and positive-definiteness of the continuum problem which allows us to use algebraic solvers with optimal complexity. We perform detailed numerical analysis of the new schemes for linear elliptic problems and a specially designed linear parabolic problem that has solution dynamics typical for nonlinear problems. We show that the new schemes are competitive with the state-of-the-art schemes for steady-state problems but provide much more accurate solution dynamics for the transient problem.
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)
A perturbational h4 exponential finite difference scheme for the convective diffusion equation
International Nuclear Information System (INIS)
A perturbational h4 compact exponential finite difference scheme with diagonally dominant coefficient matrix and upwind effect is developed for the convective diffusion equation. Perturbations of second order are exerted on the convective coefficients and source term of an h2 exponential finite difference scheme proposed in this paper based on a transformation to eliminate the upwind effect of the convective diffusion equation. Four numerical examples including one- to three-dimensional model equations of fluid flow and a problem of natural convective heat transfer are given to illustrate the excellent behavior of the present exponential schemes. Besides, the h4 accuracy of the perturbational scheme is verified using double precision arithmetic
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In this paper, a second-order Total Variation Diminishing (TVD) finite difference scheme of upwind type is employed for the numerical approximation of the classical hydrodynamic model for semiconductors proposed by Bloetekjaer and Baccarani-Wordeman. In particular, the high-order hyperbolic fluxes are evaluated by a suitable extrapolation on adjacent cells of the first-order fluxes of Roe, while total variation diminishing is achieved by limiting the slopes of the discrete Riemann invariants using the so-called Flux Corrected Transport approach. Extensive numerical simulations are performed on a submicron n+-n-n+ ballistic diode. The numerical experiments show that the spurious oscillations arising in the electron current are not completely suppressed by the TVD scheme, and can lead to serious numerical instabilities when the solution of the hydrodynamic model is non-smooth and the computational mesh is coarse. The accuracy of the numerical method is investigated in terms of conservation of the steady electron current. The obtained results show that the second-order scheme does not behave much better than a corresponding first-order one due to a poor performance of the slope limiters caused by the presence of local extrema of the Riemann invariant associated with the hyperbolic system
Nonstandard finite difference scheme for SIRS epidemic model with disease-related death
Fitriah, Z.; Suryanto, A.
2016-04-01
It is well known that SIRS epidemic with disease-related death can be described by a system of nonlinear ordinary differential equations (NL ODEs). This model has two equilibrium points where their existence and stability properties are determined by the basic reproduction number [1]. Besides the qualitative properties, it is also often needed to solve the system of NL ODEs. Euler method and 4th order Runge-Kutta (RK4) method are often used to solve the system of NL ODEs. However, both methods may produce inconsistent qualitative properties of the NL ODEs such as converging to wrong equilibrium point, etc. In this paper we apply non-standard finite difference (NSFD) scheme (see [2,3]) to approximate the solution of SIRS epidemic model with disease-related death. It is shown that the discrete system obtained by NSFD scheme is dynamically consistent with the continuous model. By our numerical simulations, we find that the solutions of NSFD scheme are always positive, bounded and convergent to the correct equilibrium point for any step size of integration (h), while those of Euler or RK4 method have the same properties only for relatively small h.
A FINITE-DIFFERENCE, DISCRETE-WAVENUMBER METHOD FOR CALCULATING RADAR TRACES
A hybrid of the finite-difference method and the discrete-wavenumber method is developed to calculate radar traces. The method is based on a three-dimensional model defined in the Cartesian coordinate system; the electromagnetic properties of the model are symmetric with respect ...
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.
Computational Aero-Acoustic Using High-order Finite-Difference Schemes
DEFF Research Database (Denmark)
Zhu, Wei Jun; Shen, Wen Zhong; Sørensen, Jens Nørkær
2007-01-01
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...... 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...
Compact finite difference schemes for the Euler and Navier-Stokes equations
Rose, M. E.
1983-01-01
Implicit compact finite difference schemes for the Euler equations are described which furnish equivalent treatment of the conservation and nonconservation forms; a simple modification yields an entropy-producing scheme. An extension of the scheme also treats the compressible Navier-Stokes equations; when the viscosity and heat conduction coefficients are negligible only the boundary data appropriate to the Euler equation influence the solution to any significant extent, a result consistent with singular perturbation theory.
Meier, David L
2003-01-01
A scheme is presented for accurately propagating the gravitational field constraints in finite difference implementations of numerical relativity. The method is based on similar techniques used in astrophysical magnetohydrodynamics and engineering electromagnetics, and has properties of a finite differential calculus on a four-dimensional manifold. It is motivated by the arguments that 1) an evolutionary scheme that naturally satisfies the Bianchi identities will propagate the constraints, an...
Accurate Simulation of Contaminant Transport Using High-Order Compact Finite Difference Schemes
Gurhan Gurarslan
2014-01-01
Numerical simulation of advective-dispersive contaminant transport is carried out by using high-order compact finite difference schemes combined with second-order MacCormack and fourth-order Runge-Kutta schemes. Both of the two schemes have accuracy of sixth-order in space. A sixth-order MacCormack scheme is proposed for the first time within this study. For the aim of demonstrating efficiency and high-order accuracy of the current methods, some numerical experiments have been done. The schem...
High Order Finite Difference Schemes for the Elastic Wave Equation in Discontinuous Media
Virta, Kristoffer
2013-01-01
Finite difference schemes for the simulation of elastic waves in materi- als with jump discontinuities are presented. The key feature is the highly accurate treatment of interfaces where media discontinuities arise. The schemes are constructed using finite difference operators satisfying a sum- mation - by - parts property together with a penalty technique to impose interface conditions at the material discontinuity. Two types of opera- tors are used, termed fully compatible or compatible. Stability is proved for the first case by bounding the numerical solution by initial data in a suitably constructed semi - norm. Numerical experiments indicate that the schemes using compatible operators are also stable. However, the nu- merical studies suggests that fully compatible operators give identical or better convergence and accuracy properties. The numerical experiments are also constructed to illustrate the usefulness of the proposed method to simulations involving typical interface phenomena in elastic materials...
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Jing Yin
2015-07-01
Full Text Available A total variation diminishing-weighted average flux (TVD-WAF-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer (HLL Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth-order monotone upstream-centered scheme for conservation laws (MUSCL. The time marching scheme based on the third-order TVD Runge-Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.
A new finite difference scheme for a dissipative cubic nonlinear Schrödinger equation
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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)
DEFF Research Database (Denmark)
Fuhrmann, David R.; Bingham, Harry B.; Madsen, Per A.; Thomsen, Per Grove
2004-01-01
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...... insight into into the numerical behavior of this rather complicated system of nonlinear PDEs.......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...
An explicit finite-difference solution of hypersonic flows using rational Runge-Kutta scheme
Satofuka, Nobuyuki; Morinishi, Koji
An explicit method of lines approach has been applied for solving hypersonic flows governed by the Euler, Navier-Stokes, and Boltzmann equations. The method is based on a finite difference approximation to spatial derivatives and subsequent time integration using the rational Runge-Kutta scheme. Numerical results are presented for the hypersonic flow over a double ellipse which is a test case of the Workshop on Hypersonic Flows for Reentry Problems, January 22-25, 1990 in Antibes (France).
Ceotto, Michele; Zhuang, Yu; Hase, William L
2013-02-01
This paper shows how a compact finite difference Hessian approximation scheme can be proficiently implemented into semiclassical initial value representation molecular dynamics. Effects of the approximation on the monodromy matrix calculation are tested by propagating initial sampling distributions to determine power spectra for analytic potential energy surfaces and for "on the fly" carbon dioxide direct dynamics. With the approximation scheme the computational cost is significantly reduced, making ab initio direct semiclassical dynamics computationally more feasible and, at the same time, properly reproducing important quantum effects inherent in the monodromy matrix and the pre-exponential factor of the semiclassical propagator. PMID:23406107
Optimal rotated staggered-grid finite-difference schemes for elastic wave modeling in TTI media
Yang, Lei; Yan, Hongyong; Liu, Hong
2015-11-01
The rotated staggered-grid finite-difference (RSFD) is an effective approach for numerical modeling to study the wavefield characteristics in tilted transversely isotropic (TTI) media. But it surfaces from serious numerical dispersion, which directly affects the modeling accuracy. In this paper, we propose two different optimal RSFD schemes based on the sampling approximation (SA) method and the least-squares (LS) method respectively to overcome this problem. We first briefly introduce the RSFD theory, based on which we respectively derive the SA-based RSFD scheme and the LS-based RSFD scheme. Then different forms of analysis are used to compare the SA-based RSFD scheme and the LS-based RSFD scheme with the conventional RSFD scheme, which is based on the Taylor-series expansion (TE) method. The contrast in numerical accuracy analysis verifies the greater accuracy of the two proposed optimal schemes, and indicates that these schemes can effectively widen the wavenumber range with great accuracy compared with the TE-based RSFD scheme. Further comparisons between these two optimal schemes show that at small wavenumbers, the SA-based RSFD scheme performs better, while at large wavenumbers, the LS-based RSFD scheme leads to a smaller error. Finally, the modeling results demonstrate that for the same operator length, the SA-based RSFD scheme and the LS-based RSFD scheme can achieve greater accuracy than the TE-based RSFD scheme, while for the same accuracy, the optimal schemes can adopt shorter difference operators to save computing time.
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 Grnwald-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 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.
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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.
A fourth-order finite difference scheme for the numerical solution of 1D linear hyperbolic equation
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Akbar Mohebbi
2013-10-01
Full Text Available In this paper, a high-order and unconditionally stable difference method is proposed for the numerical solution of one-space dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivative of this equation and a Pade approximation of fifth-order for the resulting system of ordinary differential equations. It is shown through analysis that the proposed scheme is unconditionally stable. This new method is easy to implement, produces very accurate results and needs short CPU time. Some numerical examples are included to demonstrate the validity and applicability of the technique. We compare the numerical results of this paper with the numerical results of some methods in the literature.
Czech Academy of Sciences Publication Activity Database
Augusta, Petr; Cichy, B.; Galkowski, K.; Rogers, E.
Vila Real : IEEE, 2015, s. 134-139. ISBN 978-1-4799-8739-9. [The 2015 IEEE 9th International Workshop on MultiDimensional (nD) Systems (nDS) (2015). Vila Real (PT), 09.09.2015-11.09.2015] Institutional support: RVO:67985556 Keywords : Discretization * implicit difference scheme * repetitive processes Subject RIV: BC - Control Systems Theory
The stability of numerical boundary treatments for compact high-order finite-difference schemes
Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul
1991-01-01
The stability characteristics of various compact fourth and sixth order spatial operators are assessed using the theory of Gustafsson, Kreiss and Sundstrom (G-K-S) for the semi-discrete Initial Boundary Value Problem (IBVP). These results are then generalized to the fully discrete case using a recently developed theory of Kreiss. In all cases, favorable comparisons are obtained between the G-K-S theory, eigenvalue determination, and numerical simulation. The conventional definition of stability is then sharpened to include only those spatial discretizations that are asymptotically stable. It is shown that many of the higher order schemes which are G-K-S stable are not asymptotically stable. A series of compact fourth and sixth order schemes, which are both asymptotically and G-K-S stable for the scalar case, are then developed.
Bhattacharya, Amitabh
2013-11-01
An efficient algorithm for simulating Stokes flow around particles is presented here, in which a second order Finite Difference method (FDM) is coupled to a Boundary Integral method (BIM). This method utilizes the strong points of FDM (i.e. localized stencil) and BIM (i.e. accurate representation of particle surface). Specifically, in each iteration, the flow field away from the particles is solved on a Cartesian FDM grid, while the traction on the particle surface (given the the velocity of the particle) is solved using BIM. The two schemes are coupled by matching the solution in an intermediate region between the particle and surrounding fluid. We validate this method by solving for flow around an array of cylinders, and find good agreement with Hasimoto's (J. Fluid Mech. 1959) analytical results.
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...
A simple parallel prefix algorithm for compact finite-difference schemes
Sun, Xian-He; Joslin, Ronald D.
1993-01-01
A compact scheme is a discretization scheme that is advantageous in obtaining highly accurate solutions. However, the resulting systems from compact schemes are tridiagonal systems that are difficult to solve efficiently on parallel computers. Considering the almost symmetric Toeplitz structure, a parallel algorithm, simple parallel prefix (SPP), is proposed. The SPP algorithm requires less memory than the conventional LU decomposition and is highly efficient on parallel machines. It consists of a prefix communication pattern and AXPY operations. Both the computation and the communication can be truncated without degrading the accuracy when the system is diagonally dominant. A formal accuracy study was conducted to provide a simple truncation formula. Experimental results were measured on a MasPar MP-1 SIMD machine and on a Cray 2 vector machine. Experimental results show that the simple parallel prefix algorithm is a good algorithm for the compact scheme on high-performance computers.
Mickens, Ronald E.
1996-01-01
A large class of physical phenomena can be modeled by evolution and wave type Partial Differential Equations (PDE). Few of these equations have known explicit exact solutions. Finite-difference techniques are a popular method for constructing discrete representations of these equations for the purpose of numerical integration. However, the solutions to the difference equations often contain so called numerical instabilities; these are solutions to the difference equations that do not correspond to any solution of the PDE's. For explicit schemes, the elimination of this behavior requires functional relations to exist between the time and space steps-sizes. We show that such functional relations can be obtained for certain PDE's by use of a positivity condition. The PDE's studied are the Burgers, Fisher, and linearized Euler equations.
Landing-gear noise prediction using high-order finite difference schemes
Liu, Wen; Wook Kim, Jae; Zhang, Xin; Angland, David; Caruelle, Bastien
2013-07-01
Aerodynamic noise from a generic two-wheel landing-gear model is predicted by a CFD/FW-H hybrid approach. The unsteady flow-field is computed using a compressible Navier-Stokes solver based on high-order finite difference schemes and a fully structured grid. The calculated time history of the surface pressure data is used in an FW-H solver to predict the far-field noise levels. Both aerodynamic and aeroacoustic results are compared to wind tunnel measurements and are found to be in good agreement. The far-field noise was found to vary with the 6th power of the free-stream velocity. Individual contributions from three components, i.e. wheels, axle and strut of the landing-gear model are also investigated to identify the relative contribution to the total noise by each component. It is found that the wheels are the dominant noise source in general. Strong vortex shedding from the axle is the second major contributor to landing-gear noise. This work is part of Airbus LAnding Gear nOise database for CAA validatiON (LAGOON) program with the general purpose of evaluating current CFD/CAA and experimental techniques for airframe noise prediction.
Information-based complexity applied to one-dimensional finite difference transport schemes
International Nuclear Information System (INIS)
Traditional dimensions for comparing algorithms (or computational procedures) for solving the same class of problems consist of accuracy and computational complexity. A recently developed branch of theoretical computer science, which is termed information-based complexity, adds a third dimension consisting of the type or amount of information regarding the problem data that are used by the computational procedure. Instead of viewing a problem in terms of finding and analyzing a particular algorithm to solve it, information-based complexity addresses the inherent computational characteristics of general problems (as a function of their size or the error of computed solutions). One seeks upper bounds, which emerge from looking at specific algorithms, and the often more important and difficult lower bounds on the complexity of problems. The purpose of this paper is to describe the application of the information-based theory to one-dimensional finite difference schemes in neutron transport. For cell-average information and two different solution operators, the authors obtain the corresponding radius of information and optimal error algorithm. The authors also compute the error of the step-characteristic algorithm
Momoniat, E.
2014-01-01
Two nonstandard finite difference schemes are derived to solve the regularized long wave equation. The criteria for choosing the “best” nonstandard approximation to the nonlinear term in the regularized long wave equation come from considering the modified equation. The two “best” nonstandard numerical schemes are shown to preserve conserved quantities when compared to an implicit scheme in which the nonlinear term is approximated in the usual way. Comparisons to the single solitary wave solu...
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...
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.
Dimitrov, Yuri M.; Vulkov, Lubin G.
2016-01-01
We construct a three-point compact finite difference scheme on a non-uniform mesh for the time-fractional Black-Scholes equation. We show that for special graded meshes used in finance, the Tavella-Randall and the quadratic meshes the numerical solution has a fourth-order accuracy in space. Numerical experiments are discussed.
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M.M. Arabshahi
2012-08-01
Full Text Available In this article, for a class of linear systems arising from the compact finite difference schemes, we apply Krylov subspace methods in combination the ADI, BLAGE,... preconditioners. We consider our scheme in solution of hyperbolic equations subject to appropriate initial and Dirichlet boundary conditions, where is constant. We show, the BLAGE preconditioner is extremely effective in achieving optimal convergence rates. Numerical results performed on model problems to confirm the efficiency of our approach.
International Nuclear Information System (INIS)
An algorithm for numerical solution of the Sturm-Liouville problem for a system of linear differential equations is described with an up to h6 accuracy of finite-difference approximation, where h is a finite-difference grid spacing. The differential operator was apptoximated by differences of the second order of accuracy and differences of consequtive orders over h. The differences with an order of accuracy higher than h2 are considered as a perturbation to an operator of the second order of accuracy. A differential equation with such a perturbation is solved using the iteration method. The efficiency of the described algorithm is demonstrated by the solution of a problem of finding a discrete spectrum in the Morse potential. The suggested scheme is used for solving problems dealing with bound states of ?-mesomolecular systems
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.
DEFF Research Database (Denmark)
Fuhrman, David R.; Bingham, Harry B.; Madsen, Per A.; Thomsen, Per Grove
2004-01-01
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...... insight into the numerical behaviour of this rather complicated system of non-linear PDEs.......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...
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)
An energy preserving finite difference scheme for the Poisson-Nernst-Planck system
He, Dongdong; Pan, Kejia
2015-01-01
In this paper, we construct a semi-implicit finite difference method for the time dependent Poisson-Nernst-Planck system. Although the Poisson-Nernst-Planck system is a nonlinear system, the numerical method presented in this paper only needs to solve a linear system at each time step, which can be done very efficiently. The rigorous proof for the mass conservation and electric potential energy decay are shown. Moreover, mesh refinement analysis shows that the method is second order convergen...
Harte, Philip T.
1994-01-01
Proper discretization of a ground-water-flow field is necessary for the accurate simulation of ground-water flow by models. Although discretiza- tion guidelines are available to ensure numerical stability, current guidelines arc flexible enough (particularly in vertical discretization) to allow for some ambiguity of model results. Testing of two common types of vertical-discretization schemes (horizontal and nonhorizontal-model-layer approach) were done to simulate sloping hydrogeologic units characteristic of New England. Differences of results of model simulations using these two approaches are small. Numerical errors associated with use of nonhorizontal model layers are small (4 percent). even though this discretization technique does not adhere to the strict formulation of the finite-difference method. It was concluded that vertical discretization by means of the nonhorizontal layer approach has advantages in representing the hydrogeologic units tested and in simplicity of model-data input. In addition, vertical distortion of model cells by this approach may improve the representation of shallow flow processes.
On the impact of boundary conditions on dual consistent finite difference discretizations
Berg, Jens; Nordström, Jan
2013-01-01
In this paper we derive well-posed boundary conditions for a linear incompletely parabolic system of equations, which can be viewed as a model problem for the compressible Navier{Stokes equations. We show a general procedure for the construction of the boundary conditions such that both the primal and dual equations are wellposed. The form of the boundary conditions is chosen such that reduction to rst order form with its complications can be avoided. The primal equation is discretized using ...
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.
International Nuclear Information System (INIS)
For modeling scalar-wave propagation in geophysical problems using finite-difference schemes, optimizing the coefficients of the finite-difference operators can reduce numerical dispersion. Most optimized finite-difference schemes for modeling seismic-wave propagation suppress only spatial but not temporal dispersion errors. We develop a novel optimized finite-difference scheme for numerical scalar-wave modeling to control dispersion errors not only in space but also in time. Our optimized scheme is based on a new stencil that contains a few more grid points than the standard stencil. We design an objective function for minimizing relative errors of phase velocities of waves propagating in all directions within a given range of wavenumbers. Dispersion analysis and numerical examples demonstrate that our optimized finite-difference scheme is computationally up to 2.5 times faster than the optimized schemes using the standard stencil to achieve the similar modeling accuracy for a given 2D or 3D problem. Compared with the high-order finite-difference scheme using the same new stencil, our optimized scheme reduces 50 percent of the computational cost to achieve the similar modeling accuracy. This new optimized finite-difference scheme is particularly useful for large-scale 3D scalar-wave modeling and inversion
Arsoski, V V; Cukaric, N A; Peeters, F M
2015-01-01
The electron states in axially symmetric quantum wires are computed by means of the effective-mass Schroedinger equation, which is written in cylindrical coordinates phi, rho, and z. We show that a direct discretization of the Schroedinger equation by central finite differences leads to a non-symmetric Hamiltonian matrix. Because diagonalization of such matrices is more complex it is advantageous to transform it in a symmetric form. This can be done by the Liouville-like transformation proposed by Rizea et al. (Comp. Phys. Comm. 179 (2008) 466-478), which replaces the wave function psi(rho) with the function F(rho)=psi(rho)sqrt(rho) and transforms the Hamiltonian accordingly. Even though a symmetric Hamiltonian matrix is produced by this procedure, the computed wave functions are found to be inaccurate near the origin, and the accuracy of the energy levels is not very high. In order to improve on this, we devised a finite-difference scheme which discretizes the Schroedinger equation in the first step, and the...
A convergent 2D finite-difference scheme for the Dirac-Poisson system and the simulation of graphene
Brinkman, Daniel
2014-01-01
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. © 2013 Elsevier Inc.
Bidégaray-Fesquet, Brigitte
2008-01-01
This technical report yields detailed calculations of the paper [1] (B. Bid\\'egaray-Fesquet, "Stability of FD-TD schemes for Maxwell-Debye and Maxwell-Lorentz equations", Technical report, LMC-IMAG, 2005) which have been however automated since (see http://ljk.imag.fr/membres/Brigitte.Bidegaray/NAUtil/). It deals with the stability analysis of various finite difference schemes for Maxwell--Debye and Maxwell--Lorentz equations. This work gives a systematic and rigorous continuation to Petropoulos previous work [5] (P.G. Petropoulos.,"Stability and phase error analysis of FD-TD in dispersive dielectrics", IEEE Transactions on Antennas and Propagation, 42(1):62--69, 1994).
Three-dimensional acoustic wave equation modeling based on the optimal finite-difference scheme
Cai, Xiao-Hui; Liu, Yang; Ren, Zhi-Ming; Wang, Jian-Min; Chen, Zhi-De; Chen, Ke-Yang; Wang, Cheng
2015-09-01
Generally, FD coefficients can be obtained by using Taylor series expansion (TE) or optimization methods to minimize the dispersion error. However, the TE-based FD method only achieves high modeling precision over a limited range of wavenumbers, and produces large numerical dispersion beyond this range. The optimal FD scheme based on least squares (LS) can guarantee high precision over a larger range of wavenumbers and obtain the best optimization solution at small computational cost. We extend the LS-based optimal FD scheme from two-dimensional (2D) forward modeling to three-dimensional (3D) and develop a 3D acoustic optimal FD method with high efficiency, wide range of high accuracy and adaptability to parallel computing. Dispersion analysis and forward modeling demonstrate that the developed FD method suppresses numerical dispersion. Finally, we use the developed FD method to source wavefield extrapolation and receiver wavefield extrapolation in 3D RTM. To decrease the computation time and storage requirements, the 3D RTM is implemented by combining the efficient boundary storage with checkpointing strategies on GPU. 3D RTM imaging results suggest that the 3D optimal FD method has higher precision than conventional methods.
Harten, A.; Tal-Ezer, H.
1981-01-01
This paper presents a family of two-level five-point implicit schemes for the solution of one-dimensional systems of hyperbolic conservation laws, which generalized the Crank-Nicholson scheme to fourth order accuracy (4-4) in both time and space. These 4-4 schemes are nondissipative and unconditionally stable. Special attention is given to the system of linear equations associated with these 4-4 implicit schemes. The regularity of this system is analyzed and efficiency of solution-algorithms is examined. A two-datum representation of these 4-4 implicit schemes brings about a compactification of the stencil to three mesh points at each time-level. This compact two-datum representation is particularly useful in deriving boundary treatments. Numerical results are presented to illustrate some properties of the proposed scheme.
Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul
1994-01-01
We present a systematic method for constructing boundary conditions (numerical and physical) of the required accuracy, for compact (Pade-like) high-order finite-difference schemes for hyperbolic systems. First a proper summation-by-parts formula is found for the approximate derivative. A 'simultaneous approximation term' is then introduced to treat the boundary conditions. This procedure leads to time-stable schemes even in the system case. An explicit construction of the fourth-order compact case is given. Numerical studies are presented to verify the efficacy of the approach.
Gerritsen, Margot; Olsson, Pelle
1996-12-01
We derive high-order finite difference schemes for the compressible Euler (and Navier-Stokes equations) that satisfy a semidiscrete energy estimate and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution. The formulation of the semidiscrete energy estimate is based on symmetrization of the equations, a canonical splitting of the flux derivative vector, and the use of difference operators that satisfy a discrete analogue to the integration-by-parts procedure used in the continuous energy estimate. For the Euler equations, the symmetrization is designed such as to preserve the homogeneity of the flux vectors. Around discontinuities or sharp gradients, refined grids are created on which the discrete equations are solved after adding artificial viscosity. The positioning of the subgrids and computation of the viscosity are aided by a detection algorithm which is based on a multiscale wavelet analysis of the pressure grid function. The wavelet theory provides easy-to-implement mathematical criteria to detect discontinuities, sharp gradients, and spurious oscillations quickly and efficiently. As the detection algorithm does not depend on the numerical method used, it is of general interest. The numerical method described and the detection algorithm are part of a general solution strategy for fluid flows, which is currently being developed by the authors and collaborators.
Wei, Xiao-Kun; Shao, Wei; Shi, Sheng-Bing; Zhang, Yong; Wang, Bing-Zhong
2015-07-01
An efficient conformal locally one-dimensional finite-difference time-domain (LOD-CFDTD) method is presented for solving two-dimensional (2D) electromagnetic (EM) scattering problems. The formulation for the 2D transverse-electric (TE) case is presented and its stability property and numerical dispersion relationship are theoretically investigated. It is shown that the introduction of irregular grids will not damage the numerical stability. Instead of the staircasing approximation, the conformal scheme is only employed to model the curve boundaries, whereas the standard Yee grids are used for the remaining regions. As the irregular grids account for a very small percentage of the total space grids, the conformal scheme has little effect on the numerical dispersion. Moreover, the proposed method, which requires fewer arithmetic operations than the alternating-direction-implicit (ADI) CFDTD method, leads to a further reduction of the CPU time. With the total-field/scattered-field (TF/SF) boundary and the perfectly matched layer (PML), the radar cross section (RCS) of two 2D structures is calculated. The numerical examples verify the accuracy and efficiency of the proposed method. Project supported by the National Natural Science Foundation of China (Grant Nos. 61331007 and 61471105).
An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation
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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)
An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation
Zhan, Ge
2013-02-19
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. © 2013 Sinopec Geophysical Research Institute.
Kiessling, Jonas
2014-05-06
Option prices in exponential Lévy models solve certain partial integro-differential equations. This work focuses on developing novel, computable error approximations for a finite difference scheme that is suitable for solving such PIDEs. The scheme was introduced in (Cont and Voltchkova, SIAM J. Numer. Anal. 43(4):1596-1626, 2005). The main results of this work are new estimates of the dominating error terms, namely the time and space discretisation errors. In addition, the leading order terms of the error estimates are determined in a form that is more amenable to computations. The payoff is only assumed to satisfy an exponential growth condition, it is not assumed to be Lipschitz continuous as in previous works. If the underlying Lévy process has infinite jump activity, then the jumps smaller than some (Formula presented.) are approximated by diffusion. The resulting diffusion approximation error is also estimated, with leading order term in computable form, as well as the dependence of the time and space discretisation errors on this approximation. Consequently, it is possible to determine how to jointly choose the space and time grid sizes and the cut off parameter (Formula presented.). © 2014 Springer Science+Business Media Dordrecht.
Dimitrov, Yuri M.; Vulkov, Lubin G.
2015-11-01
We construct a three-point compact finite difference scheme on a non-uniform mesh for the time-fractional Black-Scholes equation. We show that for special graded meshes used in finance, the Tavella-Randall and the quadratic meshes the numerical solution has a fourth-order accuracy in space. Numerical experiments are discussed.
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The aim of the present work is to compare and discuss the three of the most advanced two dimensional transport methods, the finite difference and nodal discrete ordinates and surface flux method, incorporated into the transport codes TWODANT, TWOTRAN-NODAL, MULTIMEDIUM and SURCU. For intercomparison the eigenvalue and the neutron flux distribution are calculated using these codes in the LWR pool reactor benchmark problem. Additionally the results are compared with some results obtained by French collision probability transport codes MARSYAS and TRIDENT. Because the transport solution of this benchmark problem is close to its diffusion solution some results obtained by the finite element diffusion code FINELM and the finite difference diffusion code DIFF-2D are included
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.
Achdou, Yves; Porretta, Alessio
2015-01-01
Mean field type models describing the limiting behavior of stochastic differential games as the number of players tends to +$\\infty$, have been recently introduced by J-M. Lasry and P-L. Lions. Under suitable assumptions, they lead to a system of two coupled partial differential equations, a forward Bellman equation and a backward Fokker-Planck equations. Finite difference schemes for the approximation of such systems have been proposed in previous works. Here, we prove the convergence of the...
Implicit discretization schemes for Langevin dynamics
Zhang, Guihua; Schlick, Tamar
We explore here several numerical schemes for Langevin dynamics in the general implicit discretization framework of the Langevin/implicit-Euler scheme, LI. Specifically, six schemes are constructed through different discretization combinations of acceleration, velocity, and position. Among them, the explicit BBK method (LE in our notation) and LI are recovered, and the other four (all implicit) are named LIM1, LIM2, MID1, and MID2. The last two correspond, respectively, to the well-known implicit-midpoint scheme and the trapezoidal rule. LI and LIM1 are first-order accurate and have intrinsic numerical damping. LIM2, MID1, and MID2 appear to have large-timestep stability as LI but overcome numerical damping. However, numerical results reveal limitations on other grounds. From simulations on a model butane, we find that the non-damping methods give similar results when the timestep is small; however, as the timestep increases, LIM2 exhibits a pronounced rise in the potential energy and produces wider distributions for the bond lengths. MID1 and MID2 appear to be the best among those implicit schemes for Langevin dynamics in terms of reasonably reproducing distributions for bond lengths, bond angles and dihedral angles (in comparison to 1 fs timestep explicit simulations), as well as conserving the total energy reasonably. However, the minimization subproblem (due to the implicit formulation) becomes difficult when the timestep increases further. In terms of computational time, all the implicit schemes are very demanding. Nonetheless, we observe that for moderate timesteps, even when the error is large for the fast motions, it is relatively small for the slow motions. This suggests that it is possible by large timestep algorithms to capture the slow motions without resolving accurately the fast motions.
Settle, Sean O.
2013-01-01
The primary aim of this paper is to answer the question, What are the highest-order five- or nine-point compact finite difference schemes? To answer this question, we present several simple derivations of finite difference schemes for the one- and two-dimensional Poisson equation on uniform, quasi-uniform, and nonuniform face-to-face hyperrectangular grids and directly prove the existence or nonexistence of their highest-order local accuracies. Our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights. For the one-dimensional problem, the derivation using the three-point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high-order local accuracy. However, for the two-dimensional problem, the derivation using the corresponding five-point stencil on uniform and quasi-uniform grids yields a scheme with at most second-order local accuracy, and on nonuniform grids yields at most first-order local accuracy. When expanding the five-point stencil to the nine-point stencil, the derivation using the nine-point stencil on uniform grids yields at most sixth-order local accuracy, but on quasi- and nonuniform grids yields at most fourth- and third-order local accuracy, respectively. © 2013 Society for Industrial and Applied Mathematics.
Finite-difference method on electrostatics field calculation in oil storage tank
International Nuclear Information System (INIS)
This paper starts with the basic principle of Finite-Difference Method, combined with the practical application of the method, focusing on the difference scheme of Poisson equation and Laplace equation in 2D Electric Field and Axisymmetrical Field under the square grid partition. This study solves the Finite-Difference Method on electrostatics field calculation and introduces discretization on the Dirichlet Problem in oil storage tanker. The Finite-Difference Method gives the analytic solutions of the potential equations by changing the continuous field problems into discrete system. The numerical solutions of discrete points could infinitely approximate the real solution of the continuous field through discrete model.
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We present a quasidiffusion (QD) method for solving neutral particle transport problems in Cartesian XY geometry on unstructured quadrilateral meshes, including local refinement capability. Neutral particle transport problems are central to many applications including nuclear reactor design, radiation safety, astrophysics, medical imaging, radiotherapy, nuclear fuel transport/storage, shielding design, and oil well-logging. The primary development is a new discretization of the low-order QD (LOQD) equations based on cell-local finite differences. The accuracy of the LOQD equations depends on proper calculation of special non-linear QD (Eddington) factors from a transport solution. In order to completely define the new QD method, a proper discretization of the transport problem is also presented. The transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic, parabolic representation of the angular flux on incoming faces. Analytic and numerical tests are used to test the accuracy and spatial convergence of the non-linear method. All tests exhibit O(h2) convergence of the scalar flux on orthogonal, random, and multi-level meshes
Energy Technology Data Exchange (ETDEWEB)
Wieselquist, William A., E-mail: wieselquiswa@ornl.gov [Oak Ridge National Laboratory, 1 Bethel Valley Rd., Oak Ridge, TN 37831 (United States); Anistratov, Dmitriy Y. [Department of Nuclear Engineering, North Carolina State University, Raleigh, NC 27695 (United States); Morel, Jim E. [Department of Nuclear Engineering, Texas A and M University, College Station, TX 77843 (United States)
2014-09-15
We present a quasidiffusion (QD) method for solving neutral particle transport problems in Cartesian XY geometry on unstructured quadrilateral meshes, including local refinement capability. Neutral particle transport problems are central to many applications including nuclear reactor design, radiation safety, astrophysics, medical imaging, radiotherapy, nuclear fuel transport/storage, shielding design, and oil well-logging. The primary development is a new discretization of the low-order QD (LOQD) equations based on cell-local finite differences. The accuracy of the LOQD equations depends on proper calculation of special non-linear QD (Eddington) factors from a transport solution. In order to completely define the new QD method, a proper discretization of the transport problem is also presented. The transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic, parabolic representation of the angular flux on incoming faces. Analytic and numerical tests are used to test the accuracy and spatial convergence of the non-linear method. All tests exhibit O(h{sup 2}) convergence of the scalar flux on orthogonal, random, and multi-level meshes.
Wieselquist, William A.; Anistratov, Dmitriy Y.; Morel, Jim E.
2014-09-01
We present a quasidiffusion (QD) method for solving neutral particle transport problems in Cartesian XY geometry on unstructured quadrilateral meshes, including local refinement capability. Neutral particle transport problems are central to many applications including nuclear reactor design, radiation safety, astrophysics, medical imaging, radiotherapy, nuclear fuel transport/storage, shielding design, and oil well-logging. The primary development is a new discretization of the low-order QD (LOQD) equations based on cell-local finite differences. The accuracy of the LOQD equations depends on proper calculation of special non-linear QD (Eddington) factors from a transport solution. In order to completely define the new QD method, a proper discretization of the transport problem is also presented. The transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic, parabolic representation of the angular flux on incoming faces. Analytic and numerical tests are used to test the accuracy and spatial convergence of the non-linear method. All tests exhibit O(h2) convergence of the scalar flux on orthogonal, random, and multi-level meshes.
International Nuclear Information System (INIS)
In this paper we suggest the linear scheme for the three-dimensional neutron diffusion equation approximation. In this scheme we use the finite element method with bi-quadratic test functions for x,y approximation. For z approximation of the equation we use the finite difference method. Theoretically, such a scheme provides convergence with a high order of accuracy: the third order or higher for x,y variables and the second order for z variable. The scheme provides simulations with space grid refinement. Our computational investigations showed that accuracy of calculations is acceptable even at the base (the coarsest) grid. It provides significant reduction of calculation time compared to simulations based on the second accuracy order schemes. Our scheme was implemented in the KORAT 3D code. The RBMK reactor was simulated as the test problem with a different detailed order. The results of the computational investigation of convergence were compared with the results obtained by the second accuracy order scheme. (authors)
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.
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)
Ackleh, Azmy S; Delcambre, Mark L; Sutton, Karyn L
2015-01-01
We present a second-order high-resolution finite difference scheme to approximate the solution of a mathematical model of the transmission dynamics of Mycobacterium marinum (Mm) in an aquatic environment. This work extends the numerical theory and continues the preliminary studies on the model first developed in Ackleh et al. [Structured models for the spread of Mycobacterium marinum: foundations for a numerical approximation scheme, Math. Biosci. Eng. 11 (2014), pp. 679-721]. Numerical simulations demonstrating the accuracy of the method are presented, and we compare this scheme to the first-order scheme developed in Ackleh et al. [Structured models for the spread of Mycobacterium marinum: foundations for a numerical approximation scheme, Math. Biosci. Eng. 11 (2014), pp. 679-721] to show that the first-order method requires significantly more computational time to provide solutions with a similar accuracy. We also demonstrated that the model can be a tool to understand surprising or nonintuitive phenomena regarding competitive advantage in the context of biologically realistic growth, birth and death rates. PMID:25271885
An implicit finite-difference operator for the Helmholtz equation
Chu, Chunlei
2012-07-01
We have developed an implicit finite-difference operator for the Laplacian and applied it to solving the Helmholtz equation for computing the seismic responses in the frequency domain. This implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost, compared with the corresponding conventional explicit finite-difference scheme. We achieved this by taking advantage of the inherently implicit nature of the Helmholtz equation and merging together the two linear systems: one from the implicit finite-difference discretization of the Laplacian and the other from the discretization of the Helmholtz equation itself. The end result of this simple yet important merging manipulation is a single linear system, similar to the one resulting from the conventional explicit finite-difference discretizations, without involving any differentiation matrix inversions. We analyzed grid dispersions of the discrete Helmholtz equation to show the accuracy of this implicit finite-difference operator and used two numerical examples to demonstrate its efficiency. Our method can be extended to solve other frequency domain wave simulation problems straightforwardly. © 2012 Society of Exploration Geophysicists.
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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)
Discretized Newton-relaxation solution of high resolution flux-difference split schemes
International Nuclear Information System (INIS)
A method is presented for the implicit solution of high resolution flux-difference split schemes for which it is impractical to obtain the Jacobian matrix analytically. The method is used frequently in numerical schemes for solving nonlinear systems of equations, but apparently has not been used to date for approximating the Jacobian of the Roe numerical flux vectors. The method consists of obtaining a discretized Jacobian by using simple finite-differences of the flux vectors, and then using this discretized Jacobian in a Newton-relaxation solution method. The method is applied to both compressible and incompressible flows. Numerical results are presented that demonstrate the method works for both inviscid and viscous flows. 21 refs
Alizadeh Nomeli, M.; Riaz, A.
2012-12-01
Increasing concentration of CO2 as a greenhouse gas in the atmosphere causes global warming and it subsequently perturbs the balance of the life cycle. In order to mitigate the concentration of CO2 in the atmosphere, the sequestration of CO2 into deep geological formations has been investigated theoretically and experimentally in recent decades. Solubility and mineral trapping are the most promising long term solutions to geologic CO2 sequestration, because they prevent its return to the atmosphere. In this study, the CO2 sequestration capacity of both aqueous and mineral phases is evaluated. Mineral alterations, however, are too slow to be modeled experimentally; therefore a numerical model is required. This study presents a model to simulate a reactive fluid within permeable porous media. The problem contains reactive transport modeling between a miscible flow and minerals in post-injection regime. Rates of dissolution and precipitation (PD) of minerals are determined by taking into account the pH of the system, in addition to the consideration of the influence of temperature. We solve fluid convection, diffusion and PD reactions inside a fracture in order to predict the amount of CO2 that can be stored as precipitation of secondary carbonates after specific period of time. The modeling of flow and transport inside the fracture for the mineral trapping purpose is based on space discretization by means of integral finite differences. Dissolution and precipitation of all minerals in simulations presented in the current study are assumed to be kinetically controlled. Therefore the model can monitor changes in porosity and permeability during the simulation from changes in the volume of the fracture.
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.
Finite difference techniques for nonlinear hyperbolic conservation laws
International Nuclear Information System (INIS)
The present study is concerned with numerical approximations to the initial value problem for nonlinear systems of conservative laws. Attention is given to the development of a class of conservation form finite difference schemes which are based on the finite volume method (i.e., the method of averages). These schemes do not fit into the classical framework of conservation form schemes discussed by Lax and Wendroff (1960). The finite volume schemes are specifically intended to approximate solutions of multidimensional problems in the absence of rectangular geometries. In addition, the development is reported of different schemes which utilize the finite volume approach for time discretization. Particular attention is given to local time discretization and moving spatial grids. 17 references
Discrete Conservation Properties of Unstructured Mesh Schemes
Perot, J. Blair
2011-01-01
Numerical methods with discrete conservation statements are useful because they cannot produce solutions that violate important physical constraints. A large number of numerical methods used in computational fluid dynamics (CFD) have either global or local conservation statements for some of the primary unknowns of the method. This review suggests that local conservation of primary unknowns often follows from global conservation of those quantities. Secondary conservation involves the conservation of derived quantities, such as kinetic energy, entropy, and vorticity, which are not directly unknowns of the numerical system. Secondary conservation can further improve physical fidelity of a numerical solution, but it is typically much harder to achieve. We consider current approaches to secondary conservation and techniques used outside of CFD that are potentially related. Finally, the review concludes with a discussion of how secondary conservation properties might be included automatically.
Optimized Discretization Schemes For Brain Images
Directory of Open Access Journals (Sweden)
USHA RANI.N,
2011-02-01
Full Text Available In medical image processing active contour method is the important technique in segmenting human organs. Geometric deformable curves known as levelsets are widely used in segmenting medical images. In this modeling , evolution of the curve is described by the basic lagrange pde expressed as a function of space and time. This pde can be solved either using continuous functions or discrete numerical methods.This paper deals with the application of numerical methods like finite diffefence and TVd-RK methods for brain scans. The stability and accuracy of these methods are also discussed. This paper also deals with the more accurate higher order non-linear interpolation techniques like ENO and WENO in reconstructing the brain scans like CT,MRI,PET and SPECT is considered.
Modified Cascade Synchronization Scheme for Discrete-Time Hyperchaotic Systems
International Nuclear Information System (INIS)
In this paper, the modified cascade synchronization scheme is proposed to investigate the synchronization in discrete-time hyperchaotic systems. By choosing a general kind of proportional scaling error functions and based on rigorous control theory, we take the discrete-time hyperchaotic system due to Wang and 3D generalized Henon map as two examples to achieve the modified cascade synchronization, respectively. Numerical simulations are used to verify the effectiveness of the proposed technique
A Transport Acceleration Scheme for Multigroup Discrete Ordinates with Upscattering
Energy Technology Data Exchange (ETDEWEB)
Evans, Thomas M [ORNL; Clarno, Kevin T [ORNL; Morel, Jim E. [Texas A& M University
2010-01-01
We have developed a modification of the two-grid upscatter acceleration scheme of Adams and Morel. The modified scheme uses a low-angular-order discrete ordinates equation to accelerate Gauss-Seidel multigroup iteration. This modification ensures that the scheme does not suffer from consistency problems that can affect diffusion-accelerated methods in multidimensional, multimaterial problems. The new transport two-grid scheme is very simple to implement for different spatial discretizations because it uses the same transport operator. The scheme has also been demonstrated to be very effective on three-dimensional, multimaterial problems. On simple one-dimensional graphite and heavy-water slabs modeled in three dimensions with reflecting boundary conditions, we see reductions in the number of Gauss-Seidel iterations by factors of 75 to 1000. We have also demonstrated the effectiveness of the new method on neutron well-logging problems. For forward problems, the new acceleration scheme reduces the number of Gauss-Seidel iterations by more than an order of magnitude with a corresponding reduction in the run time. For adjoint problems, the speedup is not as dramatic, but the new method still reduces the run time by greater than a factor of 6.
Droniou, Jerome; Gallout, Thierry; Herbin, Raphaele
2008-01-01
We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.
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].
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.
New single-carrier transceiver scheme based on the discrete sine transform
Faisal Al-kamali
2014-01-01
A discrete sine transform (DST)-based single-carrier transceiver scheme for broadband wireless communications is proposed and investigated. The proposed scheme uses a DST rather than the conventional discrete Fourier transform (DFT) as a basis function to implement the single-carrier system. The performance of the proposed scheme is studied and compared with the DFT-based single-carrier transceiver scheme and the discrete cosine transform based single-carrier transceiver scheme. Simulation re...
On Cryptographic Schemes Based on Discrete Logarithms and Factoring
Joye, Marc
At CRYPTO 2003, Rubin and Silverberg introduced the concept of torus-based cryptography over a finite field. We extend their setting to the ring of integers modulo N. We so obtain compact representations for cryptographic systems that base their security on the discrete logarithm problem and the factoring problem. This results in smaller key sizes and substantial savings in memory and bandwidth. But unlike the case of finite fields, analogous trace-based compression methods cannot be adapted to accommodate our extended setting when the underlying systems require more than a mere exponentiation. As an application, we present an improved, torus-based implementation of the ACJT group signature scheme.
International Nuclear Information System (INIS)
This report examines, and establishes the causes of, previously identified time step and mesh size dependencies. These dependencies were observed in the solution of a coupled system of heat conduction and fluid flow equations as used in the TRAC-PF1/MOD1 computer code. The report shows that a significant time step size dependency can arise in calculations of the quenching of a previously unwetted surface. The cause of this dependency is shown to be the explicit evaluation, and subsequent smoothing of the term which couples the heat transfer and fluid flow equations. An axial mesh size dependency is also identified, but this is very much smaller than the time step size dependency. The report concludes that the time step size dependency represents a potential limitation on the use of large time step sizes for types of calculation discussed. This limitation affects the present TRAC-PF-1/MOD1 computer code and may similarly affect other semi-implicit finite difference codes that employ similar techniques. It is likely to be of greatest significance in codes where multi-step techniques are used to allow the use of large time steps
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)
Applications of an exponential finite difference technique
Handschuh, Robert F.; Keith, Theo G., Jr.
1988-01-01
An exponential finite difference scheme first presented by Bhattacharya for one dimensional unsteady heat conduction problems in Cartesian coordinates was extended. The finite difference algorithm developed was used to solve the unsteady diffusion equation in one dimensional cylindrical coordinates and was applied to two and three dimensional conduction problems in Cartesian coordinates. Heat conduction involving variable thermal conductivity was also investigated. The method was used to solve nonlinear partial differential equations in one and two dimensional Cartesian coordinates. Predicted results are compared to exact solutions where available or to results obtained by other numerical methods.
Finite-difference models of ordinary differential equations - Influence of denominator functions
Mickens, Ronald E.; Smith, Arthur
1990-01-01
This paper discusses the influence on the solutions of finite-difference schemes of using a variety of denominator functions in the discrete modeling of the derivative for any ordinary differential equation. The results obtained are a consequence of using a generalized definition of the first derivative. A particular example of the linear decay equation is used to illustrate in detail the various solution possibilities that can occur.
A free energy satisfying finite difference method for Poisson--Nernst--Planck equations
Liu, Hailiang; Wang, Zhongming
2013-01-01
In this work we design and analyze a free energy satisfying finite difference method for solving Poisson-Nernst-Planck equations in a bounded domain. The algorithm is of second order in space, with numerical solutions satisfying all three desired properties: i) mass conservation, ii) positivity preserving, and iii) free energy satisfying in the sense that these schemes satisfy a discrete free energy dissipation inequality. These ensure that the computed solution is a probability density, and ...
A new coding scheme for discrete memoryless MACs with common rate-limited feedback
BELHADJ AMOR, SELMA
2015-01-01
We propose a new coding scheme for the discrete memoryless two-user multi-access channel (MAC) with rate-limited feedback. Our scheme combines ideas from the Venkataramanan-Pradhan scheme for perfect feedback with ideas from the Shaviv-Steinberg scheme for rate-limited feedback. Our achievable region includes the Shaviv-Steinberg achievable region and this inclusion can be strict. For general MACs and for sufficiently large feedback rates, our scheme outperforms the Shaviv-Steinberg scheme as...
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)
A fully-discrete Semi-Lagrangian scheme for a first order mean field game problem
Carlini, Elisabetta; Silva, Francisco
2012-01-01
In this work we propose a fully-discrete Semi-Lagrangian scheme for a {\\it first order mean field game system}. We prove that the resulting discretization admits at least one solution and, in the scalar case, we prove a convergence result for the scheme. Numerical simulations and examples are also discussed.
High-order Finite Difference Solution of Euler Equations for Nonlinear Water Waves
DEFF Research Database (Denmark)
Christiansen, Torben Robert Bilgrav; Bingham, Harry B.; Engsig-Karup, Allan Peter
2012-01-01
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...
Finite difference and finite element methods
International Nuclear Information System (INIS)
The relationships between and relative advantages of finite difference and finite element methods are discussed. The less familiar finite element methods are described first for equilibrium problems: it is shown how quadratic elements on right triangles lead to natural generalisations of the powerful, fourth order accurate nine-point difference scheme for the Laplacian. For evolutionary problems, the recent development of more accurate difference methods is considered together with that of Galerkin methods. It is shown how conservation properties are best preserved by the latter methods and, in particular, how the supression of non-linear instabilities in the advection equation is achieved by the Arakawa schemes. Finally, an error analysis is described which is applicable to both finite difference and finite element methods. (Auth.)
A Two-Scale Discretization Scheme for Mixed Variational Formulation of Eigenvalue Problems
Yidu Yang; Wei Jiang; Yu Zhang; Wenjun Wang; Hai Bi
2012-01-01
This paper discusses highly efficient discretization schemes for mixed variational formulation of eigenvalue problems. A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems. With this scheme, the solution of an eigenvalue problem on a fine grid ${K}^{h}$ is reduced to the solution of an eigenvalue problem on a much coarser grid ${K}^{H}$ and the solution of a li...
A Factorization Scheme for Some Discrete Hartley Transform Matrices
Oliveira, H M; Cintra, R. J.; de Souza, R. M. Campello
2015-01-01
Discrete transforms such as the discrete Fourier transform (DFT) and the discrete Hartley transform (DHT) are important tools in numerical analysis. The successful application of transform techniques relies on the existence of efficient fast transforms. In this paper some fast algorithms are derived. The theoretical lower bound on the multiplicative complexity for the DFT/DHT are achieved. The approach is based on the factorization of DHT matrices. Algorithms for short blocklengths such as $N...
Long-time behaviour of a fully discrete Lagrangian scheme for a family of fourth order
Osberger, Horst
2015-01-01
A fully discrete Lagrangian scheme for solving a family of fourth order equations numerically is presented. The discretization is based on the equation's underlying gradient flow structure w.r.t. the $L^2$-Wasserstein distance, and adapts numerous of its most important structural properties by construction, as conservation of mass and entropy-dissipation. In this paper, the long-time behaviour of our discretization is analyzed: We show that discrete solutions decay exponentially to equilibriu...
The Relation of Finite Element and Finite Difference Methods
Vinokur, M.
1976-01-01
Finite element and finite difference methods are examined in order to bring out their relationship. It is shown that both methods use two types of discrete representations of continuous functions. They differ in that finite difference methods emphasize the discretization of independent variable, while finite element methods emphasize the discretization of dependent variable (referred to as functional approximations). An important point is that finite element methods use global piecewise functional approximations, while finite difference methods normally use local functional approximations. A general conclusion is that finite element methods are best designed to handle complex boundaries, while finite difference methods are superior for complex equations. It is also shown that finite volume difference methods possess many of the advantages attributed to finite element methods.
Discretization of the Gabor-type scheme by sampling of the Zak transform
Zibulski, Meir; Zeevi, Yehoshua Y.
1994-09-01
The matrix algebra approach was previously applied in the analysis of the continuous Gabor representation in the Zak transform domain. In this study we analyze the discrete and finite (periodic) scheme by the same approach. A direct relation that exists between the two schemes, based on the sampling of the Zak transform, is established. Specifically, we show that sampling of the Gabor expansion in the Zak transform domain yields a discrete scheme of representation. Such a derivation yields a simple relation between the schemes by means of the periodic extension of the signal. We show that in the discrete Zak domain the frame operator can be expressed by means of a matrix-valued function which is simply the sampled version of the matrix-valued function of the continuous scheme. This result establishes a direct relation between the frame properties of the two schemes.
Simulation of Metasurfaces in Finite Difference Techniques
Vahabzadeh, Yousef; Caloz, Christophe
2016-01-01
We introduce a rigorous and simple method for analyzing metasurfaces, modeled as zero-thickness electromagnetic sheets, in Finite Difference (FD) techniques. The method consists in describing the spatial discontinuity induced by the metasurface as a virtual structure, located between nodal rows of the Yee grid, using a finite difference version of Generalized Sheet Transition Conditions (GSTCs). In contrast to previously reported approaches, the proposed method can handle sheets exhibiting both electric and magnetic discontinuities, and represents therefore a fundamental contribution in computational electromagnetics. It is presented here in the framework of the FD Frequency Domain (FDFD) method but also applies to the FD Time Domain (FDTD) scheme. The theory is supported by five illustrative examples.
Asymptotic analysis of discrete schemes for non-equilibrium radiation diffusion
Cui, Xia; Yuan, Guang-wei; Shen, Zhi-jun
2016-05-01
Motivated by providing well-behaved fully discrete schemes in practice, this paper extends the asymptotic analysis on time integration methods for non-equilibrium radiation diffusion in [2] to space discretizations. Therein studies were carried out on a two-temperature model with Larsen's flux-limited diffusion operator, both the implicitly balanced (IB) and linearly implicit (LI) methods were shown asymptotic-preserving. In this paper, we focus on asymptotic analysis for space discrete schemes in dimensions one and two. First, in construction of the schemes, in contrast to traditional first-order approximations, asymmetric second-order accurate spatial approximations are devised for flux-limiters on boundary, and discrete schemes with second-order accuracy on global spatial domain are acquired consequently. Then by employing formal asymptotic analysis, the first-order asymptotic-preserving property for these schemes and furthermore for the fully discrete schemes is shown. Finally, with the help of manufactured solutions, numerical tests are performed, which demonstrate quantitatively the fully discrete schemes with IB time evolution indeed have the accuracy and asymptotic convergence as theory predicts, hence are well qualified for both non-equilibrium and equilibrium radiation diffusion.
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.
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.
Exponential Finite-Difference Technique
Handschuh, Robert F.
1989-01-01
Report discusses use of explicit exponential finite-difference technique to solve various diffusion-type partial differential equations. Study extends technique to transient-heat-transfer problems in one dimensional cylindrical coordinates and two and three dimensional Cartesian coordinates and to some nonlinear problems in one or two Cartesian coordinates.
A new mimetic scheme for the acoustic wave equation
Solano, Freysimar; Guevara-Jordan, Juan; Rojas, Otilio; Otero Calviño, Beatriz; Rodriguez, R.
2016-01-01
A new mimetic finite difference scheme for solving the acoustic wave equation is presented. It combines a novel second order tensor mimetic discretizations in space and a leapfrog approximation in time to produce an explicit multidimensional scheme. Convergence analysis of the new scheme on a staggered grid shows that it can take larger time steps than standard finite difference schemes based on ghost points formulation. A set of numerical test problems gives evidence of the versatility of th...
The computer algebra approach of the finite difference methods for PDEs
International Nuclear Information System (INIS)
In this paper, a first attempt has been made to realize the computer algebra construction of the finite difference methods or the finite difference schemes for constant coefficient partial differential equations. (author). 9 refs, 2 tabs
Zhou, Nanrun; Yang, Jianping; Tan, Changfa; Pan, Shumin; Zhou, Zhihong
2015-11-01
A new discrete fractional random transform based on two circular matrices is designed and a novel double-image encryption-compression scheme is proposed by combining compressive sensing with discrete fractional random transform. The two random circular matrices and the measurement matrix utilized in compressive sensing are constructed by using a two-dimensional sine Logistic modulation map. Two original images can be compressed, encrypted with compressive sensing and connected into one image. The resulting image is re-encrypted by Arnold transform and the discrete fractional random transform. Simulation results and security analysis demonstrate the validity and security of the scheme.
Implicit finite-difference simulations of seismic wave propagation
Chu, Chunlei
2012-03-01
We propose a new finite-difference modeling method, implicit both in space and in time, for the scalar wave equation. We use a three-level implicit splitting time integration method for the temporal derivative and implicit finite-difference operators of arbitrary order for the spatial derivatives. Both the implicit splitting time integration method and the implicit spatial finite-difference operators require solving systems of linear equations. We show that it is possible to merge these two sets of linear systems, one from implicit temporal discretizations and the other from implicit spatial discretizations, to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm. We give the complete derivations of the implicit splitting time integration method and the implicit spatial finite-difference operators, and present the resulting discretized formulas for the scalar wave equation. We conduct a thorough numerical analysis on grid dispersions of this new implicit modeling method. We show that implicit spatial finite-difference operators greatly improve the accuracy of the implicit splitting time integration simulation results with only a slight increase in computational time, compared with explicit spatial finite-difference operators. We further verify this conclusion by both 2D and 3D numerical examples. © 2012 Society of Exploration Geophysicists.
Convergence of the Approximation scheme to American option pricing via the discrete Morse semiflow
Ishii, Katsuyuki; Omata, Seiro
2009-01-01
We consider the approximation scheme of the American call option via the discrete Morse semiflow. It is the minimizing scheme of a time-semidiscretized variational functional. In this paper we obtain a rate of convergence of approximate solutions. In addition, the convergence of approximate free boundaries is proved.
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.
Discrete unified gas kinetic scheme with force term for incompressible fluid flows
Wu, Chen; Chai, Zhenhua; Wang, Peng
2014-01-01
The discrete unified gas kinetic scheme (DUGKS) is a finite-volume scheme with discretization of particle velocity space, which combines the advantages of both lattice Boltzmann equation (LBE) method and unified gas kinetic scheme (UGKS) method, such as the simplified flux evaluation scheme, flexible mesh adaption and the asymptotic preserving properties. However, DUGKS is proposed for near incompressible fluid flows, the existing compressible effect may cause some serious errors in simulating incompressible problems. To diminish the compressible effect, in this paper a novel DUGKS model with external force is developed for incompressible fluid flows by modifying the approximation of Maxwellian distribution. Meanwhile, due to the pressure boundary scheme, which is wildly used in many applications, has not been constructed for DUGKS, the non-equilibrium extrapolation (NEQ) scheme for both velocity and pressure boundary conditions is introduced. To illustrate the potential of the proposed model, numerical simul...
Labb, Chantal; Renaud, Jean-Franois
2010-01-01
A discretization scheme for nonnegative diffusion processes is proposed and the convergence of the corresponding sequence of approximate processes is proved using the martingale problem framework. Motivations for this scheme come typically from finance, especially for path-dependent option pricing. The scheme is simple: one only needs to find a nonnegative distribution whose mean and variance satisfy a simple condition to apply it. Then, for virtually any (path-dependent) payoff, Monte Carlo option prices obtained from this scheme will converge to the theoretical price. Examples of models and diffusion processes for which the scheme applies are provided.
Ducomet, Bernard; Romanova, Alla
2013-01-01
An initial-boundary value problem for the $n$-dimensional ($n\\geq 2$) time-dependent Schr\\"odinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for $n\\geq 3$. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting method, the uniqueness of solution and the unconditional uniform in time $L^2$-stability (in particular, $L^2$-conservativeness) are proved. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applicable for general potential. Numerical results on the 2D tunnel effect for a P\\"{o}schl-Teller-like potential-barrier and a rectangular potential-well are also included.
Original Signer's Forgery Attacks on Discrete Logarithm Based Proxy Signature Schemes
Directory of Open Access Journals (Sweden)
Tianjie Cao
2007-05-01
Full Text Available A proxy signature scheme enables a proxy signer to sign messages on behalf of the original signer. In this paper, we demonstrate that a number of discrete logarithm based proxy signature schemes are vulnerable to an original signer's forgery attack. In this attack, a malicious original signer can impersonate a proxy signer and produce a forged proxy signature on a message. A third party will incorrectly believe that the proxy signer was responsible for generating the proxy signature. This contradicts the strong unforgeability property that is required of proxy signatures schemes. We show six proxy signature schemes vulnerable to this attack including Lu et al.'s proxy blind multi-signature scheme, Xue and Cao's proxy blind signature scheme, Fu et al. and Gu et al.'s anonymous proxy signature schemes, Dai et al. and Huang et al.'s nominative proxy signature schemes are all insecure against the original signer's forgery.
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.
Finite difference neuroelectric modeling software.
Dang, Hung V; Ng, Kwong T
2011-06-15
This paper describes a finite difference neuroelectric modeling software (FNS), written in C and MATLAB, which can be executed as a standalone program or integrated with other packages for electroencephalography (EEG) analysis. The package from the Oxford Center for Functional MRI of the Brain (FMRIB), FMRIB Software Library (FSL), is used to segment the anatomical magnetic resonance (MR) image for realistic head modeling. The EEG electrode array is fitted to the realistic head model using the Bioelectromagnetism MATLAB toolbox. The finite difference formulation for a general inhomogeneous anisotropic body is used to obtain the system matrix equation, which is then solved using the conjugate gradient algorithm. The reciprocity theorem is utilized to limit the number of required forward solutions to N-1, where N is the number of electrodes. Results show that the forward solver only requires 500 MB of random-access memory (RAM) for a realistic 256×256×256 head model and that the software can be conveniently combined with inverse algorithms such as beamformers and MUSIC. The software is freely available under the GNU Public License. PMID:21477619
A mimetic finite difference method for the Stokes problem with elected edge bubbles
Energy Technology Data Exchange (ETDEWEB)
Lipnikov, K [Los Alamos National Laboratory; Berirao, L [DIPARTMENTO DI MATERMATICA
2009-01-01
A new mimetic finite difference method for the Stokes problem is proposed and analyzed. The unstable P{sub 1}-P{sub 0} discretization is stabilized by adding a small number of bubble functions to selected mesh edges. A simple strategy for selecting such edges is proposed and verified with numerical experiments. The discretizations schemes for Stokes and Navier-Stokes equations must satisfy the celebrated inf-sup (or the LBB) stability condition. The stability condition implies a balance between discrete spaces for velocity and pressure. In finite elements, this balance is frequently achieved by adding bubble functions to the velocity space. The goal of this article is to show that the stabilizing edge bubble functions can be added only to a small set of mesh edges. This results in a smaller algebraic system and potentially in a faster calculations. We employ the mimetic finite difference (MFD) discretization technique that works for general polyhedral meshes and can accomodate non-uniform distribution of stabilizing bubbles.
Energy Technology Data Exchange (ETDEWEB)
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 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
Wang, Yajun; Liu, Yang; Li, Hong; Wang, Jinfeng
2016-03-01
In this article, a Galerkin finite element method combined with second-order time discrete scheme for finding the numerical solution of nonlinear time fractional Cable equation is studied and discussed. At time t_{k-α/2} , a second-order two step scheme with α -parameter is proposed to approximate the first-order derivative, and a weighted discrete scheme covering second-order approximation is used to approximate the Riemann-Liouville fractional derivative, where the approximate order is higher than the obtained results by the L1-approximation with order (2-α in the existing references. For the spatial direction, Galerkin finite element approximation is presented. The stability of scheme and the rate of convergence in L^2 -norm with O(Δ t^2+(1+Δ t^{-α})h^{m+1}) are derived in detail. Moreover, some numerical tests are shown to support our theoretical results.
Original Signer's Forgery Attacks on Discrete Logarithm Based Proxy Signature Schemes
Tianjie Cao; Xianping Mao
2007-01-01
A proxy signature scheme enables a proxy signer to sign messages on behalf of the original signer. In this paper, we demonstrate that a number of discrete logarithm based proxy signature schemes are vulnerable to an original signer's forgery attack. In this attack, a malicious original signer can impersonate a proxy signer and produce a forged proxy signature on a message. A third party will incorrectly believe that the proxy signer was responsible for generating the proxy signature. This co...
Coding Schemes for Discrete Memoryless Multicast Networks with Rate-limited Feedback
Wu, Youlong
2015-01-01
Coding schemes for discrete memoryless multicast networks with rate-limited feedback from the receivers and relays to the transmitter are proposed. The coding schemes are based on block-Markov coding, {joint backward decoding} and hybrid relaying strategy. In each block, the receivers and relays compress their channel outputs and send the compression indices to the transmitter through the feedback links. In the next block, after obtaining the compression indices, the transmitter sends them to...
Arbitrary Dimension Convection-Diffusion Schemes for Space-Time Discretizations
Energy Technology Data Exchange (ETDEWEB)
Bank, Randolph E. [Univ. of California, San Diego, CA (United States); Vassilevski, Panayot S. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Zikatanov, Ludmil T. [Bulgarian Academy of Sciences, Sofia (Bulgaria)
2016-01-20
This note proposes embedding a time dependent PDE into a convection-diffusion type PDE (in one space dimension higher) with singularity, for which two discretization schemes, the classical streamline-diffusion and the EAFE (edge average finite element) one, are investigated in terms of stability and error analysis. The EAFE scheme, in particular, is extended to be arbitrary order which is of interest on its own. Numerical results, in combined space-time domain demonstrate the feasibility of the proposed approach.
Fully discrete Galerkin schemes for the nonlinear and nonlocal Hartree equation
Directory of Open Access Journals (Sweden)
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.
An Efficient Signcryption Scheme based on The Elliptic Curve Discrete Logarithm Problem
Directory of Open Access Journals (Sweden)
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.
Phonon Boltzmann equation-based discrete unified gas kinetic scheme for multiscale heat transfer
Guo, Zhaoli
2016-01-01
Numerical prediction of multiscale heat transfer is a challenging problem due to the wide range of time and length scales involved. In this work a discrete unified gas kinetic scheme (DUGKS) is developed for heat transfer in materials with different acoustic thickness based on the phonon Boltzmann equation. With discrete phonon direction, the Boltzmann equation is discretized with a second-order finite-volume formulation, in which the time-step is fully determined by the Courant-Friedrichs-Lewy (CFL) condition. The scheme has the asymptotic preserving (AP) properties for both diffusive and ballistic regimes, and can present accurate solutions in the whole transition regime as well. The DUGKS is a self-adaptive multiscale method for the capturing of local transport process. Numerical tests for both heat transfers with different Knudsen numbers are presented to validate the current method.
Energy Technology Data Exchange (ETDEWEB)
Perron, Sebastien [ARDC, Alcan, Applied Science Research Group, 1955 Mellon Blvd, P.O. Box 1250, Quebec G7S 4K8, Jonquiere (Canada); Boivin, Sylvain [Universite du Quebec a Chicoutimi, 555 Boulevard de l' universite, Quebec G7H 2B1, Chicoutimi (Canada); Herard, Jean-Marc [DRD, Electricite de France, 6, quai Watier 78400, Chatou (France)
2004-09-01
We present a new method to solve incompressible thermal flows and the transport of scalar quantities. It is a finite volume scheme for unstructured meshes whose time discretization is based upon the fractional time step method. The governing equations are discretized using a collocated, cell-centered arrangement of velocity and pressure. The solution variables are stored at the cell-circum-centers. This scheme is convergent, stable and allows computing solutions that does not violate the maximum principle when it applies. Theoretical results and numerical properties of the scheme are provided. Predictions of Boussinesq fluid flow, flow past a cylinder and heat transport in a cylinder are performed to validate the method. (authors)
Numerical time-domain electromagnetics based on finite-difference and convolution
Lin, Yuanqu
Time-domain methods posses a number of advantages over their frequency-domain counterparts for the solution of wideband, nonlinear, and time varying electromagnetic scattering and radiation phenomenon. Time domain integral equation (TDIE)-based methods, which incorporate the beneficial properties of integral equation method, are thus well suited for solving broadband scattering problems for homogeneous scatterers. Widespread adoption of TDIE solvers has been retarded relative to other techniques by their inefficiency, inaccuracy and instability. Moreover, two-dimensional (2D) problems are especially problematic, because 2D Green's functions have infinite temporal support, exacerbating these difficulties. This thesis proposes a finite difference delay modeling (FDDM) scheme for the solution of the integral equations of 2D transient electromagnetic scattering problems. The method discretizes the integral equations temporally using first- and second-order finite differences to map Laplace-domain equations into the Z domain before transforming to the discrete time domain. The resulting procedure is unconditionally stable because of the nature of the Laplace- to Z-domain mapping. The first FDDM method developed in this thesis uses second-order Lagrange basis functions with Galerkin's method for spatial discretization. The second application of the FDDM method discretizes the space using a locally-corrected Nystrom method, which accelerates the precomputation phase and achieves high order accuracy. The Fast Fourier Transform (FFT) is applied to accelerate the marching-on-time process in both methods. While FDDM methods demonstrate impressive accuracy and stability in solving wideband scattering problems for homogeneous scatterers, they still have limitations in analyzing interactions between several inhomogenous scatterers. Therefore, this thesis devises a multi-region finite-difference time-domain (MR-FDTD) scheme based on domain-optimal Green's functions for solving sparsely-populated problems. The scheme uses a discrete Green's function (DGF) on the FDTD lattice to truncate the local subregions, and thus reduces reflection error on the local boundary. A continuous Green's function (CGF) is implemented to pass the influence of external fields into each FDTD region which mitigates the numerical dispersion and anisotropy of standard FDTD. Numerical results will illustrate the accuracy and stability of the proposed techniques.
Perot, J. B.; Subramanian, V.
2007-09-01
The Keller Box scheme is a face-based method for solving partial differential equations that has numerous attractive mathematical and physical properties. It is shown that these attractive properties collectively follow from the fact that the scheme discretizes partial derivatives exactly and only makes approximations in the algebraic constitutive relations appearing in the PDE. The exact discrete calculus associated with the Keller Box scheme is shown to be fundamentally different from all other mimetic (physics capturing) numerical methods. This suggests that a unique exact discrete calculus does not exist. It also suggests that existing analysis techniques based on concepts in algebraic topology (in particular - the discrete de Rham complex) are unnecessarily narrowly focused since they do not capture the Keller Box scheme. The discrete calculus analysis allows a generalization of the Keller Box scheme to non-simplectic meshes to be constructed. Analysis and tests of the method on the unsteady advection-diffusion equations are presented.
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.
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.
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.
A free energy satisfying finite difference method for Poisson-Nernst-Planck equations
Liu, Hailiang; Wang, Zhongming
2014-07-01
In this work we design and analyze a free energy satisfying finite difference method for solving Poisson-Nernst-Planck equations in a bounded domain. The algorithm is of second order in space, with numerical solutions satisfying all three desired properties: i) mass conservation, ii) positivity preserving, and iii) free energy satisfying in the sense that these schemes satisfy a discrete free energy dissipation inequality. These ensure that the computed solution is a probability density, and the schemes are energy stable and preserve the equilibrium solutions. Both one- and two-dimensional numerical results are provided to demonstrate the good qualities of the algorithm, as well as effects of relative size of the data given.
Robust Multiple Image Watermarking Scheme using Discrete Cosine Transform with Multiple Descriptions
Mohan B. Chandra; Kumar S. Srinivas
2009-01-01
A novel oblivious and robust multiple imagewatermarking scheme using Multiple Descriptions (MD) andQuantization Index Modulation (QIM) of the host image ispresented in this paper. Watermark embedding is done at twostages. In the first stage, Discrete Cosine Transform (DCT) ofodd description of the host image is computed. The watermarkimage is embedded in the resulting DC coefficients. In thesecond stage, a copy of the watermark image is embedded in thewatermarked image generated at the first ...
Andjar Gran, Carlos; Ayala Vallespi, Dolors; Brunet Crosa, Pere
1999-01-01
This work discusses simplification algorithms for the generation of a multiresolution family of solid representations from an initial polyhedral solid. We introduce the Discretized Polyhedra Simplification (DPS), a framework for polyhedra simplification using space decomposition models. The DPS is based on a new error measurement and provides a sound scheme for error-bounded, geometry and topology simplification while preserving the validity of the model. A method following this framework, Di...
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 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.
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 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)
A Critical Evaluation of the Resolution Properties of B-Spline and Compact Finite Difference Methods
Kwok, Wai Yip; Moser, Robert D.; Jiménez, Javier
2001-12-01
Resolution properties of B-spline and compact finite difference schemes are compared using Fourier analysis in periodic domains, and tests based on solution of the wave and heat equations in finite domains, with uniform and nonuniform grids. Results show that compact finite difference schemes have a higher convergence rate and in some cases better resolution. However, B-spline schemes have a more straightforward and robust formulation, particularly near boundaries on nonuniform meshes.
Frank, Jason
2006-01-01
In this note we show that multisymplectic Runge-Kutta box schemes, of which the Gauss-Legendre methods are the most important, preserve a discrete conservation law of wave action. The result follows by loop integration over an ensemble of flow realizations, and the local energy-momentum conservation law for continuous variables in semi-discretizations
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.
A Novel Image Encryption Scheme Based on Multi-orbit Hybrid of Discrete Dynamical System
Ruisong Ye; Huiqing Huang; Xiangbo Tan
2014-01-01
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 p...
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.
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.
A finite difference method for free boundary problems
Fornberg, Bengt
2010-04-01
Fornberg and Meyer-Spasche proposed some time ago a simple strategy to correct finite difference schemes in the presence of a free boundary that cuts across a Cartesian grid. We show here how this procedure can be combined with a minimax-based optimization procedure to rapidly solve a wide range of elliptic-type free boundary value problems. © 2009 Elsevier B.V. All rights reserved.
Second-order accurate nonoscillatory schemes for scalar conservation laws
Huynh, Hung T.
1989-01-01
Explicit finite difference schemes for the computation of weak solutions of nonlinear scalar conservation laws is presented and analyzed. These schemes are uniformly second-order accurate and nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time.
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
Continuous and discretized pursuit learning schemes: various algorithms and their comparison.
Oommen, B J; Agache, M
2001-01-01
A learning automaton (LA) is an automaton that interacts with a random environment, having as its goal the task of learning the optimal action based on its acquired experience. Many learning automata (LAs) have been proposed, with the class of estimator algorithms being among the fastest ones, Thathachar and Sastry, through the pursuit algorithm, introduced the concept of learning algorithms that pursue the current optimal action, following a reward-penalty learning philosophy. Later, Oommen and Lanctot extended the pursuit algorithm into the discretized world by presenting the discretized pursuit algorithm, based on a reward-inaction learning philosophy. In this paper we argue that the reward-penalty and reward-inaction learning paradigms in conjunction with the continuous and discrete models of computation, lead to four versions of pursuit learning automata. We contend that a scheme that merges the pursuit concept with the most recent response of the environment, permits the algorithm to utilize the LAs long-term and short-term perspectives of the environment. In this paper, we present all four resultant pursuit algorithms, prove the E-optimality of the newly introduced algorithms, and present a quantitative comparison between them. PMID:18244792
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...
A non-linear constrained optimization technique for the mimetic finite difference method
Energy Technology Data Exchange (ETDEWEB)
Manzini, Gianmarco [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Svyatskiy, Daniil [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Bertolazzi, Enrico [Univ. of Trento (Italy); Frego, Marco [Univ. of Trento (Italy)
2014-09-30
This is a strategy for the construction of monotone schemes in the framework of the mimetic finite difference method for the approximation of diffusion problems on unstructured polygonal and polyhedral meshes.
Finite difference order doubling in two dimensions
International Nuclear Information System (INIS)
An order doubling process previously used to obtain eighth-order eigenvalues from the fourth-order Numerov method is applied to the perturbed oscillator in two dimensions. A simple method of obtaining high order finite difference operators is reported and an odd parity boundary condition is found to be effective in facilitating the smooth operation of the order doubling process
Convergence of summation-by-parts finite difference methods for the wave equation
Siyang WANG; Kreiss, Gunilla
2015-01-01
In this paper, we consider finite difference approximations of the second order wave equation. We use finite difference operators satisfying the summation-by-parts property to discretize the equation in space. Boundary conditions and grid interface conditions are imposed by the simultaneous-approximation-term technique. Typically, the truncation error is larger at the grid points near a boundary or grid interface than that in the interior. Normal mode analysis can be used to analyze how the l...
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 ε2φxx+a(x)φ = 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).
Tuned Finite-Difference Diffusion Operators
Maron, Jason
2008-01-01
Finite-difference simulations of fluid dynamics and magnetohydrodynamics generally require an explicit diffusion operator, either to maintain stability by attenuating grid-scale structure, or to implement physical diffusivities such as viscosity or resistivity. If the goal is stability only, the diffusion must act at the grid scale, but should affect structure at larger scales as little as possible. For physical diffusivities the diffusion scale depends on the problem, and diffusion may act at larger scales as well. Diffusivity undesirably limits the computational timestep in both cases. We construct tuned finite-difference diffusion operators that minimally limit the timestep while acting as desired near the diffusion scale. Such operators reach peak values at the diffusion scale rather than at the grid scale, but behave as standard operators at larger scales. We focus on the specific applications of hyperdiffusivity for numerical stabilization, and high Schmidt and high Prandtl number simulations where the ...
Elementary introduction to finite difference equations
Energy Technology Data Exchange (ETDEWEB)
White, J.W.
1976-05-03
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.
Directory of Open Access Journals (Sweden)
Deepak Sharma
2014-09-01
Full Text Available Encryption along with compression is the process used to secure any multimedia content processing with minimum data storage and transmission. The transforms plays vital role for optimizing any encryption-compression systems. Earlier the original information in the existing security system based on the fractional Fourier transform (FRFT is protected by only a certain order of FRFT. In this article, a novel method for encryption-compression scheme based on multiple parameters of discrete fractional Fourier transform (DFRFT with random phase matrices is proposed. The multiple-parameter discrete fractional Fourier transform (MPDFRFT possesses all the desired properties of discrete fractional Fourier transform. The MPDFRFT converts to the DFRFT when all of its order parameters are the same. We exploit the properties of multiple-parameter DFRFT and propose a novel encryption-compression scheme using the double random phase in the MPDFRFT domain for encryption and compression data. The proposed scheme with MPDFRFT significantly enhances the data security along with image quality of decompressed image compared to DFRFT and FRFT and it shows consistent performance with different images. The numerical simulations demonstrate the validity and efficiency of this scheme based on Peak signal to noise ratio (PSNR, Compression ratio (CR and the robustness of the schemes against bruit force attack is examined.
Kovcs, M; Lindgren, F
2012-01-01
We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation.
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.
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.
The numerical solution of problems in calculus of variation using Chebyshev finite difference method
Energy Technology Data Exchange (ETDEWEB)
Saadatmandi, Abbas [Department of Mathematics, Faculty of Science, University of Kashan, Kashan (Iran, Islamic Republic of)], E-mail: saadatmandi@kashanu.ac.ir; Dehghan, Mehdi [Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran (Iran, Islamic Republic of)], E-mail: mdehghan@aut.ac.ir
2008-05-26
The Chebyshev finite difference method is used for finding the solution of the ordinary differential equations which arise from problems of calculus of variations. Our approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a non-uniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and yields very accurate results.
International Nuclear Information System (INIS)
High order resolution schemes based on the NVD and TVD boundedness criteria are applied to radiative transfer problems using the DOM in two-dimensional unstructured triangular grids. The implementation of these schemes in unstructured grids requires approximations, and two implementations reported in the literature are compared with a new one. Three different methods have been used to calculate the gradient of the radiation intensity at the center of the control volumes. The various schemes are applied to several test problems, the results are compared with those obtained using the step scheme, the mean flux interpolation scheme and another high order scheme based on a truncated Taylor series expansion, and the most accurate implementations are identified. It is concluded that although the high order schemes perform much better than the others, they are not as accurate as in Cartesian coordinates, and their order of convergence is lower than in that case. - Highlights: The radiative transfer equation is solved in unstructured grids using the DOM. Discretization schemes based on the NVD and TVD boundedness criteria are used. Several implementations relying on different approximations are compared. The order of convergence in unstructured grids is lower than in Cartesian grids. The most accurate and the fastest implementations of the schemes were identified
A parallel finite-difference method for computational aerodynamics
International Nuclear Information System (INIS)
A finite-difference scheme for solving complex three-dimensional aerodynamic flow on parallel-processing supercomputers is presented. The method consists of a basic flow solver with multigrid convergence acceleration, embedded grid refinements, and a zonal equation scheme. Multitasking and vectorization have been incorporated into the algorithm. Results obtained include multiprocessed flow simulations from the Cray X-MP and Cray-2. Speedups as high as 3.3 for the two-dimensional case and 3.5 for segments of the three-dimensional case have been achieved on the Cray-2. The entire solver attained a factor of 2.7 improvement over its unitasked version on the Cray-2. The performance of the parallel algorithm on each machine is analyzed. 14 refs
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.
Yang, L. M.; Shu, C.; Wang, Y.
2016-03-01
In this work, a discrete gas-kinetic scheme (DGKS) is presented for simulation of two-dimensional viscous incompressible and compressible flows. This scheme is developed from the circular function-based GKS, which was recently proposed by Shu and his co-workers [L. M. Yang, C. Shu, and J. Wu, J. Comput. Phys. 274, 611 (2014), 10.1016/j.jcp.2014.06.033]. For the circular function-based GKS, the integrals for conservation forms of moments in the infinity domain for the Maxwellian function-based GKS are simplified to those integrals along the circle. As a result, the explicit formulations of conservative variables and fluxes are derived. However, these explicit formulations of circular function-based GKS for viscous flows are still complicated, which may not be easy for the application by new users. By using certain discrete points to represent the circle in the phase velocity space, the complicated formulations can be replaced by a simple solution process. The basic requirement is that the conservation forms of moments for the circular function-based GKS can be accurately satisfied by weighted summation of distribution functions at discrete points. In this work, it is shown that integral quadrature by four discrete points on the circle, which forms the D2Q4 discrete velocity model, can exactly match the integrals. Numerical results showed that the present scheme can provide accurate numerical results for incompressible and compressible viscous flows with roughly the same computational cost as that needed by the Roe scheme.
Integral and finite difference inequalities and applications
Pachpatte, B G
2006-01-01
The monograph is written with a view to provide basic tools for researchers working in Mathematical Analysis and Applications, concentrating on differential, integral and finite difference equations. It contains many inequalities which have only recently appeared in the literature and which can be used as powerful tools and will be a valuable source for a long time to come. It is self-contained and thus should be useful for those who are interested in learning or applying the inequalities with explicit estimates in their studies.- Contains a variety of inequalities discovered which find numero
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 ε → 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.
The Complex-Step-Finite-Difference method
Abreu, Rafael; Stich, Daniel; Morales, Jose
2015-07-01
We introduce the Complex-Step-Finite-Difference method (CSFDM) as a generalization of the well-known Finite-Difference method (FDM) for solving the acoustic and elastic wave equations. We have found a direct relationship between modelling the second-order wave equation by the FDM and the first-order wave equation by the CSFDM in 1-D, 2-D and 3-D acoustic media. We present the numerical methodology in order to apply the introduced CSFDM and show an example for wave propagation in simple homogeneous and heterogeneous models. The CSFDM may be implemented as an extension into pre-existing numerical techniques in order to obtain fourth- or sixth-order accurate results with compact three time-level stencils. We compare advantages of imposing various types of initial motion conditions of the CSFDM and demonstrate its higher-order accuracy under the same computational cost and dispersion-dissipation properties. The introduced method can be naturally extended to solve different partial differential equations arising in other fields of science and engineering.
Finite difference methods for coupled flow interaction transport models
Directory of Open Access Journals (Sweden)
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.
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
Comparative study of numerical schemes of TVD3, UNO3-ACM and optimized compact scheme
Lee, Duck-Joo; Hwang, Chang-Jeon; Ko, Duck-Kon; Kim, Jae-Wook
1995-01-01
Three different schemes are employed to solve the benchmark problem. The first one is a conventional TVD-MUSCL (Monotone Upwind Schemes for Conservation Laws) scheme. The second scheme is a UNO3-ACM (Uniformly Non-Oscillatory Artificial Compression Method) scheme. The third scheme is an optimized compact finite difference scheme modified by us: the 4th order Runge Kutta time stepping, the 4th order pentadiagonal compact spatial discretization with the maximum resolution characteristics. The problems of category 1 are solved by using the second (UNO3-ACM) and third (Optimized Compact) schemes. The problems of category 2 are solved by using the first (TVD3) and second (UNO3-ACM) schemes. The problem of category 5 is solved by using the first (TVD3) scheme. It can be concluded from the present calculations that the Optimized Compact scheme and the UN03-ACM show good resolutions for category 1 and category 2 respectively.
Jang, Juhi; Li, Fengyan; Qiu, Jing-Mei; Xiong, Tao
2013-01-01
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-equilib...
Fang, Chung-Chieh
2012-01-01
Nonlinear systems, such as switching DC-DC boost or buck converters, have rich dynamics. A simple one-dimensional discrete-time model is used to analyze the boost or buck converter in discontinuous conduction mode. Seven different control schemes (open-loop power stage, voltage mode control, current mode control, constant power load, constant current load, constant-on-time control, and boundary conduction mode) are analyzed systematically. The linearized dynamics is obtained simply by taking ...
Helsing, Johan; Holst, Anders
2013-01-01
The incorporation of analytical kernel information is exploited in the construction of Nystr\\"om discretization schemes for integral equations modeling planar Helmholtz boundary value problems. Splittings of kernels and matrices, coarse and fine grids, high-order polynomial interpolation, product integration performed on the fly, and iterative solution are some of the numerical techniques used to seek rapid and stable convergence of computed fields in the entire computational domain.
Helsing, Johan; Holst, Anders
2015-01-01
The incorporation of analytical kernel information is exploited in the construction of Nystrm discretization schemes for integral equations modeling planar Helmholtz boundary value problems. Splittings of kernels and matrices, coarse and fine grids, high-order polynomial interpolation, product integration performed on the fly, and iterative solution are some of the numerical techniques used to seek rapid and stable convergence of computed fields in the entire computational domain.
Pencil: Finite-difference Code for Compressible Hydrodynamic Flows
Brandenburg, Axel; Dobler, Wolfgang
2010-10-01
The Pencil code is a high-order finite-difference code for compressible hydrodynamic flows with magnetic fields. It is highly modular and can easily be adapted to different types of problems. The code runs efficiently under MPI on massively parallel shared- or distributed-memory computers, like e.g. large Beowulf clusters. The Pencil code is primarily designed to deal with weakly compressible turbulent flows. To achieve good parallelization, explicit (as opposed to compact) finite differences are used. Typical scientific targets include driven MHD turbulence in a periodic box, convection in a slab with non-periodic upper and lower boundaries, a convective star embedded in a fully nonperiodic box, accretion disc turbulence in the shearing sheet approximation, self-gravity, non-local radiation transfer, dust particle evolution with feedback on the gas, etc. A range of artificial viscosity and diffusion schemes can be invoked to deal with supersonic flows. For direct simulations regular viscosity and diffusion is being used. The code is written in well-commented Fortran90.
Mostrel, M. M.
1988-01-01
New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.
Relative and Absolute Error Control in a Finite-Difference Method Solution of Poisson's Equation
Prentice, J. S. C.
2012-01-01
An algorithm for error control (absolute and relative) in the five-point finite-difference method applied to Poisson's equation is described. The algorithm is based on discretization of the domain of the problem by means of three rectilinear grids, each of different resolution. We discuss some hardware limitations associated with the algorithm,…
Chu, Chunlei
2009-01-01
We analyze the dispersion properties and stability conditions of the high‐order convolutional finite difference operators and compare them with the conventional finite difference schemes. We observe that the convolutional finite difference method has better dispersion properties and becomes more efficient than the conventional finite difference method with the increasing order of accuracy. This makes the high‐order convolutional operator a good choice for anisotropic elastic wave simulations on rotated staggered grids since its enhanced dispersion properties can help to suppress the numerical dispersion error that is inherent in the rotated staggered grid structure and its efficiency can help us tackle 3D problems cost‐effectively.
International Nuclear Information System (INIS)
The suitability of high-order accurate, centered and upwind-biased compact difference schemes for large eddy simulation (LES) is evaluated through the static and dynamic analyses. For the static error analysis, the power spectra of the finite-differencing and aliasing errors are evaluated in the discrete Fourier space, and for the dynamic error analysis LES of isotropic turbulence is performed with various dissipative and non-dissipative schemes. Results from the static analysis give a misleading conclusion that both the aliasing and finite-differencing errors increase as the numerical dissipation increases. The dynamic analysis, however, shows that the aliasing error decreases as the dissipation increases and the finite-differencing error overweighs the aliasing error. It is also shown that there exists an optimal upwind scheme of minimizing the total discretization error because the dissipative schemes decrease the aliasing error but increase the finite-differencing error. In addition, a classical issue on the treatment of nonlinear term in the Navier-Stokes equation is revisited to show that the skew-symmetric form minimizes both the finite-differencing and aliasing errors. The findings from the dynamic analysis are confirmed by the physical space simulations of turbulent channel flow at Re=23000 and flow over a circular cylinder at Re=3900
Cagnetti, Filippo
2013-11-01
We consider a numerical scheme for the one dimensional time dependent Hamilton-Jacobi equation in the periodic setting. This scheme consists in a semi-discretization using monotone approximations of the Hamiltonian in the spacial variable. From classical viscosity solution theory, these schemes are known to converge. In this paper we present a new approach to the study of the rate of convergence of the approximations based on the nonlinear adjoint method recently introduced by L.C. Evans. We estimate the rate of convergence for convex Hamiltonians and recover the O(h) convergence rate in terms of the L∞ norm and O(h) in terms of the L1 norm, where h is the size of the spacial grid. We discuss also possible generalizations to higher dimensional problems and present several other additional estimates. The special case of quadratic Hamiltonians is considered in detail in the end of the paper. © 2013 IMACS.
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.
Saarelma, Jukka; Botts, Jonathan; Hamilton, Brian; Savioja, Lauri
2016-04-01
Finite-difference time-domain (FDTD) simulation has been a popular area of research in room acoustics due to its capability to simulate wave phenomena in a wide bandwidth directly in the time-domain. A downside of the method is that it introduces a direction and frequency dependent error to the simulated sound field due to the non-linear dispersion relation of the discrete system. In this study, the perceptual threshold of the dispersion error is measured in three-dimensional FDTD schemes as a function of simulation distance. Dispersion error is evaluated for three different explicit, non-staggered FDTD schemes using the numerical wavenumber in the direction of the worst-case error of each scheme. It is found that the thresholds for the different schemes do not vary significantly when the phase velocity error level is fixed. The thresholds are found to vary significantly between the different sound samples. The measured threshold for the audibility of dispersion error at the probability level of 82% correct discrimination for three-alternative forced choice is found to be 9.1 m of propagation in a free field, that leads to a maximum group delay error of 1.8 ms at 20 kHz with the chosen phase velocity error level of 2%. PMID:27106330
Junge, Oliver; Matthes, Daniel; Osberger, Horst
2015-01-01
We introduce a novel spatio-temporal discretization for nonlinear Fokker-Planck equations on the multi-dimensional unit cube. This discretization is based on two structural properties of these equations: the first is the representation as a gradient flow of an entropy functional in the $L^2$-Wasserstein metric, the second is the Lagrangian nature, meaning that solutions can be written as the push forward transformation of the initial density under suitable flow maps. The resulting numerical s...
ON FINITE DIFFERENCES ON A STRING PROBLEM
Directory of Open Access Journals (Sweden)
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.)
Liao, Fei; Ye, Zhengyin
2015-12-01
Despite significant progress in recent computational techniques, the accurate numerical simulations, such as direct-numerical simulation and large-eddy simulation, are still challenging. For accurate calculations, the high-order finite difference method (FDM) is usually adopted with coordinate transformation from body-fitted grid to Cartesian grid. But this transformation might lead to failure in freestream preservation with the geometric conservation law (GCL) violated, particularly in high-order computations. GCL identities, including surface conservation law (SCL) and volume conservation law (VCL), are very important in discretization of high-order FDM. To satisfy GCL, various efforts have been made. An early and successful approach was developed by Thomas and Lombard [6] who used the conservative form of metrics to cancel out metric terms to further satisfy SCL. Visbal and Gaitonde [7] adopted this conservative form of metrics for SCL identities and satisfied VCL identity through invoking VCL equation to acquire the derivative of Jacobian in computation on moving and deforming grids with central compact schemes derived by Lele [5]. Later, using the metric technique from Visbal and Gaitonde [7], Nonomura et al. [8] investigated the freestream and vortex preservation properties of high-order WENO and WCNS on stationary curvilinear grids. A conservative metric method (CMM) was further developed by Deng et al. [9] with stationary grids, and detailed discussion about the innermost difference operator of CMM was shown with proof and corresponding numerical test cases. Noticing that metrics of CMM is asymmetrical without coordinate-invariant property, Deng et al. proposed a symmetrical CMM (SCMM) [12] by using the symmetric forms of metrics derived by Vinokur and Yee [10] to further eliminate asymmetric metric errors with stationary grids considered only. The research from Abe et al. [11] presented new asymmetric and symmetric conservative forms of time metrics and Jacobian on three-dimensional moving and deforming mesh. Moreover, Abe et al. [14] discussed the symmetrical and asymmetrical geometric interpretations of metrics and Jacobian. By deriving sufficient conditions for the conservative form of VCL, Sjögreen et al. [13] generalized their previous GCL treatment for stationary grids to moving and deforming grids with a new form of time metrics and Jacobian. Recently, Liao et al. [1] focused on the discretization and geometric interpretations of metrics and Jacobian in cell-centered finite difference methods (CCFDM), where the geometric conservation of multiblock interfaces, the treatment of singular axis and simplification of multiblock boundary condition are discussed in detail.
Luneva, Maria; Holt, Jason; Harle, James; Liu, Hedong
2013-04-01
The results of a recently developed NEMO-shelf pan-Arctic Ocean model coupled with LIM2 ice model are presented. This pan Arctic model has a hybrid s-z vertical discretization with terrain following coordinates on the shelf, condensing towards the bottom and surface boundary layer, and partial step z-coordinates in the abyss. This allows (a) processes near the surface to be resolved (b) Cascading (shelf convection), which contributes to the formation of halocline and deep dense water, to be well reproduced; and (c) minimize pressure gradient errors peculiar to terrain following coordinates. Horizontal grid and topography corresponds to global NEMO -ORCA 0.25 model (which uses a tripolar grid) with seamed slit between the western and eastern parts. In the Arctic basin this horizontal resolution corresponds to 15-10km with 5-7 km in the Canadian Archipelago. The model uses the General Length Scale vertical turbulent mixing scheme with (K- ?) closure and Kantha and Clayson type structural functions. Smagorinsky type Laplacian diffusivity and viscosity are employed for the description of a horizontal mixing. Vertical Piecewise Parabolic Method has been implemented with the aim to reduce an artificial vertical mixing. Boundary conditions are taken from the 5-days mean output of NOCS version of the global ORCA-025 model and OTPS/tpxo7 for 9 tidal harmonics . For freshwater runoff we employed two different forcings: a climatic one, used in global ORCA-0.25 model, and a recently available data base from Dai and Trenberth (Feb2011) 1948-2007, which takes in account inter-annual variability and includes 1200 river guages for the Arctic ocean coast. The simulations have been performed for two intervals: 1978-1988 and 1997-2007. The model adequately reproduces the main features of dynamics, tides and ice volume/concentration. The analysis shows that the main effects of tides occur at the ice-water interface and bottom boundary layers due to mesoscale Ekman pumping , generated by nonlinear shear tidal stresses, acting as a 'tidal winds' on the surfaces. Harmonic analysis shows, that at least five harmonics should be taken in account: three semidiurnal M2, S2, N2 and two diurnal K1 and O1. We present results from the following experiments: (a) with tidal forcing and without tidal forcing; (b) with climatic runoff and with Dai and Trenberth database. To examine the effects of summer ice openings on the formation of brine rejection and dense water cascades, additional idealised experiments have been performed: (c) for initial conditions of hydrographic fields and fluxes for 1978 with initial summer ice concentration of 2000; (d) opposite case of initial ocean conditions for 2000 and ice concentration of 1978. The comparisons with global ORCA-025 simulations and available data are discussed.
Determination of finite-difference weights using scaled binomial windows
Chu, Chunlei
2012-05-01
The finite-difference method evaluates a derivative through a weighted summation of function values from neighboring grid nodes. Conventional finite-difference weights can be calculated either from Taylor series expansions or by Lagrange interpolation polynomials. The finite-difference method can be interpreted as a truncated convolutional counterpart of the pseudospectral method in the space domain. For this reason, we also can derive finite-difference operators by truncating the convolution series of the pseudospectral method. Various truncation windows can be employed for this purpose and they result in finite-difference operators with different dispersion properties. We found that there exists two families of scaled binomial windows that can be used to derive conventional finite-difference operators analytically. With a minor change, these scaled binomial windows can also be used to derive optimized finite-difference operators with enhanced dispersion properties. © 2012 Society of Exploration Geophysicists.
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
Combination of finite-difference and finite-volume techniques in global reactor calculations
International Nuclear Information System (INIS)
The paper addresses the problems encountered in reconstructing, based on the nodal coarsemesh solution, the interior of pin values which are needed in local safety margin analysis. Finite-difference techniques are applied to find flux corner values using the results of the nodal diffusion calculation. The advantage of this new approach is that it can be used for an arbitrary number of neutron energy groups and in any geometry. As examples, corner formulae for Cartesian and hexagonal geometry are derived and analysed with respect to computational expenditure and atainable accuracy. Its shown that the fine-mesh flux distribution in a node can be directly reconstructed by successive use of higher-order standard and rotated finite-difference formulae. In this way an approximation to the discrete solution is computed up to an error comparable to the truncation error of a much more expensive standard finite-difference calculation. (orig./HP)
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 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
Operto, S.; VIRIEUX, J; Ribodetti, Alessandra; Anderson, J E
2009-01-01
A 2D finite-difference, frequency-domain method was developed for modeling viscoacoustic seismic waves in transversely isotropic media with a tilted symmetry axis. The medium is parameterized by the P-wave velocity on the symmetry axis, the density, the attenuation factor, Thomsen's anisotropic parameters delta and epsilon, and the tilt angle. The finite-difference discretization relies on a parsimonious mixed-grid approach that designs accurate yet spatially compact stencils. The system of l...
Discrete level schemes and their gamma radiation branching ratios (CENPL-DLS): Pt.2
International Nuclear Information System (INIS)
The DLS data files contains the data and information of nuclear discrete levels and gamma rays. At present, it has 79461 levels and 93177 gamma rays for 1908 nuclides. The DLS sub-library has been set up at the CNDC, and widely used for nuclear model calculation and other field. the DLS management retrieval code DLS is introduced and an example is given for 56Fe. (1 tab.)
An analytical discrete ordinates solution for two-dimensional problems based on nodal schemes
International Nuclear Information System (INIS)
In this work, the ADO method is used to solve the integrated one dimensional equations generated by the application of a nodal scheme on the two dimensional transport problem in cartesian geometry. Particularly, relations between the averaged fluxes and the unknown fluxes at the boundary are introduced as the usually needed auxiliary equations. The ADO approach, along with a level symmetric quadrature scheme, lead to an important reduction in the order of the associated eigenvalue systems. Numerical results are presented for a two dimensional problem in order to compare with available results in the literature. (author)
Discrete memory schemes for finite strain thermoplasticity and application to shape memory alloys
International Nuclear Information System (INIS)
A theory of finite strain plasticity has been proposed: The scheme of pure hysteresis with mixed transport has been extended to the case of non-rotational kinematics. Secondly, the simple shear case has been studied, taking into account Drucker's recent analysis regarding the 'appropriate simple idealizations for finite plasticity'. Illustrations are provided for general stress/strain paths. Also a new theory of isotropic hyperelasticity has been proposed. The 'reversible' relative Cauchy stress tensor (of type (1,1) and weight one) is defined in the dragged along coordinates as a tensorial isotropic function of the Almansi tensor and of its invariants (through the partial derivatives of the actual scalar density of elastic energy per unit extent of dragged along coordinates). The correspondance between strain and stress paths is then defined in a general form which is particularly convenient for the study of first order effects, limit behaviours, coupling and second order effects. Illustrations are provided. The addition of the pure hysteresis stress contribution σa and of the reversible contribution σrev leads to a scheme of 'superelasticity' departure to obtain a provisional scheme of shape memory effects. Some remarks are given regarding some of the possible generalizations of the scheme. (orig./GL)
Elements of Polya-Schur theory in finite difference setting
Brndn, P; Shapiro, B
2012-01-01
In this note we attempt to develop an analog of P\\'olya-Schur theory describing the class of univariate hyperbolicity preservers in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i.e. the minimal distance between the roots) is at least one. In particular, finite difference versions of the classical Hermite-Poulain theorem and generalized Laguerre inequalities are obtained.
Linear rational finite differences from derivatives of barycentric rational interpolants
Klein, Georges; Berrut, Jean-Paul
2012-01-01
Derivatives of polynomial interpolants lead in a natural way to approximations of derivatives of the interpolated function, e.g., through finite differences. We extend a study of the approximation of derivatives of linear barycentric rational interpolants and present improved finite difference formulas arising from these interpolants. The formulas contain the classical finite differences as a special case and are more stable for calculating one-sided derivatives as well as derivatives close t...
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.
A Scheme to Share Information via Employing Discrete Algorithm to Quantum States
International Nuclear Information System (INIS)
We propose a protocol for information sharing between two legitimate parties (Bob and Alice) via public-key cryptography. In particular, we specialize the protocol by employing discrete algorithm under mod that maps integers to quantum states via photon rotations. Based on this algorithm, we find that the protocol is secure under various classes of attacks. Specially, owe to the algorithm, the security of the classical privacy contained in the quantum public-key and the corresponding cipher text is guaranteed. And the protocol is robust against the impersonation attack and the active wiretapping attack by designing particular checking processing, thus the protocol is valid. (general)
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; Kytýř, Daniel; Doktor, Tomáš; Fíla, Tomáš; Jiroušek, Ondřej
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 ČR(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
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
Roč. 48, č. 2 (2012), s. 247-254. ISSN 0005-1098 R&D Projects: GA ČR(CZ) GAP103/11/0517; GA ČR 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, 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.
A spherical higher-order finite-difference time-domain algorithm with perfectly matched layer
International Nuclear Information System (INIS)
A higher-order finite-difference time-domain (HO-FDTD) in the spherical coordinate is presented in this paper. The stability and dispersion properties of the proposed scheme are investigated and an air-filled spherical resonator is modeled in order to demonstrate the advantage of this scheme over the finite-difference time-domain (FDTD) and the multiresolution time-domain (MRTD) schemes with respect to memory requirements and CPU time. Moreover, the Berenger's perfectly matched layer (PML) is derived for the spherical HO-FDTD grids, and the numerical results validate the efficiency of the PML. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)
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...
An Image Hiding Scheme Using 3D Sawtooth Map and Discrete Wavelet Transform
Directory of Open Access Journals (Sweden)
Ruisong Ye
2012-07-01
Full Text Available An image encryption scheme based on the 3D sawtooth map is proposed in this paper. The 3D sawtooth map is utilized to generate chaotic orbits to permute the pixel positions and to generate pseudo-random gray value sequences to change the pixel gray values. The image encryption scheme is then applied to encrypt the secret image which will be imbedded in one host image. The encrypted secret image and the host image are transformed by the wavelet transform and then are merged in the frequency domain. Experimental results show that the stego-image looks visually identical to the original host one and the secret image can be effectively extracted upon image processing attacks, which demonstrates strong robustness against a variety of attacks.
Implicit-Explicit Runge-Kutta schemes for numerical discretization of optimal control problems
Herty, Michael; Steffensen, Sonja
2012-01-01
Implicit-explicit (IMEX) Runge-Kutta methods play a major rule in the numerical treatment of differential systems governed by stiff and non-stiff terms. This paper discusses order conditions and symplecticity properties of a class of IMEX Runge-Kutta methods in the context of optimal control problems. The analysis of the schemes is based on the continuous optimality system. Using suitable transformations of the adjoint equation, order conditions up to order three are proven as well as the relation between adjoint schemes obtained through different transformations is investigated. Conditions for the IMEX Runge-Kutta methods to be symplectic are also derived. A numerical example illustrating the theoretical properties is presented.
Generalized pursuit learning schemes: new families of continuous and discretized learning automata.
Agache, M; Oommen, B J
2002-01-01
The fastest learning automata (LA) algorithms currently available fall in the family of estimator algorithms introduced by Thathachar and Sastry (1986). The pioneering work of these authors was the pursuit algorithm, which pursues only the current estimated optimal action. If this action is not the one with the minimum penalty probability, this algorithm pursues a wrong action. In this paper, we argue that a pursuit scheme that generalizes the traditional pursuit algorithm by pursuing all the actions with higher reward estimates than the chosen action, minimizes the probability of pursuing a wrong action, and is a faster converging scheme. To attest this, we present two new generalized pursuit algorithms (GPAs) and also present a quantitative comparison of their performance against the existing pursuit algorithms. Empirically, the algorithms proposed here are among the fastest reported LA to date. PMID:18244880
Implicit-Explicit Runge-Kutta schemes for numerical discretization of optimal control problems
Herty, Michael; Pareschi, Lorenzo; Steffensen, Sonja
2012-01-01
Implicit-explicit (IMEX) Runge-Kutta methods play a major rule in the numerical treatment of differential systems governed by stiff and non-stiff terms. This paper discusses order conditions and symplecticity properties of a class of IMEX Runge-Kutta methods in the context of optimal control problems. The analysis of the schemes is based on the continuous optimality system. Using suitable transformations of the adjoint equation, order conditions up to order three are proven as well as the rel...
Tam, Christopher K. W.; Webb, Jay C.
1994-01-01
In this paper finite-difference solutions of the Helmholtz equation in an open domain are considered. By using a second-order central difference scheme and the Bayliss-Turkel radiation boundary condition, reasonably accurate solutions can be obtained when the number of grid points per acoustic wavelength used is large. However, when a smaller number of grid points per wavelength is used excessive reflections occur which tend to overwhelm the computed solutions. Excessive reflections are due to the incompability between the governing finite difference equation and the Bayliss-Turkel radiation boundary condition. The Bayliss-Turkel radiation boundary condition was developed from the asymptotic solution of the partial differential equation. To obtain compatibility, the radiation boundary condition should be constructed from the asymptotic solution of the finite difference equation instead. Examples are provided using the improved radiation boundary condition based on the asymptotic solution of the governing finite difference equation. The computed results are free of reflections even when only five grid points per wavelength are used. The improved radiation boundary condition has also been tested for problems with complex acoustic sources and sources embedded in a uniform mean flow. The present method of developing a radiation boundary condition is also applicable to higher order finite difference schemes. In all these cases no reflected waves could be detected. The use of finite difference approximation inevita bly introduces anisotropy into the governing field equation. The effect of anisotropy is to distort the directional distribution of the amplitude and phase of the computed solution. It can be quite large when the number of grid points per wavelength used in the computation is small. A way to correct this effect is proposed. The correction factor developed from the asymptotic solutions is source independent and, hence, can be determined once and for all. The effectiveness of the correction factor in providing improvements to the computed solution is demonstrated in this paper.
Improving sub-grid scale accuracy of boundary features in regional finite-difference models
Panday, Sorab; Langevin, Christian D.
2012-01-01
As an alternative to grid refinement, the concept of a ghost node, which was developed for nested grid applications, has been extended towards improving sub-grid scale accuracy of flow to conduits, wells, rivers or other boundary features that interact with a finite-difference groundwater flow model. The formulation is presented for correcting the regular finite-difference groundwater flow equations for confined and unconfined cases, with or without Newton Raphson linearization of the nonlinearities, to include the Ghost Node Correction (GNC) for location displacement. The correction may be applied on the right-hand side vector for a symmetric finite-difference Picard implementation, or on the left-hand side matrix for an implicit but asymmetric implementation. The finite-difference matrix connectivity structure may be maintained for an implicit implementation by only selecting contributing nodes that are a part of the finite-difference connectivity. Proof of concept example problems are provided to demonstrate the improved accuracy that may be achieved through sub-grid scale corrections using the GNC schemes.
Improving sub-grid scale accuracy of boundary features in regional finite-difference models
Panday, Sorab; Langevin, Christian D.
2012-06-01
As an alternative to grid refinement, the concept of a ghost node, which was developed for nested grid applications, has been extended towards improving sub-grid scale accuracy of flow to conduits, wells, rivers or other boundary features that interact with a finite-difference groundwater flow model. The formulation is presented for correcting the regular finite-difference groundwater flow equations for confined and unconfined cases, with or without Newton Raphson linearization of the nonlinearities, to include the Ghost Node Correction (GNC) for location displacement. The correction may be applied on the right-hand side vector for a symmetric finite-difference Picard implementation, or on the left-hand side matrix for an implicit but asymmetric implementation. The finite-difference matrix connectivity structure may be maintained for an implicit implementation by only selecting contributing nodes that are a part of the finite-difference connectivity. Proof of concept example problems are provided to demonstrate the improved accuracy that may be achieved through sub-grid scale corrections using the GNC schemes.
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.
International Nuclear Information System (INIS)
The majority of the existing reactor system analysis codes were developed using low-order numerical schemes in both space and time. In many nuclear thermal—hydraulics applications, it is desirable to use higher-order numerical schemes to reduce numerical errors. High-resolution spatial discretization schemes provide high order spatial accuracy in smooth regions and capture sharp spatial discontinuity without nonphysical spatial oscillations. In this work, we adapted an existing high-resolution spatial discretization scheme on staggered grids in two-phase flow applications. Fully implicit time integration schemes were also implemented to reduce numerical errors from operator-splitting types of time integration schemes. The resulting nonlinear system has been successfully solved using the Jacobian-free Newton—Krylov (JFNK) method. The high-resolution spatial discretization and high-order fully implicit time integration numerical schemes were tested and numerically verified for several two-phase test problems, including a two-phase advection problem, a two-phase advection with phase appearance/disappearance problem, and the water faucet problem. Numerical results clearly demonstrated the advantages of using such high-resolution spatial and high-order temporal numerical schemes to significantly reduce numerical diffusion and therefore improve accuracy. Our study also demonstrated that the JFNK method is stable and robust in solving two-phase flow problems, even when phase appearance/disappearance exists.
Energy Technology Data Exchange (ETDEWEB)
Zou, Ling [Idaho National Lab. (INL), Idaho Falls, ID (United States); Zhao, Haihua [Idaho National Lab. (INL), Idaho Falls, ID (United States); Zhang, Hongbin [Idaho National Lab. (INL), Idaho Falls, ID (United States)
2015-09-01
The majority of the existing reactor system analysis codes were developed using low-order numerical schemes in both space and time. In many nuclear thermal–hydraulics applications, it is desirable to use higher-order numerical schemes to reduce numerical errors. High-resolution spatial discretization schemes provide high order spatial accuracy in smooth regions and capture sharp spatial discontinuity without nonphysical spatial oscillations. In this work, we adapted an existing high-resolution spatial discretization scheme on staggered grids in two-phase flow applications. Fully implicit time integration schemes were also implemented to reduce numerical errors from operator-splitting types of time integration schemes. The resulting nonlinear system has been successfully solved using the Jacobian-free Newton–Krylov (JFNK) method. The high-resolution spatial discretization and high-order fully implicit time integration numerical schemes were tested and numerically verified for several two-phase test problems, including a two-phase advection problem, a two-phase advection with phase appearance/disappearance problem, and the water faucet problem. Numerical results clearly demonstrated the advantages of using such high-resolution spatial and high-order temporal numerical schemes to significantly reduce numerical diffusion and therefore improve accuracy. Our study also demonstrated that the JFNK method is stable and robust in solving two-phase flow problems, even when phase appearance/disappearance exists.
Indian Academy of Sciences (India)
Bilge Inan; Ahmet Refik Bahadir
2013-10-01
This paper describes two new techniques which give improved exponential finite difference solutions of Burgers’ equation. These techniques are called implicit exponential finite difference method and fully implicit exponential finite difference method for solving Burgers’ equation. As the Burgers’ equation is nonlinear, the scheme leads to a system of nonlinear equations. At each time-step, Newton’s method is used to solve this nonlinear system. The results are compared with exact values and it is clearly shown that results obtained using both the methods are precise and reliable.
Investigation of Calculation Techniques of Finite Difference Method
Directory of Open Access Journals (Sweden)
Audrius Krukonis
2011-03-01
Full Text Available Finite difference method used for microstrip transmission line analysis is considered in this article. Paper mainly deals with iterative and bound matrix calculation techniques of finite difference method. Mathematical model for microstrip transmission line electrical potential calculations using both techniques is described. Results of characteristic impedance calculation using iterative and bound matrix techniques are presented and analyzed.Article in Lithuanian
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 (?X2/?? = 2 was found to be reliable and accurate for prediction purposes."
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.
Discrete Wavelet Transform Method: A New Optimized Robust Digital Image Watermarking Scheme
Directory of Open Access Journals (Sweden)
Hassan Talebi
2012-11-01
Full Text Available 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. Robustness of the proposed algorithm was tested against the following attacks: JPEG2000 and old JPEG compression, adding salt and pepper noise, median filtering, rotating, cropping and scaling. The promising experimental results are reported and discussed.
International Nuclear Information System (INIS)
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
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
THE SOLUTION OF THE CABLE EQUATIONS BY MEANS OF FINITE DIFFERENCE TIME DOMAIN METHOD
Directory of Open Access Journals (Sweden)
Patsiuk V.I.
2010-04-01
Full Text Available The analysis and comparison of accuracy of numerical solutions received by Finite Difference Time Domain (FDTD method and Godunov's method at the solution of the cable equations is carried out. It is demonstrated, that at sudden short circuits and at transition to idling mode in numerical solutions received by means of FDTD method for long lines with the distributed parameters appear strong nonphysical oscillations. It is shown, that the settlement scheme offered by authors on the basis of Godunov's method is deprived these lacks and provides high accuracy for the numerical solutions received at the analysis of dynamic modes in long lines, caused by sudden short circuits and line transitions in an idling mode. Key words: cable equations, finite difference time domain method, Godunov’s scheme.
Optimal variable-grid finite-difference modeling for porous media
International Nuclear Information System (INIS)
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. (paper)
Chu, Chunlei
2012-01-01
Discrete earth models are commonly represented by uniform structured grids. In order to ensure accurate numerical description of all wave components propagating through these uniform grids, the grid size must be determined by the slowest velocity of the entire model. Consequently, high velocity areas are always oversampled, which inevitably increases the computational cost. A practical solution to this problem is to use nonuniform grids. We propose a nonuniform grid implicit spatial finite difference method which utilizes nonuniform grids to obtain high efficiency and relies on implicit operators to achieve high accuracy. We present a simple way of deriving implicit finite difference operators of arbitrary stencil widths on general nonuniform grids for the first and second derivatives and, as a demonstration example, apply these operators to the pseudo-acoustic wave equation in tilted transversely isotropic (TTI) media. We propose an efficient gridding algorithm that can be used to convert uniformly sampled models onto vertically nonuniform grids. We use a 2D TTI salt model to demonstrate its effectiveness and show that the nonuniform grid implicit spatial finite difference method can produce highly accurate seismic modeling results with enhanced efficiency, compared to uniform grid explicit finite difference implementations. © 2011 Elsevier B.V.
AnisWave2D: User's Guide to the 2d Anisotropic Finite-Difference Code
Toomey, Aoife
2005-01-01
This document describes a parallel finite-difference code for modeling wave propagation in 2D, fully anisotropic materials. The code utilizes a mesh refinement scheme to improve computational efficiency. Mesh refinement allows the grid spacing to be tailored to the velocity model, so that fine grid spacing can be used in low velocity zones where the seismic wavelength is short, and coarse grid spacing can be used in zones with higher material velocities. Over-sampling of the seismic wave...
Cristea, Artur; Sofonea, Victor
2003-01-01
The origin of the spurious interface velocity in finite difference lattice Boltzmann models for liquid - vapor systems is related to the first order upwind scheme used to compute the space derivatives in the evolution equations. A correction force term is introduced to eliminate the spurious velocity. The correction term helps to recover sharp interfaces and sets the phase diagram close to the one derived using the Maxwell construction.
Vincenti, H.; Vay, J.-L.
2016-03-01
Very high order or pseudo-spectral Maxwell solvers are the method of choice to reduce discretization effects (e.g. numerical dispersion) that are inherent to low order Finite-Difference Time-Domain (FDTD) schemes. However, due to their large stencils, these solvers are often subject to truncation errors in many electromagnetic simulations. These truncation errors come from non-physical modifications of Maxwell's equations in space that may generate spurious signals affecting the overall accuracy of the simulation results. Such modifications for instance occur when Perfectly Matched Layers (PMLs) are used at simulation domain boundaries to simulate open media. Another example is the use of arbitrary order Maxwell solver with domain decomposition technique that may under some condition involve stencil truncations at subdomain boundaries, resulting in small spurious errors that do eventually build up. In each case, a careful evaluation of the characteristics and magnitude of the errors resulting from these approximations, and their impact at any frequency and angle, requires detailed analytical and numerical studies. To this end, we present a general analytical approach that enables the evaluation of numerical errors of fully three-dimensional arbitrary order finite-difference Maxwell solver, with arbitrary modification of the local stencil in the simulation domain. The analytical model is validated against simulations of domain decomposition technique and PMLs, when these are used with very high-order Maxwell solver, as well as in the infinite order limit of pseudo-spectral solvers. Results confirm that the new analytical approach enables exact predictions in each case. It also confirms that the domain decomposition technique can be used with very high-order Maxwell solvers and a reasonably low number of guard cells with negligible effects on the whole accuracy of the simulation.
International Nuclear Information System (INIS)
Highlights: • Using high-resolution spatial scheme in solving two-phase flow problems. • Fully implicit time integrations scheme. • Jacobian-free Newton–Krylov method. • Analytical solution for two-phase water faucet problem. - Abstract: The majority of the existing reactor system analysis codes were developed using low-order numerical schemes in both space and time. In many nuclear thermal–hydraulics applications, it is desirable to use higher-order numerical schemes to reduce numerical errors. High-resolution spatial discretization schemes provide high order spatial accuracy in smooth regions and capture sharp spatial discontinuity without nonphysical spatial oscillations. In this work, we adapted an existing high-resolution spatial discretization scheme on staggered grids in two-phase flow applications. Fully implicit time integration schemes were also implemented to reduce numerical errors from operator-splitting types of time integration schemes. The resulting nonlinear system has been successfully solved using the Jacobian-free Newton–Krylov (JFNK) method. The high-resolution spatial discretization and high-order fully implicit time integration numerical schemes were tested and numerically verified for several two-phase test problems, including a two-phase advection problem, a two-phase advection with phase appearance/disappearance problem, and the water faucet problem. Numerical results clearly demonstrated the advantages of using such high-resolution spatial and high-order temporal numerical schemes to significantly reduce numerical diffusion and therefore improve accuracy. Our study also demonstrated that the JFNK method is stable and robust in solving two-phase flow problems, even when phase appearance/disappearance exists
Techniques for correcting approximate finite difference solutions. [considering transonic flow
Nixon, D.
1978-01-01
A method of correcting finite-difference solutions for the effect of truncation error or the use of an approximate basic equation is presented. Applications to transonic flow problems are described and examples are given.
Optimization of plate thickness using finite different method
International Nuclear Information System (INIS)
A finite difference numerical method of solving biharmonic equation is presented. The biharmonic equation and plate theory are used to solve a classical engineering problem involving the optimisation of plate thickness to minimise deformations and stresses in the plate. Matlab routines were developed to solve the resulting finite difference equations. The results from the finite difference method were compared with results obtained using ANSYS finite element formulation. Using the finite difference method, a plate thickness of 277 mm was obtained with a mesh size of 3 m and a plate thickness of 271 mm was obtained with a mesh size of 1 m., whiles using ANSYS finite element formulation, a plate thickness of 270 mm was obtained. The significance of these results is that, by using off-the-shelf general application tool and without resorting to expensive dedicated application tool, simple engineering problems could be solved. (au)
exponential finite difference technique for solving partial differential equations
International Nuclear Information System (INIS)
An exponential finite difference algorithm, as first presented by Bhattacharya for one-dimensianal steady-state, heat conduction in Cartesian coordinates, has been extended. The finite difference algorithm developed was used to solve the diffusion equation in one-dimensional cylindrical coordinates and applied to two- and three-dimensional problems in Cartesian coordinates. The method was also used to solve nonlinear partial differential equations in one (Burger's equation) and two (Boundary Layer equations) dimensional Cartesian coordinates. Predicted results were compared to exact solutions where available, or to results obtained by other numerical methods. It was found that the exponential finite difference method produced results that were more accurate than those obtained by other numerical methods, especially during the initial transient portion of the solution. Other applications made using the exponential finite difference technique included unsteady one-dimensional heat transfer with temperature varying thermal conductivity and the development of the temperature field in a laminar Couette flow
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)
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
International Nuclear Information System (INIS)
An extension of the finite difference time domain is applied to solve the Schroedinger equation. A systematic analysis of stability and convergence of this technique is carried out in this article. The numerical scheme used to solve the Schroedinger equation differs from the scheme found in electromagnetics. Also, the unit cell employed to model quantum devices is different from the Yee cell used by the electrical engineering community. A bound for the time step is derived to ensure stability. Several numerical experiments in quantum structures demonstrate the accuracy of a second order, comparable to the analysis of electromagnetic devices with the Yee cell
Jang, Jihyeon; Hong, Song-You
2015-10-01
The spectral method is generally assumed to provide better numerical accuracy than the finite difference method. However, the majority of regional models use finite discretization methods due to the difficulty of specifying time-dependent lateral boundary conditions in spectral models. This study evaluates the behavior of nonhydrostatic dynamics with a spectral discretization. To this end, Juang's nonhydrostatic dynamical core for the National Centers for Environmental Prediction (NCEP) regional spectral model has been implemented into the Regional Model Program (RMP) of the Global/Regional Integrated Model system (GRIMs). The behavior of the nonhydrostatic RMP is validated, and compared with that of the hydrostatic core in 2-D idealized experiments: the mountain wave, rising thermal bubble, and density current experiments. The nonhydrostatic effect in the RMP is further validated in comparison with the results from the Weather Research and Forecasting (WRF) model, which uses a finite difference method. The analyses of the experimental results from the RMP generally follow the characteristics found in previous studies without any discernible difference. For example, in both the RMP and the WRF model, the eastward-tilted propagation of mountain waves is very similar in the nonhydrostatic core experiments. Both nonhydrostatic models also efficiently reproduce the motion and deformation of the warm and cold bubbles, but the RMP results contain more small-scale noise. In a 1-km real-case simulation testbed, the lee waves that originate over the eastern flank of the Korean peninsula travel further eastward in the WRF model than in the RMP. It is found that differences of small-scale wave characteristics between the RMP and WRF model are mainly from the numerical techniques used, such as the accuracy of the advection scheme and the magnitude of the numerical diffusion, rather than from discrepancies in the spatial discretization.
Jang, Jihyeon; Hong, Song-You
2016-04-01
The spectral method is generally assumed to provide better numerical accuracy than the finite difference method. However, the majority of regional models use finite discretization methods due to the difficulty of specifying time-dependent lateral boundary conditions in spectral models. This study evaluates the behavior of nonhydrostatic dynamics with a spectral discretization. To this end, Juang's nonhydrostatic dynamical core for the National Centers for Environmental Prediction (NCEP) regional spectral model has been implemented into the Regional Model Program (RMP) of the Global/Regional Integrated Model system (GRIMs). The behavior of the nonhydrostatic RMP is validated, and compared with that of the hydrostatic core in 2-D idealized experiments: the mountain wave, rising thermal bubble, and density current experiments. The nonhydrostatic effect in the RMP is further validated in comparison with the results from the Weather Research and Forecasting (WRF) model, which uses a finite difference method. The analyses of the experimental results from the RMP generally follow the characteristics found in previous studies without any discernible difference. For example, in both the RMP and the WRF model, the eastward-tilted propagation of mountain waves is very similar in the nonhydrostatic core experiments. Both nonhydrostatic models also efficiently reproduce the motion and deformation of the warm and cold bubbles, but the RMP results contain more small-scale noise. In a 1-km real-case simulation testbed, the lee waves that originate over the eastern flank of the Korean peninsula travel further eastward in the WRF model than in the RMP. It is found that differences of small-scale wave characteristics between the RMP and WRF model are mainly from the numerical techniques used, such as the accuracy of the advection scheme and the magnitude of the numerical diffusion, rather than from discrepancies in the spatial discretization.
On the Definition of Surface Potentials for Finite-Difference Operators
Tsynkov, S. V.; Bushnell, Dennis M. (Technical Monitor)
2001-01-01
For a class of linear constant-coefficient finite-difference operators of the second order, we introduce the concepts similar to those of conventional single- and double-layer potentials for differential operators. The discrete potentials are defined completely independently of any notion related to the approximation of the continuous potentials on the grid. We rather use all approach based on differentiating, and then inverting the differentiation of a function with surface discontinuity of a particular kind, which is the most general way of introducing surface potentials in the theory of distributions. The resulting finite-difference "surface" potentials appear to be solutions of the corresponding continuous potentials. Primarily, this pertains to the possibility of representing a given solution to the homogeneous equation on the domain as a variety of surface potentials, with the density defined on the domain's boundary. At the same time the discrete surface potentials can be interpreted as one specific realization of the generalized potentials of Calderon's type, and consequently, their approximation properties can be studied independently in the framework of the difference potentials method by Ryaben'kii. The motivation for introducing and analyzing the discrete surface potentials was provided by the problems of active shielding and control of sound, in which the aforementioned source terms that drive the potentials are interpreted as the acoustic control sources that cancel out the unwanted noise on a predetermined region of interest.
Directory of Open Access Journals (Sweden)
A. Caserta
1998-06-01
Full Text Available This paper deals with the antiplane wave propagation in a 2D heterogeneous dissipative medium with complex layer interfaces and irregular topography. The initial boundary value problem which represents the viscoelastic dynamics driving 2D antiplane wave propagation is formulated. The discretization scheme is based on the finite-difference technique. Our approach presents some innovative features. First, the introduction of the forcing term into the equation of motion offers the advantage of an easier handling of different inputs such as general functions of spatial coordinates and time. Second, in the case of a straight-line source, the symmetry of the incident plane wave allows us to solve the problem of oblique incidence simply by rotating the 2D model. This artifice reduces the oblique incidence to the vertical one. Third, the conventional rheological model of the generalized Maxwell body has been extended to include the stress-free boundary condition. For this reason we solve explicitly the stress-free boundary condition, not following the most popular technique called vacuum formalism. Finally, our numerical code has been constructed to model the seismic response of complex geological structures: real geological interfaces are automatically digitized and easily introduced in the input model. Three numerical applications are discussed. To validate our numerical model, the first test compares the results of our code with others shown in the literature. The second application rotates the input model to simulate the oblique incidence. The third one deals with a real high-complexity 2D geological structure.
Stability and non-standard finite difference method of the generalized Chua's circuit
Radwan, Ahmed
2011-08-01
In this paper, we develop a framework to obtain approximate numerical solutions of the fractional-order Chua\\'s circuit with Memristor using a non-standard finite difference method. Chaotic response is obtained with fractional-order elements as well as integer-order elements. Stability analysis and the condition of oscillation for the integer-order system are discussed. In addition, the stability analyses for different fractional-order cases are investigated showing a great sensitivity to small order changes indicating the poles\\' locations inside the physical s-plane. The GrnwaldLetnikov method is used to approximate the fractional derivatives. Numerical results are presented graphically and reveal that the non-standard finite difference scheme is an effective and convenient method to solve fractional-order chaotic systems, and to validate their stability. © 2011 Elsevier Ltd. All rights reserved.
Convergence of finite difference transient response computations for thin shells.
Sobel, L. H.; Geers, T. L.
1973-01-01
Numerical studies pertaining to the limits of applicability of the finite difference method in the solution of linear transient shell response problems are performed, and a computational procedure for the use of the method is recommended. It is found that the only inherent limitation of the finite difference method is its inability to reproduce accurately response discontinuities. This is not a serious limitation in view of natural constraints imposed by the extension of Saint Venant's principle to transient response problems. It is also found that the short wavelength limitations of thin shell (Bernoulli-Euler) theory create significant convergence difficulties in computed response to certain types of transverse excitations. These difficulties may be overcome, however, through proper selection of finite difference mesh dimensions and temporal smoothing of the excitation.
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.
Finite-Difference Frequency-Domain Method in Nanophotonics
DEFF Research Database (Denmark)
Ivinskaya, Aliaksandra
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...
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...
Higher-order finite-difference formulation of periodic Orbital-free Density Functional Theory
Ghosh, Swarnava; Suryanarayana, Phanish
2016-02-01
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 for performing OF-DFT simulations with different variants of the electronic kinetic energy. In particular, we propose a self-consistent field (SCF) type fixed-point method for calculations involving linear-response kinetic energy functionals. In this framework, evaluation of both the electronic ground-state and forces on the nuclei are amenable to computations that scale linearly with the number of atoms. We develop a parallel implementation of this formulation using the finite-difference discretization. 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 converges rapidly, and that it can be further accelerated using extrapolation techniques like Anderson's mixing. We validate the accuracy of the results by comparing the energies and forces with plane-wave methods for selected examples, including the vacancy formation energy in Aluminum. Overall, the suitability of the proposed formulation for scalable high performance computing makes it an attractive choice for large-scale OF-DFT calculations consisting of thousands of atoms.
Hejranfar, Kazem; Ezzatneshan, Eslam
2015-11-01
A high-order compact finite-difference lattice Boltzmann method (CFDLBM) is extended and applied to accurately simulate two-phase liquid-vapor flows with high density ratios. Herein, the He-Shan-Doolen-type lattice Boltzmann multiphase model is used and the spatial derivatives in the resulting equations 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 two-phase flow solver. A high-order spectral-type low-pass compact nonlinear filter is used to regularize the numerical solution and remove spurious waves generated by flow nonlinearities in smooth regions and at the same time to remove the numerical oscillations in the interfacial region between the two phases. Three discontinuity-detecting sensors for properly switching between a second-order and a higher-order filter are applied and assessed. It is shown that the filtering technique used can be conveniently adopted to reduce the spurious numerical effects and improve the numerical stability of the CFDLBM implemented. A sensitivity study is also conducted to evaluate the effects of grid size and the filtering procedure implemented on the accuracy and performance of the solution. The accuracy and efficiency of the proposed solution procedure based on the compact finite-difference LBM are examined by solving different two-phase systems. Five test cases considered herein for validating the results of the two-phase flows are an equilibrium state of a planar interface in a liquid-vapor system, a droplet suspended in the gaseous phase, a liquid droplet located between two parallel wettable surfaces, the coalescence of two droplets, and a phase separation in a liquid-vapor system at different conditions. Numerical results are also presented for the coexistence curve and the verification of the Laplace law. Results obtained are in good agreement with the analytical solutions and also the numerical results reported in the literature. The study shows that the present solution methodology is robust, efficient, and accurate for solving two-phase liquid-vapor flow problems even at high density ratios.
Hejranfar, Kazem; Ezzatneshan, Eslam
2015-11-01
A high-order compact finite-difference lattice Boltzmann method (CFDLBM) is extended and applied to accurately simulate two-phase liquid-vapor flows with high density ratios. Herein, the He-Shan-Doolen-type lattice Boltzmann multiphase model is used and the spatial derivatives in the resulting equations 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 two-phase flow solver. A high-order spectral-type low-pass compact nonlinear filter is used to regularize the numerical solution and remove spurious waves generated by flow nonlinearities in smooth regions and at the same time to remove the numerical oscillations in the interfacial region between the two phases. Three discontinuity-detecting sensors for properly switching between a second-order and a higher-order filter are applied and assessed. It is shown that the filtering technique used can be conveniently adopted to reduce the spurious numerical effects and improve the numerical stability of the CFDLBM implemented. A sensitivity study is also conducted to evaluate the effects of grid size and the filtering procedure implemented on the accuracy and performance of the solution. The accuracy and efficiency of the proposed solution procedure based on the compact finite-difference LBM are examined by solving different two-phase systems. Five test cases considered herein for validating the results of the two-phase flows are an equilibrium state of a planar interface in a liquid-vapor system, a droplet suspended in the gaseous phase, a liquid droplet located between two parallel wettable surfaces, the coalescence of two droplets, and a phase separation in a liquid-vapor system at different conditions. Numerical results are also presented for the coexistence curve and the verification of the Laplace law. Results obtained are in good agreement with the analytical solutions and also the numerical results reported in the literature. The study shows that the present solution methodology is robust, efficient, and accurate for solving two-phase liquid-vapor flow problems even at high density ratios. PMID:26651814
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.
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 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
Algorithmic vs. finite difference Jacobians for infrared atmospheric radiative transfer
International Nuclear Information System (INIS)
Jacobians, i.e. partial derivatives of the radiance and transmission spectrum with respect to the atmospheric state parameters to be retrieved from remote sensing observations, are important for the iterative solution of the nonlinear inverse problem. Finite difference Jacobians are easy to implement, but computationally expensive and possibly of dubious quality; on the other hand, analytical Jacobians are accurate and efficient, but the implementation can be quite demanding. GARLIC, our “Generic Atmospheric Radiation Line-by-line Infrared Code”, utilizes algorithmic differentiation (AD) techniques to implement derivatives w.r.t. atmospheric temperature and molecular concentrations. In this paper, we describe our approach for differentiation of the high resolution infrared and microwave spectra and provide an in-depth assessment of finite difference approximations using “exact” AD Jacobians as a reference. The results indicate that the “standard” two-point finite differences with 1 K and 1% perturbation for temperature and volume mixing ratio, respectively, can exhibit substantial errors, and central differences are significantly better. However, these deviations do not transfer into the truncated singular value decomposition solution of a least squares problem. Nevertheless, AD Jacobians are clearly recommended because of the superior speed and accuracy. - Highlights: • Line-by-line infrared radiative transfer modeling. • Temperature and concentration Jacobians by algorithmic/automatic differentiation. • Two-point finite differences can have large errors, central differences are much better. • Singular value analysis reveals minor sensitivity
A Finite Difference Element Method for thin elastic Shells
Cho, Daniel
2009-01-01
We present, in this paper, a four nodes quadrangular shell element (FDEM4) based on a Finite Difference Element Method that we introduce. Its stability and robustness with respect to shear locking and membrane locking problems is discussed. Numerical tests including inhibited and non-inhibited cases of thin linear shells are presented and compared with widely used DKT and MITC4 elements.
High-order compact difference scheme for the numerical solution of time fractional heat equations.
Karatay, Ibrahim; Bayramoglu, Serife R
2014-01-01
A high-order finite difference scheme is proposed for solving time fractional heat equations. The time fractional derivative is described in the Riemann-Liouville sense. In the proposed scheme a new second-order discretization, which is based on Crank-Nicholson method, is applied for the time fractional part and fourth-order accuracy compact approximation is applied for the second-order space derivative. The spectral stability and the Fourier stability analysis of the difference scheme are shown. Finally a detailed numerical analysis, including tables, figures, and error comparison, is given to demonstrate the theoretical results and high accuracy of the proposed scheme. PMID:24696040
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 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.
Georgios S Stamatakos; Dionysiou, Dimitra D.
2009-01-01
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 mathe...
Camporeale, Enrico; Delzanno, Gian Luca; Bergen, Benjamin K.; Moulton, J. David
2013-01-01
We discuss a spectral method for the numerical solution of the Vlasov-Poisson system where the velocity space is decomposed by means of an Hermite basis. We describe a semi-implicit time discretization that extends the range of numerical stability relative to an explicit scheme. We also introduce and discuss the effects of an artificial collisional operator, which is necessary to take care of the velocity space filamentation problem, unavoidable in collisionless plasmas. The computational eff...
International Nuclear Information System (INIS)
As an effort to achieve efficient yet accurate transport transient calculations for power reactors, the conditional transport update scheme in method of characteristics (MOC) based coarse mesh finite difference (CMFD) transient calculation is developed. In this scheme, the transport calculations serves as an online group constant generator for the 3-D CMFD transient calculation and the solution of 3-D transient problem is mainly obtained from the 3-D CMFD transient calculation. In order to reduce the computational burden of the intensive transport calculation, the transport updates is conditionally performed by monitoring change of composition and core condition. This efficient transient transport method is applied to 3x3 assembly rod ejection problem to examine the effectiveness and accuracy of the conditional transport calculation scheme. (author)
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
International Nuclear Information System (INIS)
A code is presented for the numerical solution of the Boussinesq equations by means of finite differences. To deal with general complex geometries non orthogonal boundary fitted coordinates are used, which allow an arbitrary choice of the coordinate lines. It does not yield the loss of accuracy, inherent in classical finite difference schemes. Boundary conditions were examined in detail for velocity and temperature. The report describes two first applications with or without heat transfer: the flow in a cooling-tower (Navier-Stokes) and the flow in a pool of a fast breeder (Boussinesq with natural convection)
Tadghighi, Hormoz; Hassan, Ahmed A.; Charles, Bruce
1990-01-01
The present numerical finite-difference scheme for helicopter blade-load prediction during realistic, self-generated three-dimensional blade-vortex interactions (BVI) derives the velocity field through a nonlinear superposition of the rotor flow-field yielded by the full potential rotor flow solver RFS2 for BVI, on the one hand, over the rotational vortex flow field computed with the Biot-Savart law. Despite the accurate prediction of the acoustic waveforms, peak amplitudes are found to have been persistently underpredicted. The inclusion of BVI noise source in the acoustic analysis significantly improved the perceived noise level-corrected tone prediction.
High-order finite difference solution for 3D nonlinear wave-structure interaction
DEFF Research Database (Denmark)
Ducrozet, Guillaume; Bingham, Harry B.; Engsig-Karup, Allan Peter; Ferrant, Pierre
2010-01-01
This contribution presents our recent progress on developing an efficient fully-nonlinear potential flow model for simulating 3D wave-wave and wave-structure interaction over arbitrary depths (i.e. in coastal and offshore environment). The model is based on a high-order finite difference scheme...... OceanWave3D presented in [1, 2]. A nonlinear decomposition of the solution into incident and scattered fields is used to increase the efficiency of the wave-structure interaction problem resolution. Application of the method to the diffraction of nonlinear waves around a fixed, bottom mounted circular...
Osher, S.
1984-01-01
The construction of a reliable, shock capturing finite difference method to solve the Euler equations for inviscid, supersonic flow past fighter and missile type configurations is highly desirable. The numerical method must have a firm theoretical foundation and must be robust and efficient. It should be able to treat subsonic pockets in a predominantly supersonic flow. The method must also be easily applicable to the complex topologies of the aerodynamic configuration under consideration. The ongoing approach to this task is described and for steady supersonic flows is presented. This scheme is the basic numerical method. Results of work obtained during previous years are presented.
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
Steger, J. L.
1978-01-01
Although the Navier-Stokes equations describe most flows of interest in aerodynamics, the inviscid conservation law equations may be used for small regions with viscous forces. Thus, Euler equations and several time-accurate finite difference procedures, explicit and implicit, are discussed. Although implicit techniques require more computational work, they permit larger time steps to be taken without instability. It is noted that the Jacobian matrices for Euler equations in conservation-law form have certain eigenvalue-eigenvector properties which may be used to construct conservative-form coefficient matrices. This reduces the computation time of several implicit and semiimplicit schemes. Extensions of the basic approach to other areas are suggested.
Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups
Givental, Alexander; Lee, Yuan-Pin
2001-01-01
We conjecture that appropriate K-theoretic Gromov-Witten invariants of complex flag manifolds G/B are governed by finite-difference versions of Toda systems constructed in terms of the Langlands-dual quantized universal enveloping algebras U_q(g'). The conjecture is proved in the case of classical flag manifolds of the series A. The proof is based on a refinement of the famous Atiyah-Hirzebruch argument for rigidity of arithmetical genus applied to hyperquot-scheme compactifications of spaces...
Mattila, Keijo Kalervo; Hegele Júnior, Luiz Adolfo; Philippi, Paulo Cesar
2014-01-01
We propose isotropic finite differences for high-accuracy approximation of high-rank derivatives. These finite differences are based on direct application of lattice-Boltzmann stencils. The presented finite-difference expressions are valid in any dimension, particularly in two and three dimensions, and any lattice-Boltzmann stencil isotropic enough can be utilized. A theoretical basis for the proposed utilization of lattice-Boltzmann stencils in the approximation of high-rank derivatives is established. In particular, the isotropy and accuracy properties of the proposed approximations are derived directly from this basis. Furthermore, in this formal development, we extend the theory of Hermite polynomial tensors in the case of discrete spaces and present expressions for the discrete inner products between monomials and Hermite polynomial tensors. In addition, we prove an equivalency between two approaches for constructing lattice-Boltzmann stencils. For the numerical verification of the presented finite differences, we introduce 5th-, 6th-, and 8th-order two-dimensional lattice-Boltzmann stencils. PMID:24688360
Coelho, Pedro J.
2014-08-01
High order resolution schemes based on the NVD and TVD boundedness criteria are applied to radiative transfer problems using the DOM in two-dimensional unstructured triangular grids. The implementation of these schemes in unstructured grids requires approximations, and two implementations reported in the literature are compared with a new one. Three different methods have been used to calculate the gradient of the radiation intensity at the center of the control volumes. The various schemes are applied to several test problems, the results are compared with those obtained using the step scheme, the mean flux interpolation scheme and another high order scheme based on a truncated Taylor series expansion, and the most accurate implementations are identified. It is concluded that although the high order schemes perform much better than the others, they are not as accurate as in Cartesian coordinates, and their order of convergence is lower than in that case.
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.
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.
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 aplicao de um esquema de discretizao mais eficiente para volumes finitos, denominado Flux-Spline utilizando-se, para tal, de dois casos de transporte difusivo de umidade e calor, atravs de um meio poroso capilar. Os resultados da soluo numrica do sistema de equaes formado pelas equaes de Luikov mostram desempenho adequado do esquema para este tipo de problema, quando comparado ao tradicional esquema de diferena central e ao mtodo 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.
Gladden, Herbert J.; Ko, Ching L.; Boddy, Douglas E.
1995-01-01
A higher-order finite-difference technique is developed to calculate the developing-flow field of steady incompressible laminar flows in the entrance regions of circular pipes. Navier-Stokes equations governing the motion of such a flow field are solved by using this new finite-difference scheme. This new technique can increase the accuracy of the finite-difference approximation, while also providing the option of using unevenly spaced clustered nodes for computation such that relatively fine grids can be adopted for regions with large velocity gradients. The velocity profile at the entrance of the pipe is assumed to be uniform for the computation. The velocity distribution and the surface pressure drop of the developing flow then are calculated and compared to existing experimental measurements reported in the literature. Computational results obtained are found to be in good agreement with existing experimental correlations and therefore, the reliability of the new technique has been successfully tested.
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 properties with a h...
Calculating rotordynamic coefficients of seals by finite-difference techniques
Dietzen, F. J.; Nordmann, R.
1987-01-01
For modelling the turbulent flow in a seal the Navier-Stokes equations in connection with a turbulence (kappa-epsilon) model are solved by a finite-difference method. A motion of the shaft round the centered position is assumed. After calculating the corresponding flow field and the pressure distribution, the rotor-dynamic coefficients of the seal can be determined. These coefficients are compared with results obtained by using the bulk flow theory of Childs and with experimental results.
Finite difference methods applied to biot theory in porous medium
Shen, Jonah Wai
1995-01-01
Finite difference methods are used to solve the Biot equations for wave propagation in a porous medium. The computational domain is a two dimensional grid of uniform spacing where truncation of the grid on all sides is accomplished by applying homogeneous Dirichlet boundary conditions. The difference method is second order in space and time, and is seen to accurately predict phase speeds of the primary compressional and shear waves. (AN)
Algorithmic vs. finite difference Jacobians for infrared atmospheric radiative transfer
Schreier, Franz; Gimeno García, Sebastián; Vasquez, Mayte; Xu, Jian
2015-10-01
Jacobians, i.e. partial derivatives of the radiance and transmission spectrum with respect to the atmospheric state parameters to be retrieved from remote sensing observations, are important for the iterative solution of the nonlinear inverse problem. Finite difference Jacobians are easy to implement, but computationally expensive and possibly of dubious quality; on the other hand, analytical Jacobians are accurate and efficient, but the implementation can be quite demanding. GARLIC, our "Generic Atmospheric Radiation Line-by-line Infrared Code", utilizes algorithmic differentiation (AD) techniques to implement derivatives w.r.t. atmospheric temperature and molecular concentrations. In this paper, we describe our approach for differentiation of the high resolution infrared and microwave spectra and provide an in-depth assessment of finite difference approximations using "exact" AD Jacobians as a reference. The results indicate that the "standard" two-point finite differences with 1 K and 1% perturbation for temperature and volume mixing ratio, respectively, can exhibit substantial errors, and central differences are significantly better. However, these deviations do not transfer into the truncated singular value decomposition solution of a least squares problem. Nevertheless, AD Jacobians are clearly recommended because of the superior speed and accuracy.
Introduction to finite-difference methods for numerical fluid dynamics
Energy Technology Data Exchange (ETDEWEB)
Scannapieco, E.; Harlow, F.H.
1995-09-01
This work is intended to be a beginner`s exercise book for the study of basic finite-difference techniques in computational fluid dynamics. It is written for a student level ranging from high-school senior to university senior. Equations are derived from basic principles using algebra. Some discussion of partial-differential equations is included, but knowledge of calculus is not essential. The student is expected, however, to have some familiarity with the FORTRAN computer language, as the syntax of the computer codes themselves is not discussed. Topics examined in this work include: one-dimensional heat flow, one-dimensional compressible fluid flow, two-dimensional compressible fluid flow, and two-dimensional incompressible fluid flow with additions of the equations of heat flow and the {Kappa}-{epsilon} model for turbulence transport. Emphasis is placed on numerical instabilities and methods by which they can be avoided, techniques that can be used to evaluate the accuracy of finite-difference approximations, and the writing of the finite-difference codes themselves. Concepts introduced in this work include: flux and conservation, implicit and explicit methods, Lagrangian and Eulerian methods, shocks and rarefactions, donor-cell and cell-centered advective fluxes, compressible and incompressible fluids, the Boussinesq approximation for heat flow, Cartesian tensor notation, the Boussinesq approximation for the Reynolds stress tensor, and the modeling of transport equations. A glossary is provided which defines these and other terms.
Zhao, Jia; Yang, Xiaofeng; Shen, Jie; Wang, Qi
2016-01-01
We develop a linear, first-order, decoupled, energy-stable scheme for a binary hydrodynamic phase field model of mixtures of nematic liquid crystals and viscous fluids that satisfies an energy dissipation law. We show that the semi-discrete scheme in time satisfies an analogous, semi-discrete energy-dissipation law for any time-step and is therefore unconditionally stable. We then discretize the spatial operators in the scheme by a finite-difference method and implement the fully discrete scheme in a simplified version using CUDA on GPUs in 3 dimensions in space and time. Two numerical examples for rupture of nematic liquid crystal filaments immersed in a viscous fluid matrix are given, illustrating the effectiveness of this new scheme in resolving complex interfacial phenomena in free surface flows of nematic liquid crystals.
Baumeister, Kenneth J.; Kreider, Kevin L.
1996-01-01
An explicit finite difference iteration scheme is developed to study harmonic sound propagation in aircraft engine nacelles. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.
Baumeister, K. J.; Kreider, K. L.
1996-01-01
An explicit finite difference iteration scheme is developed to study harmonic sound propagation in ducts. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.
ATLAS: A Real-Space Finite-Difference Implementation of Orbital-Free Density Functional Theory
Mi, Wenhui; Sua, Chuanxun; Zhoua, Yuanyuan; Zhanga, Shoutao; Lia, Quan; Wanga, Hui; Zhang, Lijun; Miao, Maosheng; Wanga, Yanchao; Ma, Yanming
2015-01-01
Orbital-free density functional theory (OF-DFT) is a promising method for large-scale quantum mechanics simulation as it provides a good balance of accuracy and computational cost. Its applicability to large-scale simulations has been aided by progress in constructing kinetic energy functionals and local pseudopotentials. However, the widespread adoption of OF-DFT requires further improvement in its efficiency and robustly implemented software. Here we develop a real-space finite-difference method for the numerical solution of OF-DFT in periodic systems. Instead of the traditional self-consistent method, a powerful scheme for energy minimization is introduced to solve the Euler--Lagrange equation. Our approach engages both the real-space finite-difference method and a direct energy-minimization scheme for the OF-DFT calculations. The method is coded into the ATLAS software package and benchmarked using periodic systems of solid Mg, Al, and Al$_{3}$Mg. The test results show that our implementation can achieve ...
ATLAS: A real-space finite-difference implementation of orbital-free density functional theory
Mi, Wenhui; Shao, Xuecheng; Su, Chuanxun; Zhou, Yuanyuan; Zhang, Shoutao; Li, Quan; Wang, Hui; Zhang, Lijun; Miao, Maosheng; Wang, Yanchao; Ma, Yanming
2016-03-01
Orbital-free density functional theory (OF-DFT) is a promising method for large-scale quantum mechanics simulation as it provides a good balance of accuracy and computational cost. Its applicability to large-scale simulations has been aided by progress in constructing kinetic energy functionals and local pseudopotentials. However, the widespread adoption of OF-DFT requires further improvement in its efficiency and robustly implemented software. Here we develop a real-space finite-difference (FD) method for the numerical solution of OF-DFT in periodic systems. Instead of the traditional self-consistent method, a powerful scheme for energy minimization is introduced to solve the Euler-Lagrange equation. Our approach engages both the real-space finite-difference method and a direct energy-minimization scheme for the OF-DFT calculations. The method is coded into the ATLAS software package and benchmarked using periodic systems of solid Mg, Al, and Al3Mg. The test results show that our implementation can achieve high accuracy, efficiency, and numerical stability for large-scale simulations.
Schwarz, S.; Kempe, T.; Frhlich, 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 Frhlich (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.
International Nuclear Information System (INIS)
A code called COMESH based on corner mesh finite difference scheme has been developed to solve multigroup diffusion theory equations. One can solve 1-D, 2-D or 3-D problems in Cartesian geometry and 1-D (r) or 2-D (r-z) problem in cylindrical geometry. On external boundary one can use either homogeneous Dirichlet (θ-specified) or Neumann (∇θ specified) type boundary conditions or a linear combination of the two. Internal boundaries for control absorber simulations are also tackled by COMESH. Many an acceleration schemes like successive line over-relaxation, two parameter Chebyschev acceleration for fission source, generalised coarse mesh rebalancing etc., render the code COMESH a very fast one for estimating eigenvalue and flux/power profiles in any type of reactor core configuration. 6 refs. (author)
Five-point form of the nodal diffusion method and comparison with finite-difference
International Nuclear Information System (INIS)
Nodal Methods have been derived, implemented and numerically tested for several problems in physics and engineering. In the field of nuclear engineering, many nodal formalisms have been used for the neutron diffusion equation, all yielding results which were far more computationally efficient than conventional Finite Difference (FD) and Finite Element (FE) methods. However, not much effort has been devoted to theoretically comparing nodal and FD methods in order to explain the very high accuracy of the former. In this summary we outline the derivation of a simple five-point form for the lowest order nodal method and compare it to the traditional five-point, edge-centered FD scheme. The effect of the observed differences on the accuracy of the respective methods is established by considering a simple test problem. It must be emphasized that the nodal five-point scheme derived here is mathematically equivalent to previously derived lowest order nodal methods. 7 refs., 1 tab
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)
Santos, Frederico P.; Filho, Hermes Alves; Barros, Ricardo C.
2013-10-01
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 (low leakage). 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 (forward and backward) across the spatial grid. We consider a number of numerical experiments to illustrate the efficiency of the offered diffusion synthetic acceleration (DSA) technique.
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 time domain simulation for the brass instrument bore.
Bilbao, Stefan; Chick, John
2013-11-01
In this article, interleaved finite difference time domain methods are developed for the purpose of simulating the dynamics of the acoustic bore, using, as a starting point, an impedance formulation of wave propagation in an acoustic tube; attention is focused here on modeling of viscothermal and radiation losses in the time domain. In particular, in contrast to other methods, the bore, including the mouth-piece and bell, is treated as a unit, and is not subdivided into smaller units such as cylindrical or conical segments. Numerical simulations of input impedances are then compared with measurement for a variety of brass instruments. PMID:24180795
Finite element and finite difference methods in electromagnetic scattering
Morgan, MA
2013-01-01
This second volume in the Progress in Electromagnetic Research series examines recent advances in computational electromagnetics, with emphasis on scattering, as brought about by new formulations and algorithms which use finite element or finite difference techniques. Containing contributions by some of the world's leading experts, the papers thoroughly review and analyze this rapidly evolving area of computational electromagnetics. Covering topics ranging from the new finite-element based formulation for representing time-harmonic vector fields in 3-D inhomogeneous media using two coupled sca
Mimetic Finite Differences for Flow in Fractures from Microseismic Data
Al-Hinai, Omar
2015-01-01
We present a method for porous media flow in the presence of complex fracture networks. The approach uses the Mimetic Finite Difference method (MFD) and takes advantage of MFD\\'s ability to solve over a general set of polyhedral cells. This flexibility is used to mesh fracture intersections in two and three-dimensional settings without creating small cells at the intersection point. We also demonstrate how to use general polyhedra for embedding fracture boundaries in the reservoir domain. The target application is representing fracture networks inferred from microseismic analysis.
Application of a finite difference technique to thermal wave propagation
Baumeister, K. J.
1975-01-01
A finite difference formulation is presented for thermal wave propagation resulting from periodic heat sources. The numerical technique can handle complex problems that might result from variable thermal diffusivity, such as heat flow in the earth with ice and snow layers. In the numerical analysis, the continuous temperature field is represented by a series of grid points at which the temperature is separated into real and imaginary terms. Next, computer routines previously developed for acoustic wave propagation are utilized in the solution for the temperatures. The calculation procedure is illustrated for the case of thermal wave propagation in a uniform property semi-infinite medium.
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 (FD), finite-element (FE), and hybrid FD-FE 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. We present alternative formulations of equation of motion for a smooth elastic continuum. We then develop alternative formulations for a canonical problem with a welded material interface and free surface. We continue with a model of an earthquake source. We complete the general theoretical introduction by a chapter on the constitutive laws for elastic and viscoelastic media, and brief review of strong formulations of the equation of motion. What follows is a block of chapters on the finite-difference and finite-element methods. We develop FD targets for the free surface and welded material interface. We then present various FD schemes for a smooth continuum, free surface, and welded interface. We focus on the staggered-grid and mainly optimally-accurate FD schemes. We also present alternative formulations of the FE method. We include the FD and FE implementations of the traction-at-split-nodes method for simulation of dynamic rupture propagation. The FD modeling is applied to the model of the deep sedimentary Grenoble basin, France. The FD and FE methods are combined in the hybrid FD-FE method. The hybrid method is then applied to two earthquake scenarios for the Grenoble basin. Except chapters 1, 3, 5, and 12, all chapters include new, previously unpublished material and results. (author)
Energy conserving and potential-enstrophy dissipating schemes for the shallow water equations
Arakawa, Akio; Hsu, Yueh-Jiuan G.
1990-01-01
To incorporate potential enstrophy dissipation into discrete shallow water equations with no or arbitrarily small energy dissipation, a family of finite-difference schemes have been derived with which potential enstrophy is guaranteed to decrease while energy is conserved (when the mass flux is nondivergent and time is continuous). Among this family of schemes, there is a member that minimizes the spurious impact of infinite potential vorticities associated with infinitesimal fluid depth. The scheme is, therefore, useful for problems in which the free surface may intersect with the lower boundary.
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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.
Zehner, Björn; Hellwig, Olaf; Linke, Maik; Görz, Ines; Buske, Stefan
2016-01-01
3D geological underground models are often presented by vector data, such as triangulated networks representing boundaries of geological bodies and geological structures. Since models are to be used for numerical simulations based on the finite difference method, they have to be converted into a representation discretizing the full volume of the model into hexahedral cells. Often the simulations require a high grid resolution and are done using parallel computing. The storage of such a high-resolution raster model would require a large amount of storage space and it is difficult to create such a model using the standard geomodelling packages. Since the raster representation is only required for the calculation, but not for the geometry description, we present an algorithm and concept for rasterizing geological models on the fly for the use in finite difference codes that are parallelized by domain decomposition. As a proof of concept we implemented a rasterizer library and integrated it into seismic simulation software that is run as parallel code on a UNIX cluster using the Message Passing Interface. We can thus run the simulation with realistic and complicated surface-based geological models that are created using 3D geomodelling software, instead of using a simplified representation of the geological subsurface using mathematical functions or geometric primitives. We tested this set-up using an example model that we provide along with the implemented library.
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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
Finite-difference analysis of shells impacting rigid barriers
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The present investigation represents an initial attempt to develop an efficient numerical procedure for predicting the deformations and impact force time-histories of shells which impact upon a rigid target. The general large-deflection equations of motion of the shell are expressed in finite-difference form in space and integrated in time through application of the central-difference temporal operator. The effect of material nonlinearities is treated by a mechanical-sublayer material model which handles the strain-hardening, Bauschinger, and strain-rate effects. The general adequacy of this shell treatment has been validated by comparing predictions with the results of various experiments in which structures have been subjected to well-defined transient forcing functions (typically high-explosive impulse loading). The 'new' ingredient addressed in the present study involves an accounting for impact interaction and response of both the target structure and the attacking body. The impact capability of the code consists of two basic components: (a) an inspection technique which determines the occurrence and location of a collision between the shell and the target. (b) an impact force application technique which determines impact pressure based on shell penetration and penetration stiffness of the shell through the equilibrium equations to influence the response of the shell. By this procedure, the local collision analysis is combined simply in an efficient manner with the spatial and temporal finite-different solution procedure to predict the resulting transient nonlinear response of impacting shells
FDEHMT: a finite difference electromagnetic head modelling toolbox
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Hung Dang
2010-03-01
Full Text Available In EEG source imaging, the forward solution is used to obtain the scalp potentials given the source distribution in the brain. For accurate source localization, the forward solution must be performed numerically using a realistic head model. This study introduces the Finite Difference Electromagnetic Head Modelling Toolbox (FDEHMT, written in both C and MATLAB, which can be run as a standalone program or within other packages through command line interface. The FDEHMT is interfaced to FSL for anatomical MRI image segmentation using the BET and FAST commands. The segmented image is then used to construct the realistic head model and the electrode system is fitted to the scalp using the Bioelectroemagnetism toolbox. The finite difference formulation for the general inhomogeneous anisotropic bioelectric problem is implemented and the resulting matrix equation is solved using the iterative conjugate gradient algorithm. A new compression technique is used to reduce the memory required for the forward solver to less than 500 Mbytes for a realistic isotropic 256x256x256 head model. The reciprocity theorem is also utilized to reduce the number of forward calculations required in an inverse solution. Once all the required forward solutions are obtained and stored, the lead field matrix can then be computed in about a microsecond on a state-of-the-art PC. With the generic C and MATLAB routines, the FDEHMT can readily be integrated with EEG inverse algorithms such as beamforming, MUSIC and RAP-MUSIC.
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.
A RBF Based Local Gridfree Scheme for Unsteady Convection-Diffusion Problems
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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.
Finite difference method to find period-one gait cycles of simple passive walkers
Dardel, Morteza; Safartoobi, Masoumeh; Pashaei, Mohammad Hadi; Ghasemi, Mohammad Hassan; Navaei, Mostafa Kazemi
2015-01-01
Passive dynamic walking refers to a class of bipedal robots that can walk down an incline with no actuation or control input. These bipeds are sensitive to initial conditions due to their style of walking. According to small basin of attraction of passive limit cycles, it is important to start with an initial condition in the basin of attraction of stable walking (limit cycle). This paper presents a study of the simplest passive walker with point and curved feet. A new approach is proposed to find proper initial conditions for a pair of stable and unstable period-one gait limit cycles. This methodology is based on finite difference method which can solve the nonlinear differential equations of motion on a discrete time. Also, to investigate the physical configurations of the walkers and the environmental influence such as the slope angle, the parameter analysis is applied. Numerical simulations reveal the performance of the presented method in finding two stable and unstable gait patterns.
S. S. Das, M. Mohanty, R. K. Padhy, M. Sahu
2012-01-01
This paper considers the magnetohydrodynamic flow and heat transfer in a viscous incompressible fluid between two parallel porous plates experiencing a discontinuous change in wall temperature. An explicit finite difference scheme has been employed to solve the coupled non-linear equations governing the flow. The flow phenomenon has been characterized by Hartmann number, suction Reynolds number, channel Reynolds number and Prandtl number. The effects of these parameters on the velocity and te...
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 first computes unper...
Finite-difference solution of the space-angle-lethargy-dependent slowing-down transport equation
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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)
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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 demagnetizing tensor terms using a sparse grid integration scheme. This method improves the accuracy of computation at intermediate distances from the origin. We compute and report the accuracy of (i) the numerical evaluation of the exact tensor expression which is best for short distances, (ii) the asymptotic expansion best suited for large distances, and (iii) the new method based on numerical integration, which is superior to methods (i) and (ii) for intermediate distances. For all three methods, we show the measurements of accuracy and execution time as a function of distance, for calculations using single precision (4-byte) and double precision (8-byte) floating point arithmetic. We make recommendations for the choice of scheme order and integrating coefficients for the numerical integration method (iii). - Highlights: • We study the accuracy of demagnetization in finite difference micromagnetics. • We introduce a new sparse integration method to compute the tensor more accurately. • Newell, sparse integration and asymptotic method are compared for all ranges. • We provide a recommendation for optimal sparse integration scheme and parameters
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Chernyshenko, Dmitri; Fangohr, Hans
2015-05-01
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 demagnetizing tensor terms using a sparse grid integration scheme. This method improves the accuracy of computation at intermediate distances from the origin. We compute and report the accuracy of (i) the numerical evaluation of the exact tensor expression which is best for short distances, (ii) the asymptotic expansion best suited for large distances, and (iii) the new method based on numerical integration, which is superior to methods (i) and (ii) for intermediate distances. For all three methods, we show the measurements of accuracy and execution time as a function of distance, for calculations using single precision (4-byte) and double precision (8-byte) floating point arithmetic. We make recommendations for the choice of scheme order and integrating coefficients for the numerical integration method (iii). - Highlights: • We study the accuracy of demagnetization in finite difference micromagnetics. • We introduce a new sparse integration method to compute the tensor more accurately. • Newell, sparse integration and asymptotic method are compared for all ranges. • We provide a recommendation for optimal sparse integration scheme and parameters.
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Shielding calculations of advanced nuclear facilities such as accelerator based neutron sources or fusion devices of the tokamak type are complicated due to their complex geometries and their large dimensions, including bulk shields of several meters thickness. While the complexity of the geometry in the shielding calculation can be hardly handled by the discrete ordinates method, the deep penetration of radiation through bulk shields is a severe challenge for the Monte Carlo particle transport simulation technique. This work proposes a dedicated computational approach for coupled Monte Carlo - deterministic transport calculations to handle this kind of shielding problems. The Monte Carlo technique is used to simulate the particle generation and transport in the target region with both complex geometry and reaction physics, and the discrete ordinates method is used to treat the deep penetration problem in the bulk shield. To enable the coupling of these two different computational methods, a mapping approach has been developed for calculating the discrete ordinates angular flux distribution from the scored data of the Monte Carlo particle tracks crossing a specified surface. The approach has been implemented in an interface program and validated by means of test calculations using a simplified three-dimensional geometric model. Satisfactory agreement was obtained for the angular fluxes calculated by the mapping approach using the MCNP code for the Monte Carlo calculations and direct three-dimensional discrete ordinates calculations using the TORT code. In the next step, a complete program system has been developed for coupled three-dimensional Monte Carlo deterministic transport calculations by integrating the Monte Carlo transport code MCNP, the three-dimensional discrete ordinates code TORT and the mapping interface program. Test calculations with two simple models have been performed to validate the program system by means of comparison calculations using the Monte Carlo technique directly. The good agreement of the results obtained demonstrates that the program system is suitable to treat three-dimensional shielding problems with satisfactory accuracy. Finally the program system has been applied to the shielding analysis of the accelerator based IFMIF (International Fusion Materials Irradiation Facility) neutron source facility. In this application, the IFMIF-dedicated Monte Carlo code McDeLicious was used for the neutron generation and transport simulation in the target and the test cell region using a detailed geometrical model. The neutron/photon fluxes, spectra and dose rates across the back wall and in the access/maintenance room were calculated and are discussed. (orig.)
Energy Technology Data Exchange (ETDEWEB)
Wang, Wei [Deprartment of Mathematics. Florida Intl Univ., Miami, FL (United States); Shu, Chi-Wang [Division of Applied Mathematics. Brown Univ., Providence, RI (United States); Yee, H.C. [NASA Ames Research Center (ARC), Moffett Field, Mountain View, CA (United States); Sjgreen, Bjrn [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
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.
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This paper describes the the next evolution step in development of the direct method for solving systems of Nonlinear Algebraic Equations (SNAE). These equations arise from the finite difference approximation of original nonlinear partial differential equations (PDE). This method has been extended on the SNAE with three variables. The solving SNAE bases on Reiterating General Singular Value Decomposition of rectangular matrix pencils (RGSVD-algorithm). In contrast to the computer algebra algorithm in integer arithmetic based on the reduction to the Groebner's basis that algorithm is working in floating point arithmetic and realizes the reduction to the Kronecker's form. The possibilities of the method are illustrated on the example of solving the one-dimensional diffusion equation for 3-component model isotope mixture in a ga centrifuge. The implicit scheme for the finite difference equations without simplifying the nonlinear properties of the original equations is realized. The technique offered provides convergence to the solution for the single run. The Toolbox SNAE is developed in the framework of the high performance numeric computation and visualization software MATLAB. It includes more than 30 modules in MATLAB language for solving SNAE with two and three variables. (author)
A coarse-mesh nodal method-diffusive-mesh finite difference method
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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
Kamiński, M.; Supeł, Ł.
2016-02-01
It is widely known that lateral-torsional buckling of a member under bending and warping restraints of its cross-sections in the steel structures are crucial for estimation of their safety and durability. Although engineering codes for steel and aluminum structures support the designer with the additional analytical expressions depending even on the boundary conditions and internal forces diagrams, one may apply alternatively the traditional Finite Element or Finite Difference Methods (FEM, FDM) to determine the so-called critical moment representing this phenomenon. The principal purpose of this work is to compare three different ways of determination of critical moment, also in the context of structural sensitivity analysis with respect to the structural element length. Sensitivity gradients are determined by the use of both analytical and the central finite difference scheme here and contrasted also for analytical, FEM as well as FDM approaches. Computational study is provided for the entire family of the steel I- and H - beams available for the practitioners in this area, and is a basis for further stochastic reliability analysis as well as durability prediction including possible corrosion progress.
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Kamiński M.
2016-02-01
Full Text Available It is widely known that lateral-torsional buckling of a member under bending and warping restraints of its cross-sections in the steel structures are crucial for estimation of their safety and durability. Although engineering codes for steel and aluminum structures support the designer with the additional analytical expressions depending even on the boundary conditions and internal forces diagrams, one may apply alternatively the traditional Finite Element or Finite Difference Methods (FEM, FDM to determine the so-called critical moment representing this phenomenon. The principal purpose of this work is to compare three different ways of determination of critical moment, also in the context of structural sensitivity analysis with respect to the structural element length. Sensitivity gradients are determined by the use of both analytical and the central finite difference scheme here and contrasted also for analytical, FEM as well as FDM approaches. Computational study is provided for the entire family of the steel I- and H - beams available for the practitioners in this area, and is a basis for further stochastic reliability analysis as well as durability prediction including possible corrosion progress.
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.
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.
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.
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.
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...
Variational finite-difference representation of the kinetic energy operator
Maragakis, P; Kaxiras, E; 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 properties with a harmonic oscillator potential. For a more realistic application, we study the convergence of the total energy of bulk silicon using a real-space-grid density-functional code and employing both the conventional and the alternative representations of the kinetic energy operator.
Acoustic, finite-difference, time-domain technique development
International Nuclear Information System (INIS)
A close analog exists between the behavior of sound waves in an ideal gas and the radiated waves of electromagnetics. This analog has been exploited to obtain an acoustic, finite-difference, time-domain (AFDTD) technique capable of treating small signal vibrations in elastic media, such as air, water, and metal, with the important feature of bending motion included in the behavior of the metal. This bending motion is particularly important when the metal is formed into sheets or plates. Bending motion does not have an analog in electromagnetics, but can be readily appended to the acoustic treatment since it appears as a single additional term in the force equation for plate motion, which is otherwise analogous to the electromagnetic wave equation. The AFDTD technique has been implemented in a code architecture that duplicates the electromagnetic, finite-difference, time-domain technique code. The main difference in the implementation is the form of the first-order coupled differential equations obtained from the wave equation. The gradient of pressure and divergence of velocity appear in these equations in the place of curls of the electric and magnetic fields. Other small changes exist as well, but the codes are essentially interchangeable. The pre- and post-processing for model construction and response-data evaluation of the electromagnetic code, in the form of the TSAR code at Lawrence Livermore National Laboratory, can be used for the acoustic version. A variety of applications is possible, pending validation of the bending phenomenon. The applications include acoustic-radiation-pattern predictions for a submerged object; mine detection analysis; structural noise analysis for cars; acoustic barrier analysis; and symphonic hall/auditorium predictions and speaker enclosure modeling
Ghosh, Swarnava
2016-01-01
As the second component of SPARC (Simulation Package for Ab-initio Real-space Calculations), we present an accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory (DFT) for periodic systems. Specifically, employing a local formulation of the electrostatics, the Chebyshev polynomial filtered self-consistent field iteration, and a reformulation of the non-local force component, we develop a finite-difference framework wherein both the energy and atomic forces can be efficiently calculated to within chemical accuracies. We demonstrate using a wide variety of materials systems that SPARC obtains high convergence rates in energy and forces with respect to spatial discretization to reference plane-wave result; energies and forces that are consistent and display negligible `egg-box' effect; and accurate ground-state properties. We also demonstrate that the weak and strong scaling behavior of SPARC is similar to well-established and optimized plane-wave implementa...
International Nuclear Information System (INIS)
The finite-difference (FD) method is a powerful tool in seismic wave field modelling for understanding seismic wave propagation in the Earth's interior and interpreting the real seismic data. The accuracy of FD modelling partly depends on the implementation of the free-surface (i.e. traction-free) condition. In the past 40 years, at least six kinds of free-surface boundary condition approximate schemes (such as one-sided, centred finite-difference, composed, new composed, implicit and boundary-modified approximations) have been developed in FD second-order elastodynamic simulation. Herein we simulate seismic wave fields in homogeneous and lateral heterogeneous models using these free-surface boundary condition approximate schemes and evaluate their stability and applicability by comparing with corresponding analytical solutions, and then quantitatively evaluate the accuracies of different approximate schemes from the misfit of the amplitude and phase between the numerical and analytical results. Our results confirm that the composed scheme becomes unstable for the Vs/Vp ratio less than 0.57, and suggest that (1) the one-sided scheme is only accurate to first order and therefore introduces serious errors for the shorter wavelengths, other schemes are all of second-order precision; (2) the new composed, implicit and boundary-modified schemes are stable even when the Vs/Vp ratio is less than 0.2; (3) the implicit and boundary-modified schemes are able to deal with laterally varying (heterogeneous) free surface; (4) in the corresponding stability range, the one-sided scheme shows remarkable errors in both phase and amplitude compared to analytical solution (which means larger errors in travel-time and reflection strength), the other five approximate schemes show better performance in travel-time (phase) than strength (amplitude)
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 8302, thus strengthening the reliability of the model developed.
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.
Directory of Open Access Journals (Sweden)
Deb Nath S.K.
2015-12-01
Full Text Available In the present study, we have developed a code using Matlab software for solving a rectangular aluminum plate having void, notch, at different boundary conditions discretizing a two dimensional (2D heat conduction equation by the finite difference technique. We have solved a 2D mixed boundary heat conduction problem analytically using Fourier integrals (Deb Nath et al., 2006; 2007; 2007; Deb Nath and Ahmed, 2008; Deb Nath, 2008; Deb Nath and Afsar, 2009; Deb Nath and Ahmed, 2009; 2009; Deb Nath et al., 2010; Deb Nath, 2013 and the same problem is also solved using the present code developed by the finite difference technique (Ahmed et al., 2005; Deb Nath, 2002; Deb Nath et al., 2008; Ahmed and Deb Nath, 2009; Deb Nath et al., 2011; Mohiuddin et al., 2012. To verify the soundness of the present heat conduction code results using the finite difference method, the distribution of temperature at some sections of a 2D heated plate obtained by the analytical method is compared with those of the plate obtained by the present finite difference method. Interpolation technique is used as an example when the boundary of the plate does not pass through the discretized grid points of the plate. Sometimes hot and cold fluids are passed through rectangular channels in industries and many types of technical equipment. The distribution of temperature of plates including notches, slots with different temperature boundary conditions are studied. Transient heat transfer in several pure metallic plates is also studied to find out the required time to reach equilibrium temperature. So, this study will help find design parameters of such structures.
Deb Nath, S. K.; Peyada, N. K.
2015-12-01
In the present study, we have developed a code using Matlab software for solving a rectangular aluminum plate having void, notch, at different boundary conditions discretizing a two dimensional (2D) heat conduction equation by the finite difference technique. We have solved a 2D mixed boundary heat conduction problem analytically using Fourier integrals (Deb Nath et al., 2006; 2007; 2007; Deb Nath and Ahmed, 2008; Deb Nath, 2008; Deb Nath and Afsar, 2009; Deb Nath and Ahmed, 2009; 2009; Deb Nath et al., 2010; Deb Nath, 2013) and the same problem is also solved using the present code developed by the finite difference technique (Ahmed et al., 2005; Deb Nath, 2002; Deb Nath et al., 2008; Ahmed and Deb Nath, 2009; Deb Nath et al., 2011; Mohiuddin et al., 2012). To verify the soundness of the present heat conduction code results using the finite difference method, the distribution of temperature at some sections of a 2D heated plate obtained by the analytical method is compared with those of the plate obtained by the present finite difference method. Interpolation technique is used as an example when the boundary of the plate does not pass through the discretized grid points of the plate. Sometimes hot and cold fluids are passed through rectangular channels in industries and many types of technical equipment. The distribution of temperature of plates including notches, slots with different temperature boundary conditions are studied. Transient heat transfer in several pure metallic plates is also studied to find out the required time to reach equilibrium temperature. So, this study will help find design parameters of such structures.
A New Time-Dependent Finite Difference Method for Relativistic Shock Acceleration
Delaney, Sean; Duffy, Peter; Downes, Turlough P
2011-01-01
We present a new approach to calculate the particle distribution function about relativistic shocks including synchrotron losses using the method of lines with an explicit finite difference scheme. A steady, continuous, one dimensional plasma flow is considered to model thick (modified) shocks, leading to a calculation in three dimensions plus time, the former three being momentum, pitch angle and position. The method accurately reproduces the expected power law behaviour in momentum at the shock for upstream flow speeds ranging from 0.1c to 0.995c (1 < \\Gamma < 10). It also reproduces approximate analytical results for the synchrotron cutoff shape for a non-relativistic shock, demonstrating that the loss process is accurately represented. The algorithm has been implemented as a hybrid OpenMP--MPI parallel algorithm to make efficient use of SMP cluster architectures and scales well up to many hundreds of CPUs.
Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance
Giles, Michael B
2012-01-01
In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary-value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretisation of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives.
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...
Black-Scholes finite difference modeling in forecasting of call warrant prices in Bursa Malaysia
Mansor, Nur Jariah; Jaffar, Maheran Mohd
2014-07-01
Call warrant is a type of structured warrant in Bursa Malaysia. It gives the holder the right to buy the underlying share at a specified price within a limited period of time. The issuer of the structured warrants usually uses European style to exercise the call warrant on the maturity date. Warrant is very similar to an option. Usually, practitioners of the financial field use Black-Scholes model to value the option. The Black-Scholes equation is hard to solve analytically. Therefore the finite difference approach is applied to approximate the value of the call warrant prices. The central in time and central in space scheme is produced to approximate the value of the call warrant prices. It allows the warrant holder to forecast the value of the call warrant prices before the expiry date.
Samak, M. Mosleh E. Abu; Bakar, A. Ashrif A.; Kashif, Muhammad; Zan, Mohd Saiful Dzulkifly
2016-01-01
This paper discusses numerical analysis methods for different geometrical features that have limited interval values for typically used sensor wavelengths. Compared with existing Finite Difference Time Domain (FDTD) methods, the alternating direction implicit (ADI)-FDTD method reduces the number of sub-steps by a factor of two to three, which represents a 33% time savings in each single run. The local one-dimensional (LOD)-FDTD method has similar numerical equation properties, which should be calculated as in the previous method. Generally, a small number of arithmetic processes, which result in a shorter simulation time, are desired. The alternating direction implicit technique can be considered a significant step forward for improving the efficiency of unconditionally stable FDTD schemes. This comparative study shows that the local one-dimensional method had minimum relative error ranges of less than 40% for analytical frequencies above 42.85 GHz, and the same accuracy was generated by both methods.
Bakhouche, A.; Doghmane, N.
2008-06-01
In this paper, a new adaptive watermarking algorithm is proposed for still image based on the wavelet transform. The two major applications for watermarking are protecting copyrights and authenticating photographs. Our robust watermarking [3] [22] is used for copyright protection owners. The main reason for protecting copyrights is to prevent image piracy when the provider distributes the image on the Internet. Embed watermark in low frequency band is most resistant to JPEG compression, blurring, adding Gaussian noise, rescaling, rotation, cropping and sharpening but embedding in high frequency is most resistant to histogram equalization, intensity adjustment and gamma correction. In this paper, we extend the idea to embed the same watermark in two bands (LL and HH bands or LH and HL bands) at the second level of Discrete Wavelet Transform (DWT) decomposition. Our generalization includes all the four bands (LL, HL, LH, and HH) by modifying coefficients of the all four bands in order to compromise between acceptable imperceptibility level and attacks' resistance.
Comparison of measured and predicted thermal mixing tests using improved finite difference technique
International Nuclear Information System (INIS)
The numerical diffusion introduced by the use of upwind formulations in the finite difference solution of the flow and energy equations for thermal mixing problems (cold water injection after small break LOCA in a PWR) was examined. The relative importance of numerical diffusion in the flow equations, compared to its effect on the energy equation was demonstrated. The flow field equations were solved using both first order accurate upwind, and second order accurate differencing schemes. The energy equation was treated using the conventional upwind and a mass weighted skew upwind scheme. Results presented for a simple test case showed that, for thermal mixing problems, the numerical diffusion was most significant in the energy equation. The numerical diffusion effect in the flow field equations was much less significant. A comparison of predictions using the skew upwind and the conventional upwind with experimental data from a two dimensional thermal mixing text are presented. The use of the skew upwind scheme showed a significant improvement in the accuracy of the steady state predicted temperatures. (orig./HP)
Finite-difference time-domain analysis for the dynamics and diffraction of exciton-polaritons.
Chen, Minfeng; Chang, Yia-Chung; Hsieh, Wen-Feng
2015-10-01
We adopted a finite-difference time-domain (FDTD) scheme to simulate the dynamics and diffraction of exciton-polaritons, governed by the coupling of polarization waves with electromagnetic waves. The polarization wave, an approximate solution to the Schrödinger's equation at low frequencies, essentially captures the exciton behavior. Numerical stability of the scheme is analyzed and simple examples are provided to prove its validity. The system considered is both temporally and spatially dispersive, for which the FDTD analysis has attracted less attention in the literature. Here, we demonstrate that the FDTD scheme could be useful for studying the optical response of the exciton-polariton and its dynamics. The diffraction of a polariton wave from a polaritonic grating is also considered, and many sharp resonances are found, which manifest the interference effect of polariton waves. This illustrates that the measurement of transmittance or reflectance near polariton resonance can reveal subwavelength features in semiconductors, which are sensitive to polariton scattering. PMID:26479940
Finite-difference modeling of Biot's poroelastic equations across all frequencies
Energy Technology Data Exchange (ETDEWEB)
Masson, Y.J.; Pride, S.R.
2009-10-22
An explicit time-stepping finite-difference scheme is presented for solving Biot's equations of poroelasticity across the entire band of frequencies. In the general case for which viscous boundary layers in the pores must be accounted for, the time-domain version of Darcy's law contains a convolution integral. It is shown how to efficiently and directly perform the convolution so that the Darcy velocity can be properly updated at each time step. At frequencies that are low enough compared to the onset of viscous boundary layers, no memory terms are required. At higher frequencies, the number of memory terms required is the same as the number of time points it takes to sample accurately the wavelet being used. In practice, we never use more than 20 memory terms and often considerably fewer. Allowing for the convolution makes the scheme even more stable (even larger time steps might be used) than it is when the convolution is entirely neglected. The accuracy of the scheme is confirmed by comparing numerical examples to exact analytic results.
FLUOMEG: a planar finite difference mesh generator for fluid flow problems with parallel boundaries
International Nuclear Information System (INIS)
A two- or three-dimensional finite difference mesh generator capable of discretizing subrectangular flow regions (planar coordinates) with arbitrarily shaped bottom contours (vertical dimension) was developed. This economical, interactive computer code, written in FORTRAN IV and employing DISSPLA software together with graphics terminal, generates first a planar rectangular grid of variable element density according to the geometry and local kinematic flow patterns of a given fluid flow problem. Then subrectangular areas are deleted to produce canals, tributaries, bays, and the like. For three-dimensional problems, arbitrary bathymetric profiles (river beds, channel cross section, ocean shoreline profiles, etc.) are approximated with grid lines forming steps of variable spacing. Furthermore, the code works as a preprocessor numbering the discrete elements and the nodal points. Prescribed values for the principal variables can be automatically assigned to solid as well as kinematic boundaries. Cabinet drawings aid in visualizing the complete flow domain. Input data requirements are necessary only to specify the spacing between grid lines, determine land regions that have to be excluded, and to identify boundary nodes. 15 figures, 2 tables
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.
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.
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.
Energy Technology Data Exchange (ETDEWEB)
Botelho, Marco A.B.; Santos, Roberto H.M. dos; Silva, Marcelo S. [Universidade Federal da Bahia (UFBA), Salvador, BA (Brazil). Centro de Pesquisa em Geofisica e Geologia
2004-07-01
The numerical simulation of shot gathers over a (2D) velocity field, which corresponds to a model of Atlantic continental shelf, at the continental break area, using a typical model of the Brazilian Atlantic coast, suggested by PETROBRAS. The finite difference technique (FD) is used to solve the second derivatives in time and space of the acoustic wave equation, using fourth order operators to solve the spatial derivatives and second order operators to solve the time derivative. It is applied an explicitly scheme to calculate the pressure field values at a future instant. The use of rectangular mesh helps to generate data less noisy, since we can control better the numerical dispersion. The source functions (wavelets), as the first and the second derivatives of the gaussian function, are proper to generate synthetic seismograms with the FD method, because they allow an easy discretization. On the forward modeling, which is the simulation of wave fields, allows to control the stability limit of the method, wherever be the given velocity field, just employing compatible small values of the sample rate. The algorithm developed here, which uses only the FD technique, is able to perform the forward modeling, saving the image times, which can be used latter to perform the retropropagation of the wave field and thus migrate the source-gathers the reverse time extrapolation is able to test the used velocity model, and detect determine errors up to 5% on the used velocity model. (author)
International Nuclear Information System (INIS)
FDM3D is a three-dimensional(3-D) two group finite difference method diffusion equation solver adopting accelerated iterative schemes such as successive overrelaxation(SOR) or bi-conjugate gradient stabilization method(BICG-STAB) method as inner iteration schemes and Chebyshev two-parameter method or Wielant method as outer iteration schemes. It is designed to achieve an improved efficiency of the CANDU-PHWR analysis by the current RFSP code. For the efficiency test of FDM3D code, the FDM3D with SOR/Chebyshev two-parameter schemes is incorporated into the RFSP code and the benchmark problems have been analyzed by using the physics test data [6] of Wolsong units 2 and 3 and the calculation results of CANFLEX-NU physics design. It is shown that the FDM3D can reduce the CPU time of the current RFSP by 2 to 5 times. (author)
Camporeale, Enrico; Bergen, Benjamin K; Moulton, J David
2013-01-01
We discuss a spectral method for the numerical solution of the Vlasov-Poisson system where the velocity space is decomposed by means of an Hermite basis. We describe a semi-implicit time discretization that extends the range of numerical stability relative to an explicit scheme. We also introduce and discuss the effects of an artificial collisional operator, which is necessary to take care of the velocity space filamentation problem unavoidable in collisionless plasmas. The computational efficiency and the cost-effectiveness of this method are compared to a Particle-in-Cell (PIC) method in the case of a two-dimensional phase space. The following examples are discussed: Langmuir wave, Landau damping, ion-acoustic wave, two-stream instability, and plasma echo. The Hermite spectral method can achieve solutions that are several orders of magnitude more accurate at a fraction of the cost with respect to the PIC.
International Nuclear Information System (INIS)
Verification that a numerical method performs as intended is an integral part of code development. Semi-analytical benchmarks enable one such verification modality. Unfortunately, a semi-analytical benchmark requires some degree of analytical forethought and treats only relatively idealized cases making it of limited diagnostic value. In the first part of our investigation (Part I, in these proceedings), we established the theory of a straightforward finite difference scheme for the 1D, monoenergetic neutron diffusion equation in plane media. We also demonstrated an analytically enhanced version that leads to the analytical solution. The second part of our presentation (Part II) concerns the numerical implementation and application of the finite difference solutions of Part I. Here, we demonstrate how the numerical schemes themselves provide the semi-analytical benchmark. With the analytical solution known, we therefore have a test for accuracy of the proposed finite difference algorithms designed for high order. (authors)
Field Test of a Hybrid Finite-Difference and Analytic Element Regional Model.
Abrams, D B; Haitjema, H M; Feinstein, D T; Hunt, R J
2016-01-01
Regional finite-difference models often have cell sizes that are too large to sufficiently model well-stream interactions. Here, a steady-state hybrid model is applied whereby the upper layer or layers of a coarse MODFLOW model are replaced by the analytic element model GFLOW, which represents surface waters and wells as line and point sinks. The two models are coupled by transferring cell-by-cell leakage obtained from the original MODFLOW model to the bottom of the GFLOW model. A real-world test of the hybrid model approach is applied on a subdomain of an existing model of the Lake Michigan Basin. The original (coarse) MODFLOW model consists of six layers, the top four of which are aggregated into GFLOW as a single layer, while the bottom two layers remain part of MODFLOW in the hybrid model. The hybrid model and a refined "benchmark" MODFLOW model simulate similar baseflows. The hybrid and benchmark models also simulate similar baseflow reductions due to nearby pumping when the well is located within the layers represented by GFLOW. However, the benchmark model requires refinement of the model grid in the local area of interest, while the hybrid approach uses a gridless top layer and is thus unaffected by grid discretization errors. The hybrid approach is well suited to facilitate cost-effective retrofitting of existing coarse grid MODFLOW models commonly used for regional studies because it leverages the strengths of both finite-difference and analytic element methods for predictions in mildly heterogeneous systems that can be simulated with steady-state conditions. PMID:25628100
Cartesian Coordinate, Oblique Boundary, Finite Differences and Interpolation
Hutchinson, Ian H
2011-01-01
A numerical scheme is described for accurately accommodating oblique, non-aligned, boundaries, on a three-dimensional cartesian grid. The scheme gives second-order accuracy in the solution for potential of Poisson's equation using compact difference stencils involving only nearest neighbors. Implementation for general "Robin" boundary conditions and for boundaries between media of different dielectric constant for arbitrary-shaped regions is described in detail. The scheme also provides for the interpolation of field (potential gradient) which, despite first-order peak errors immediately adjacent to the boundaries, has overall second order accuracy, and thus provides with good accuracy what is required in particle-in-cell codes: the force. Numerical tests on the implementation confirm the scalings and the accuracy.
Sprague, Mark W; Luczkovich, Joseph J
2016-01-01
This finite-difference time domain (FDTD) model for sound propagation in very shallow water uses pressure and velocity grids with both 3-dimensional Cartesian and 2-dimensional cylindrical implementations. Parameters, including water and sediment properties, can vary in each dimension. Steady-state and transient signals from discrete and distributed sources, such as the surface of a vibrating pile, can be used. The cylindrical implementation uses less computation but requires axial symmetry. The Cartesian implementation allows asymmetry. FDTD calculations compare well with those of a split-step parabolic equation. Applications include modeling the propagation of individual fish sounds, fish aggregation sounds, and distributed sources. PMID:26611072
Ransom, Jonathan B.
2002-01-01
A multifunctional interface method with capabilities for variable-fidelity modeling and multiple method analysis is presented. The methodology provides an effective capability by which domains with diverse idealizations can be modeled independently to exploit the advantages of one approach over another. The multifunctional method is used to couple independently discretized subdomains, and it is used to couple the finite element and the finite difference methods. The method is based on a weighted residual variational method and is presented for two-dimensional scalar-field problems. A verification test problem and a benchmark application are presented, and the computational implications are discussed.
Hierarchical Parallelism in Finite Difference Analysis of Heat Conduction
Padovan, Joseph; Krishna, Lala; Gute, Douglas
1997-01-01
Based on the concept of hierarchical parallelism, this research effort resulted in highly efficient parallel solution strategies for very large scale heat conduction problems. Overall, the method of hierarchical parallelism involves the partitioning of thermal models into several substructured levels wherein an optimal balance into various associated bandwidths is achieved. The details are described in this report. Overall, the report is organized into two parts. Part 1 describes the parallel modelling methodology and associated multilevel direct, iterative and mixed solution schemes. Part 2 establishes both the formal and computational properties of the scheme.
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.
International Nuclear Information System (INIS)
Highlights: • The stiffness confinement method is combined with multigroup CMFD with SENM nodal kernel. • The systematic methods for determining the shape and amplitude frequencies are established. • Eigenvalue problems instead of fixed source problems are solved in the transient calculation. • It is demonstrated that much larger time step sizes can be used with the SCM–CMFD method. - Abstract: An improved Stiffness Confinement Method (SCM) is formulated within the framework of the coarse mesh finite difference (CMFD) formulation for efficient multigroup spatial kinetics calculation. The algorithm for searching for the amplitude frequency that makes the dynamic eigenvalue unity is developed in a systematic way along with the methods for determining the shape and precursor frequencies. A nodal calculation scheme is established within the CMFD framework to incorporate the cross section changes due to thermal feedback and dynamic frequency update. The conditional nodal update scheme is employed such that the transient calculation is performed mostly with the CMFD formulation and the CMFD parameters are conditionally updated by intermittent nodal calculations. A quadratic representation of amplitude frequency is introduced as another improvement. The performance of the improved SCM within the CMFD framework is assessed by comparing the solution accuracy and computing times for the NEACRP control rod ejection benchmark problems with those obtained with the Crank–Nicholson method with exponential transform (CNET). It is demonstrated that the improved SCM is beneficial for large time step size calculations with stability and accuracy enhancement
Dynamic ultimate load analysis using a finite difference method
International Nuclear Information System (INIS)
The method of numerical integration is explained on a one-degree-of-freedom system. A generalization to systems with several degrees of freedom is given. The conditions for numerical stability and for getting a sufficient approximation to the exact solution of the differential equations are dealt with. Not only a time discretization but also a geometric discretization is necessary. This may be anticipated by a lumped-mass dynamic model, or, with continuous bodies, it could be performed, e.g., by a mesh pattern of finite coordinate differences. Examples are given for the numerical treatment especially of beams and plates. Starting from the corresponding differential equations describing a process of wave propagation, the rotational inertia of single beam or plate elements as well as the transverse shear deformations are included. By this numerical method of dynamic analysis suitable for computer programming, point-by-point time-history solutions are obtained for deterministic excitations and for material properties, both varying arbitrarily with time and space. Applications for practical dynamic problems of nuclear structural design taking into account a defined material ductility are discussed. (orig./HP)
Vertical Discretization of Hydrostatic Primitive Equations with Finite Element Method
Yi, Tae-Hyeong; Park, Ja-Rin
2014-05-01
A vertical finite element (VFE) discretization of hydrostatic primitive equations is developed for the dynamical core of a numerical weather prediction (NWP) system at KIAPS, which is horizontally discretized by a spectral element on a cubed-sphere grid. The governing equations are discretized on a hybrid pressure-based vertical coordinate [1]. Compared with a vertical finite difference (VFD) discretization, which is only first order accurate for non-uniform grids, the VFE has many advantages such that it gives more accurate results, all variables are defined in the same full level, the level of vertical noise might be reduced [2], and it is easily coupled with existing physics packages, developed for a Lorentz staggering grid system. Due to these reasons, we adopted the VFE scheme presented by Untch [2] for the vertical discretization. Instead of using semi-Lagrangian and semi-implicit schemes of ECMWF, we use the Eulerian equations and second-order Runge-Kutta scheme as the first step in implementing the VFE for the dynamical core of the KIAPS's NWP model. Since the Eulerian hydrostatic equations are used in this study, both integral and derivative operators are required to implement the VFE using the Galerkin method with b-splines as basis functions. To compare the accuracy of the VFE with the VFD, the two-dimensional test case of mountain waves is used where physical configuration and initial conditions are the same as that of Durran [3]. In this case, the horizontal and vertical velocities obtained by the analytical solution, VFD, VFE-linear and VFE-cubic are compared to understand their numerical features and the vertical flux of horizontal momentum is also presented as the measurement of solution accuracy since it is sensitive to errors in a solution [3]. It is shown that the VFE with a cubic b-spline function is more accurate than the VFD and VFE with a linear b-spline function as the vertical flux is closer to unity, which will be presented in the conference. Reference Simmons, A. J., Burridge, D. M., 1981: An energy and angular momentum conserving vertical finite difference scheme and hybrid vertical coordinates. Mon. Wea. Rev., 109, 758-766. Untch, A., Hortal, M., 2004: A finite-element schemes for the vertical discretization of the semi-Lagrangian version of the ECMWF forecast model. Q. J. R. Meteorol. Soc., 130, 1505-1530. Durran, D. R., Klemp, J. B., 1983: A compressible model for the simulation of moist mountain waves. Mon. Wea. Rev., 111, 2341-2361.
Craig, Elena
Perfectly Matched Layer (PML) absorbing boundary conditions were first proposed by Berenger in 1994 for the Maxwell's equations of electromagnetics. Since Hu first applied the method to Euler's equations in 1996, progress made in the application of PML to Computational Aeroacoustics (CAA) includes linearized Euler equations with non-uniform mean flow, non-linear Euler equations, flows with an arbitrary mean flow direction, and non-linear clavier-Stokes equations. Although Boltzmann-BGK methods have appeared in the literature and have been shown capable of simulating aeroacoustics phenomena, very little has been done to develop absorbing boundary conditions for these methods. The purpose of this work was to extend the PML methodology to the discrete velocity Boltzmann-BGK equation (DVBE) for the case of a horizontal mean flow in two and three dimensions. The proposed extension of the PML has been accomplished in this dissertation. Both split and unsplit PML absorbing boundary conditions are presented in two and three dimensions. A finite difference and a lattice model are considered for the solution of the PML equations. The linear stability of the PML equations is investigated for both models. The small relaxation time needed for the discrete velocity Boltzmann-BC4K model to solve the Euler equations renders the explicit Runge-Kutta schemes impractical. Alternatively, implicit-explicit Runge-Kutta (IMEX) schemes are used in the finite difference model and are implemented explicitly by exploiting the special structure of the Boltzmann-BGK equation. This yields a numerically stable solution by the finite difference schemes. As the lattice model proves to be unstable, a coupled model consisting of a lattice Boltzmann (LB) method for the Ulterior domain and an IMEX finite difference method for the PML domains is proposed and investigated. Numerical examples of acoustic and vorticity waves are included to support the validity of the PML equations. In each example, accurate solutions are obtained, supporting the conclusion that PML is an effective absorbing boundary condition.
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
Energy Technology Data Exchange (ETDEWEB)
Deupree, R.G.
1977-01-01
Finite difference techniques were used to examine the coupling of radial pulsation and convection in stellar models having comparable time scales. Numerical procedures are emphasized, including diagnostics to help determine the range of free parameters.
International Nuclear Information System (INIS)
Graphene has been considered as a promising material which may find applications in the THz science. In this work, we numerically investigate tunable photonic crystals in the THz range based on stacked graphene/dielelctric layers, a complex pole—residue pair model is used to find the effective permittivity of graphene, which could be easily incorporated into the finite-difference time domain (FDTD) algorithm. Two different schemes of photonic crystal used for extending the bandgap have been simulated through this FDTD technique. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)
Ouyang, Chaojun; He, Siming; Xu, Qiang; Luo, Yu; Zhang, Wencheng
2013-03-01
A two-dimensional mountainous mass flow dynamic procedure solver (Massflow-2D) using the MacCormack-TVD finite difference scheme is proposed. The solver is implemented in Matlab on structured meshes with variable computational domain. To verify the model, a variety of numerical test scenarios, namely, the classical one-dimensional and two-dimensional dam break, the landslide in Hong Kong in 1993 and the Nora debris flow in the Italian Alps in 2000, are executed, and the model outputs are compared with published results. It is established that the model predictions agree well with both the analytical solution as well as the field observations.
Mackie, Randall L.; Smith, J. Torquil; Madden, Theodore R.
1994-07-01
We have developed a robust and efficient finite difference algorithm for computing the magnetotelluric response of general three-dimensional (3-D) models using the minimum residual relaxation method. The difference equations that we solve are second order in H and are derived from the integral forms of Maxwell's equations on a staggered grid. The boundary H field values are obtained from two-dimensional transverse magnetic mode calculations for the vertical planes in the 3-D model. An incomplete Cholesky decomposition of the diagonal subblocks of the coefficient matrix is used as a preconditioner, and corrections are made to the H fields every few iterations to ensure there are no H divergences in the solution. For a plane wave source field, this algorithm reduces the errors in the H field for simple 3-D models to around the 0.01% level compared to their fully converged values in a modest number of iterations, taking only a few minutes of computation time on our desktop workstation. The E fields can then be determined from discretized versions of the curl of H equations.
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.
Mimetic finite difference method for the stokes problem on polygonal meshes
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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.
International Nuclear Information System (INIS)
The lowest order Nodal Integral Method (NIM) which belongs to a large class of nodal methods, the Lawrence-Dorning class, is written in a five-point, weighted-difference form and contrasted against the edge-centered Finite Difference Method (FDM). The final equations for the two methods exhibit three differences: the NIM employs almost three times as many discrete-variables (which are node- and surface-averaged values of the flux) as the FDM; the spatial weights in the NIM include hyperbolic functions opposed to the algebraic weights in the FDM; the NIM explicitly imposes continuity of the net current across cell edges. A homogeneous model problem is devised to enable an analytical study of the numerical solutions accuracy. The analysis shows that on a given mesh the FDM calculated fundamental mode eigenvalue is more accurate than that calculated by the NIM. However, the NIM calculated flux distribution is more accurate, especially when the problem size is several times as thick as the diffusion length. Numerical results for a nonhomogeneous test problem indicate the very high accuracy of the NIM for fixed source problems in such cases. 18 refs., 1 fig., 1 tab
Application of a Strong Tracking Finite-Difference Extended Kalman Filter to Eye Tracking
Zhang, Zutao; Zhang, Jiashu
2010-01-01
This paper proposes a new eye tracking method using strong finite-difference Kalman filter. Firstly, strong tracking factor is introduced to modify priori covariance matrix to improve the accuracy of the eye tracking algorithm. Secondly, the finite-difference method is proposed to replace partial derivatives of nonlinear functions to eye tracking. From above deduction, the new strong finite-difference Kalman filter becomes very simple because of replacing partial derivatives calculation using...
El-Amin, Mohamed
2012-01-01
The problem of coupled structural deformation with two-phase flow in porous media is solved numerically using cellcentered finite difference (CCFD) method. In order to solve the system of governed partial differential equations, the implicit pressure explicit saturation (IMPES) scheme that governs flow equations is combined with the the implicit displacement scheme. The combined scheme may be called IMplicit Pressure-Displacement Explicit Saturation (IMPDES). The pressure distribution for each cell along the entire domain is given by the implicit difference equation. Also, the deformation equations are discretized implicitly. Using the obtained pressure, velocity is evaluated explicitly, while, using the upwind scheme, the saturation is obtained explicitly. Moreover, the stability analysis of the present scheme has been introduced and the stability condition is determined.
Accurate finite difference beam propagation method for complex integrated optical structures
DEFF Research Database (Denmark)
Rasmussen, Thomas; Povlsen, Jørn Hedegaard; Bjarklev, Anders Overgaard
1993-01-01
A simple and effective finite-difference beam propagation method in a z-varying nonuniform mesh is developed. The accuracy and computation time for this method are compared with a standard finite-difference method for both the 3-D and 2-D versions......A simple and effective finite-difference beam propagation method in a z-varying nonuniform mesh is developed. The accuracy and computation time for this method are compared with a standard finite-difference method for both the 3-D and 2-D versions...
Energy Technology Data Exchange (ETDEWEB)
Appelo, D; Petersson, N A
2007-12-17
The isotropic elastic wave equation governs the propagation of seismic waves caused by earthquakes and other seismic events. It also governs the propagation of waves in solid material structures and devices, such as gas pipes, wave guides, railroad rails and disc brakes. In the vast majority of wave propagation problems arising in seismology and solid mechanics there are free surfaces. These free surfaces have, in general, complicated shapes and are rarely flat. Another feature, characterizing problems arising in these areas, is the strong heterogeneity of the media, in which the problems are posed. For example, on the characteristic length scales of seismological problems, the geological structures of the earth can be considered piecewise constant, leading to models where the values of the elastic properties are also piecewise constant. Large spatial contrasts are also found in solid mechanics devices composed of different materials welded together. The presence of curved free surfaces, together with the typical strong material heterogeneity, makes the design of stable, efficient and accurate numerical methods for the elastic wave equation challenging. Today, many different classes of numerical methods are used for the simulation of elastic waves. Early on, most of the methods were based on finite difference approximations of space and time derivatives of the equations in second order differential form (displacement formulation), see for example [1, 2]. The main problem with these early discretizations were their inability to approximate free surface boundary conditions in a stable and fully explicit manner, see e.g. [10, 11, 18, 20]. The instabilities of these early methods were especially bad for problems with materials with high ratios between the P-wave (C{sub p}) and S-wave (C{sub s}) velocities. For rectangular domains, a stable and explicit discretization of the free surface boundary conditions is presented in the paper [17] by Nilsson et al. In summary, they introduce a discretization, that use boundary-modified difference operators for the mixed derivatives in the governing equations. Nilsson et al. show that the method is second order accurate for problems with smoothly varying material properties and stable under standard CFL constraints, for arbitrarily varying material properties. In this paper we generalize the results of Nilsson et al. to curvilinear coordinate systems, allowing for simulations on non-rectangular domains. Using summation by parts techniques, we show that there exists a corresponding stable discretization of the free surface boundary condition on curvilinear grids. We also prove that the discretization is stable and energy conserving both in semi-discrete and fully discrete form. As for the Cartesian method in, [17], the stability and conservation results holds for arbitrarily varying material properties. By numerical experiments it is established that the method is second order accurate.
Sui, Liansheng; Duan, Kuaikuai; Liang, Junli
2016-05-01
A secure double-image sharing scheme is proposed by using the Shamir's three-pass protocol in the discrete multiple-parameter fractional angular transform domain. First, an enlarged image is formed by assembling two plain images successively in the horizontal direction and scrambled in the chaotic permutation process, in which the sequences of chaotic pairs are generated by the two-dimensional Sine Logistic modulation map. Second, the scrambled image is divided into two components which are used to constitute a complex image. One component is normalized and regarded as the phase part of the complex image as well as other is considered as the amplitude part. Finally, the complex image is shared between the sender and the receiver by using the Shamir's three-pass protocol, in which the discrete multiple-parameter fractional angular transform is used as the encryption function due to its commutative property. The proposed double-image sharing scheme has an obvious advantage that the key management is convenient without distributing the random phase mask keys in advance. Moreover, the security of the image sharing scheme is enhanced with the help of extra parameters of the discrete multiple-parameter fractional angular transform. To the best of our knowledge, this is the first report on integrating the Shamir's three-pass protocol with double-image sharing scheme in the information security field. Simulation results and security analysis verify the feasibility and effectiveness of the proposed scheme.
On an explicit finite difference method for fractional diffusion equations
Yuste, S B
2003-01-01
A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous transport characterized by non-Markovian kinetics and the breakdown of Fick's law. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald-Letnikov definition of the fractional derivative operator to obtain an explicit fractional FTCS scheme for solving the fractional diffusion equation. The resulting method is amenable to a stability analysis a la von Neumann. We show that the analytical stability bounds are in excellent agreement with numerical tests. Comparison between exact analytical solutions and numerical predictions are made.
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
Treatment of late time instabilities in finite difference EMP scattering codes
International Nuclear Information System (INIS)
Time-domain solutions to the finite-differenced Maxwell's equations give rise to several well-known nonphysical propagation anomalies. In particular, when a radiative electric-field look back scheme is employed to terminate the calculation, a high-frequency, growing, numerical instability is introduced. This paper describes the constraints made on the mesh to minimize this instability, and a technique of applying an absorbing sheet to damp out this instability without altering the early time solution. Also described are techniques to extend the data record in the presence of high-frequency noise through application of a low-pass digital filter and the fitting of a damped sinusoid to the late-time tail of the data record. An application of these techniques is illustrated with numerical models of the FB-111 aircraft and the B-52 aircraft in the in-flight refueling configuration using the THREDE finite difference computer code. Comparisons are made with experimental scale model measurements with agreement typically on the order of 3 to 6 dB near the fundamental resonances
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.
Kim, J. S.; Chang, K. S.
1984-06-01
Transient as well as oscillating two-dimensional boundary layers are solved numerically by using a noniterative implicit finite difference scheme which is second-order accurate both in time and space. To obtain the exact spatial initial condition, the solution is obtained of parabolic partial differential equations at the initial plane which are reduced from the full biparabolic equations valid in the main time-space domain. Formulations are made first for incompressible flow, and then for compressible boundary layers so that the effect of temperature-induced compressibility can be considered. The method is applied to the unsteady laminar boundary layers with large temporal flow disturbances. Examples are transition to Falkner-Skan flow, oscillatory Blasius flow, constantly accelerated stagnation point flow and harmonically fluctuating flow past a circular cylinder, with or without the compressibility effect taken into account for the last two cases. Comparison with the existing data has demonstrated the excellency of the present method both in accuracy and computer-time economy.
Parallel 3d Finite-Difference Time-Domain Method on Multi-Gpu Systems
Du, Liu-Ge; Li, Kang; Kong, Fan-Min; Hu, Yuan
Finite-difference time-domain (FDTD) is a popular but computational intensive method to solve Maxwell's equations for electrical and optical devices simulation. This paper presents implementations of three-dimensional FDTD with convolutional perfect match layer (CPML) absorbing boundary conditions on graphics processing unit (GPU). Electromagnetic fields in Yee cells are calculated in parallel millions of threads arranged as a grid of blocks with compute unified device architecture (CUDA) programming model and considerable speedup factors are obtained versus sequential CPU code. We extend the parallel algorithm to multiple GPUs in order to solve electrically large structures. Asynchronous memory copy scheme is used in data exchange procedure to improve the computation efficiency. We successfully use this technique to simulate pointwise source radiation and validate the result by comparison to high precision computation, which shows favorable agreements. With four commodity GTX295 graphics cards on a single personal computer, more than 4000 million Yee cells can be updated in one second, which is hundreds of times faster than traditional CPU computation.
Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs
International Nuclear Information System (INIS)
This paper presents parallelization strategies for the radial basis function-finite difference (RBF-FD) method. As a generalized finite differencing scheme, the RBF-FD method functions without the need for underlying meshes to structure nodes. It offers high-order accuracy approximation and scales as O(N) per time step, with N being with the total number of nodes. To our knowledge, this is the first implementation of the RBF-FD method to leverage GPU accelerators for the solution of PDEs. Additionally, this implementation is the first to span both multiple CPUs and multiple GPUs. OpenCL kernels target the GPUs and inter-processor communication and synchronization is managed by the Message Passing Interface (MPI). We verify our implementation of the RBF-FD method with two hyperbolic PDEs on the sphere, and demonstrate up to 9x speedup on a commodity GPU with unoptimized kernel implementations. On a high performance cluster, the method achieves up to 7x speedup for the maximum problem size of 27,556 nodes.
Modelling natural gas hydrate emplacement : a mixed finite-element finite-difference simulator
Energy Technology Data Exchange (ETDEWEB)
Schnurle, P.; Liu, C.S. [National Taiwan Univ., Taipei (China). Inst. of Oceanography; Wang, Y. [MOEA, Taipei (China). Central Geological Survey
2008-07-01
A framework for modelling the distribution of dissolved methane, methane gas, methane hydrates, and seawater was presented. Darcy's law and energy and mass constitutive equations were integrated into a 2-D finite element model to account for space, and a finite difference model to account for time. The scheme allowed for the coupling of thermodynamics, kinetics, and transport processes from the microscopic to the geologic scale. The formation and distribution of the methane hydrates and free gas were modelled in a simple geologic framework that considered geothermal gradients, dewatering; fluid flow; methane in-situ production; and basal flux. A case study of a 6250 m wide anticlinal structure at 1000 meter water depth was used to evaluate the model. Results of the modelling study showed that methane and hydrate solubility is a critical parameter. Decreases in methane solubility beneath the hydrate stability zone (BHSZ) had a significant impact on the presence of free gas at the base of the BHSZ. Sharp decreases in hydrate solubility above the BHSZ had an impact on the amounts of methane available for hydrate emplacement and methane seeps into the water column. It was concluded that anaerobic methane oxidation and sulfate reduction parameters will be introduced into the model in order to constrain methane flux near the sea floor. 18 refs., 1 tab., 4 figs.
International Nuclear Information System (INIS)
Highlights: • Mixed convection heat transfer in 2D lid-driven cavity is studied numerically. • Hybrid scheme with multiple relaxation time lattice Boltzmann method is used to obtain the velocity field. • Finite difference method is used to compute the temperature. • Effect of both Richardson and Reynolds numbers for mixed convection is studied. - Abstract: 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
Kreider, Kevin L.; Baumeister, Kenneth J.
1996-01-01
An explicit finite difference real time iteration scheme is developed to study harmonic sound propagation in aircraft engine nacelles. To reduce storage requirements for future large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable for a harmonic monochromatic sound field, a parabolic (in time) approximation is introduced to reduce the order of the governing equation. The analysis begins with a harmonic sound source radiating into a quiescent duct. This fully explicit iteration method then calculates stepwise in time to obtain the 'steady state' harmonic solutions of the acoustic field. For stability, applications of conventional impedance boundary conditions requires coupling to explicit hyperbolic difference equations at the boundary. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.
Discrete spectrum of the two-center problem of p bar He+ atomcule
International Nuclear Information System (INIS)
A discrete spectrum of the two-center Coulomb problem of p bar He+ system is studied. For solving this problem the finite-difference scheme of the 4th-order and the continuous analog of Newton's method are applied. The algorithm for calculation of eigenvalues and eigenfunctions with optimization of the parameter of the fractional-rational transformation of the quasiradial variable to a finite interval is developed. The specific behaviour of the solutions in a vicinity of the united and separated atoms is discussed
Beam, R. M.; Warming, R. F.
1976-01-01
An implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation-law form. The algorithm is second-order time-accurate, noniterative, and in a spatially factored form. Second- or fourth-order central and second-order one-sided spatial differencing are accommodated within the solution of a block tridiagonal system of equations. Significant conceptual and computational simplifications are made for systems whose flux vectors are homogeneous functions (of degree one), e.g., the Eulerian gasdynamic equations. Conservative hybrid schemes, which switch from central to one-sided spatial differencing whenever the local characteristic speeds are of the same sign, are constructed to improve the resolution of weak solutions. Numerical solutions are presented for a nonlinear scalar model equation and the two-dimensional Eulerian gasdynamic equations.
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
Cui, Mingrong
2015-01-01
High-order compact exponential finite difference scheme for solving the time fractional convection-diffusion reaction equation with variable coefficients is considered in this paper. The convection, diffusion and reaction coefficients can depend on both the spatial and temporal variables. We begin with the one dimensional problem, and after transforming the original equation to one with diffusion coefficient unity, the new equation is discretized by a compact exponential finite difference scheme, with a high-order approximation for the Caputo time derivative. We prove the solvability of this fully discrete implicit scheme, and analyze its local truncation error. For the fractional equation with constant coefficients, we use Fourier method to prove the stability and utilize matrix analysis as a tool for the error estimate. Then we discuss the two dimensional problem, give the compact ADI scheme with the restriction that besides the time variable, the convection coefficients can only depend on the corresponding spatial variables, respectively. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.
Unconditionally stable time marching scheme for Reynolds stress models
Mor-Yossef, Y.
2014-11-01
Progress toward a stable and efficient numerical treatment for the compressible Favre-Reynolds-averaged Navier-Stokes equations with a Reynolds-stress model (RSM) is presented. The mean-flow and the Reynolds stress model equations are discretized using finite differences on a curvilinear coordinates mesh. The convective flux is approximated by a third-order upwind biased MUSCL scheme. The diffusive flux is approximated using second-order central differencing, based on a full-viscous stencil. The novel time-marching approach relies on decoupled, implicit time integration, that is, the five mean-flow equations are solved separately from the seven Reynolds-stress closure equations. The key idea is the use of the unconditionally positive-convergent implicit scheme (UPC), originally developed for two-equation turbulence models. The extension of the UPC scheme for RSM guarantees the positivity of the normal Reynolds-stress components and the turbulence (specific) dissipation rate for any time step. Thanks to the UPC matrix-free structure and the decoupled approach, the resulting computational scheme is very efficient. Special care is dedicated to maintain the implicit operator compact, involving only nearest neighbor grid points, while fully supporting the larger discretized residual stencil. Results obtained from two- and three-dimensional numerical simulations demonstrate the significant progress achieved in this work toward optimally convergent solution of Reynolds stress models. Furthermore, the scheme is shown to be unconditionally stable and positive.
High resolution schemes for hyperbolic conservation laws
Harten, A.
1983-01-01
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 appropriately 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.
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 appropriately 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. 19 refs., 9 figs
Lansing, F. L.
1980-01-01
A numerical procedure was established using the finite-difference technique in the determination of the time-varying temperature distribution of a tubular solar collector under changing solar radiancy and ambient temperature. Three types of spatial discretization processes were considered and compared for their accuracy of computations and for selection of the shortest computer time and cost. The stability criteria of this technique were analyzed in detail to give the critical time increment to ensure stable computations. The results of the numerical analysis were in good agreement with the analytical solution previously reported. The numerical method proved to be a powerful tool in the investigation of the collector sensitivity to two different flow patterns and several flow control mechanisms.
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 Jlich supercomputer center (Germany)
Construction of weighted upwind compact scheme
Wang, Zhengjie
Enormous endeavor has been devoted in spatial high order high resolution schemes in more than twenty five years previously, like total variation diminishing (TVD), essentially non-oscillatory scheme, weighted essentially non-oscillatory scheme for finite difference, and Discontinuous Galerkin methods for finite element and the finite volume. In this dissertation, a high order finite difference Weighted Upwind Compact Scheme has been constructed by dissipation and dispersion analysis. Secondly, a new method to construct global weights has been tested. Thirdly, a methodology to compromise dissipation and dispersion in constructing Weighted Upwind Compact Scheme has been derived. Finally, several numerical test cases have been shown.
Acho Zuppa, Leonardo
2015-01-01
This paper presents a modified discrete-time chaotic system obtained from the standard logistic map model. Then, a secure communication system is given and numerical experiments are carry out using the conceived discrete-time chaotic oscillator. Moreover, an experiment of our chaotic model is realized using the Arduino-UNO board.
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.
Compact finite difference method for calculating magnetic field components of cyclotrons
International Nuclear Information System (INIS)
The compact finite difference method was developed for calculating the off median plane magnetic field components of cyclotrons when only the measured midplane field data are available. It has been shown that the proposed compact finite difference differentiators are better than the finite difference differentiators previously reported by the author. The proposed compact finite difference method was tested by comparing the frequency response, by applying to an analytical magnetic field, and by applying to measured magnetic field data of the K 1200 superconducting cyclotron at the National Superconducting Cyclotron Laboratory. It should be pointed out that this improvement was obtained at the expense of more complicated machinery of mathematics, namely solving matrix problems. 8 refs., 8 figs., 1 tab
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)
Techniques for correcting approximate finite difference solutions. [applied to transonic flow
Nixon, D.
1979-01-01
A method of correcting finite-difference solutions for the effect of truncation error or the use of an approximate basic equation is presented. Applications to transonic flow problems are described and examples given.
APPLICATION OF A FINITE-DIFFERENCE TECHNIQUE TO THE HUMAN RADIOFREQUENCY DOSIMETRY PROBLEM
A powerful finite difference numerical technique has been applied to the human radiofrequency dosimetry problem. The method possesses inherent advantages over the method of moments approach in that its implementation requires much less computer memory. Consequently, it has the ca...
Lisitsa, Vadim; Tcheverda, Vladimir; Botter, Charlotte
2016-04-01
We present an algorithm for the numerical simulation of seismic wave propagation in models with a complex near surface part and free surface topography. The approach is based on the combination of finite differences with the discontinuous Galerkin method. The discontinuous Galerkin method can be used on polyhedral meshes; thus, it is easy to handle the complex surfaces in the models. However, this approach is computationally intense in comparison with finite differences. Finite differences are computationally efficient, but in general, they require rectangular grids, leading to the stair-step approximation of the interfaces, which causes strong diffraction of the wavefield. In this research we present a hybrid algorithm where the discontinuous Galerkin method is used in a relatively small upper part of the model and finite differences are applied to the main part of the model.
Hybrid lattice-Boltzmann and finite-difference simulation of electroosmotic flow in a microchannel
International Nuclear Information System (INIS)
A three-dimensional (3D) transient mathematical model is developed to simulate electroosmotic flows (EOFs) in a homogeneous, square cross-section microchannel, with and without considering the effects of axial pressure gradients. The general governing equations for electroosmotic transport are incompressible Navier-Stokes equations for fluid flow and the nonlinear Poisson-Boltzmann (PB) equation for electric potential distribution within the channel. In the present numerical approach, the hydrodynamic equations are solved using a lattice-Boltzmann (LB) algorithm and the PB equation is solved using a finite-difference (FD) method. The hybrid LB-FD numerical scheme is implemented on an iterative framework solving the system of coupled time-dependent partial differential equations subjected to the pertinent boundary conditions. Transient behavior of the EOF and effects due to the variations of different physicochemical parameters on the electroosmotic velocity profile are investigated. Transport characteristics for the case of combined electroosmotic- and pressure-driven microflows are also examined with the present model. For the sake of comparison, the cases of both favorable and adverse pressure gradients are considered. EOF behaviors of the non-Newtonian fluid are studied through implementation of the power-law model in the 3D LB algorithm devised for the fluid flow analysis. Numerical simulations reveal that the rheological characteristic of the fluid changes the EOF pattern to a considerable extent and can have significant consequences in the design of electroosmotically actuated bio-microfluidic systems. To improve the performance of the numerical solver, the proposed algorithm is implemented for parallel computing architectures and the overall parallel performance is found to improve with the number of processors.
Determination of electromagnetic cavity modes using the Finite Difference Frequency-Domain Method
J. Manzanares-Martnez; D. Moctezuma-Enriquez; R. Archuleta-Garca
2010-01-01
In this communication we propose a numerical determination of the electromagnetic modes in a cavity by using the Finite Difference Frequency-Domain Method. We first derive the analytical solution of the system and subsequently we introduce the numerical approximation. The cavity modes are obtained by solving an eigenvalue equation where the eigenvectors describe the eigenfunctions on the real space. It is found that this finite difference method can efficiently and accurately determine the re...
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.
Discrete time Bogoyavlensky lattices
Suris, Yu B
1995-01-01
Discretizations of the Bogoyavlensky lattices are introduced, belonging to the same hierarchies as the continuous--time systems. The construction exemplifies the general scheme for integrable discretization of systems on Lie algebras with r--matrix Poisson brackets. An initial value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flow is found.
Multiscale analysis of discrete nonlinear evolution equations
International Nuclear Information System (INIS)
The method of multi scale analysis is constructed for discrete systems of evolution equations for which the problem is that of the far behaviour of an input boundary datum. Discrete slow space variables are introduced in a general setting and the related finite differences are constructed. The method is applied to a series of representative examples: the Toda lattice, the nonlinear Klein-Gordon chain, the Takeno system and a discrete version of the Benjamin-Bona-Mahoney-Peregrini equation. Among the resulting limit models we find a discrete nonlinear Schroedinger equation (with reversed spacetime), a three-wave resonant interaction system and a discrete modified Volterra model. (author)
International Nuclear Information System (INIS)
An orifice flow system is developed and could be used for calibration of high vacuum gauges ranging from 10.3 mbar to 10.6 mbar with argon gas. A standard transfer gauge (SRG) confirms the applicability of the experimental set-up and obtained results. It is shown that this standard system could be used for calibration of high vacuum gauges with in the experimental error reported. The experimental results obtained .in two pressure zones are numerically modeled. A three dimensional system of Navier-Stokes equations is reduced to one dimension governing equations using symmetric technique. A finite difference method is applied to this reduced system together with prescribed boundary conditions. An SaR iterative scheme is employed to solve the system up to a fair accuracy. Numerical solutions of ID system are compared and reported to close accuracy with the experimental data. (author)
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 finite difference solution to 2-dimensional radiative transfer equation for small-animal imaging
Jin, Meng; Jiao, Yuting; Gao, Feng; Zhao, Huijuan
2010-02-01
Diffuse optical tomography (DOT) has been increasingly studied in the past decades. In DOT, the radiative transfer equation (RTE) and its P1 approximation, i.e. the diffuse equation (DE), have been used as the forward models. Since the DE-based DOT fails where biological tissue has a void-like region and when the source-detector separation is less than 5 mean free pathlengths, as in the situations of small animal imaging, the RTE-based DOT methodology has become a focus of investigation. Therefore, the complete formalism of the RTE is attracting more and more interest. It is clear that the quality of the reconstructed image depends strongly on the accuracy of the forward model. In this paper, A FDM was developed for solving two-dimensional RTE in a 2cm×2cm square homogeneous tissue with two groups of the optical properties and different schemes of the spatial and solid angle discretization. The results of the FDM are compared with the MC simulations. It is shown that when the step size of the spatial mesh becomes small, more discretized angle number is needed.
E. Camporeale; Delzanno, G. L.; Bergen, B. K.; Moulton, J. D.
2013-01-01
We describe a spectral method for the numerical solution of the Vlasov-Poisson system where the velocity space is decomposed by means of an Hermite basis, and the configuration space is discretized via a Fourier decomposition. The novelty of our approach is an implicit time discretization that allows exact conservation of charge, momentum and energy. The computational efficiency and the cost-effectiveness of this method are compared to the fully-implicit PIC method recently introduced by Mark...
Energy Technology Data Exchange (ETDEWEB)
Kleinstreuer, C.; Patterson, M.R.
1980-05-01
A two- or three-dimensional finite difference mesh generator capable of discretizing subrectangular flow regions (planar coordinates) with arbitrarily shaped bottom contours (vertical dimension) was developed. This economical, interactive computer code, written in FORTRAN IV and employing DISSPLA software together with graphics terminal, generates first a planar rectangular grid of variable element density according to the geometry and local kinematic flow patterns of a given fluid flow problem. Then subrectangular areas are deleted to produce canals, tributaries, bays, and the like. For three-dimensional problems, arbitrary bathymetric profiles (river beds, channel cross section, ocean shoreline profiles, etc.) are approximated with grid lines forming steps of variable spacing. Furthermore, the code works as a preprocessor numbering the discrete elements and the nodal points. Prescribed values for the principal variables can be automatically assigned to solid as well as kinematic boundaries. Cabinet drawings aid in visualizing the complete flow domain. Input data requirements are necessary only to specify the spacing between grid lines, determine land regions that have to be excluded, and to identify boundary nodes. 15 figures, 2 tables.
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...
International Nuclear Information System (INIS)
The problem of representing discontinuous properties in a finite difference approximation to the Neutron Diffusion Theory is considered. The exact interface conditions in one dimension are approximated with the second order finite differences and integration over a second order expansion of the flux either side of the interface. Through this approach the problem of abrupt changes in the diffusion coefficient D is addressed. Non-uniformity of properties between regions is described by a five point equation, instead of the usual three point interior equation for uniform properties. The resulting equation is not exactly equivalent to the conventional three point finite difference equation for uniform properties. The difference between them is shown to be negligible if sufficiently small steps are taken with explicit results for selected accuracy. Hence with sufficient nodal points and by averaging the coefficient D and the source term D B2 at an interface, the method is able to model flux variation between two different homogeneous regions. (author)
3D Staggered-Grid Finite-Difference Simulation of Acoustic Waves in Turbulent Moving Media
Symons, N. P.; Aldridge, D. F.; Marlin, D.; Wilson, D. K.; Sullivan, P.; Ostashev, V.
2003-12-01
Acoustic wave propagation in a three-dimensional heterogeneous moving atmosphere is accurately simulated with a numerical algorithm recently developed under the DOD Common High Performance Computing Software Support Initiative (CHSSI). Sound waves within such a dynamic environment are mathematically described by a set of four, coupled, first-order partial differential equations governing small-amplitude fluctuations in pressure and particle velocity. The system is rigorously derived from fundamental principles of continuum mechanics, ideal-fluid constitutive relations, and reasonable assumptions that the ambient atmospheric motion is adiabatic and divergence-free. An explicit, time-domain, finite-difference (FD) numerical scheme is used to solve the system for both pressure and particle velocity wavefields. The atmosphere is characterized by 3D gridded models of sound speed, mass density, and the three components of the wind velocity vector. Dependent variables are stored on staggered spatial and temporal grids, and centered FD operators possess 2nd-order and 4th-order space/time accuracy. Accurate sound wave simulation is achieved provided grid intervals are chosen appropriately. The gridding must be fine enough to reduce numerical dispersion artifacts to an acceptable level and maintain stability. The algorithm is designed to execute on parallel computational platforms by utilizing a spatial domain-decomposition strategy. Currently, the algorithm has been validated on four different computational platforms, and parallel scalability of approximately 85% has been demonstrated. Comparisons with analytic solutions for uniform and vertically stratified wind models indicate that the FD algorithm generates accurate results with either a vanishing pressure or vanishing vertical-particle velocity boundary condition. Simulations are performed using a kinematic turbulence wind profile developed with the quasi-wavelet method. In addition, preliminary results are presented using high-resolution 3D dynamic turbulent flowfields generated by a large-eddy simulation model of a stably stratified planetary boundary layer. Sandia National Laboratories is a operated by Sandia Corporation, a Lockheed Martin Company, for the USDOE under contract 94-AL85000.
Kokurin, M. M.
2015-12-01
The convergence of the quasi-reversibility method and two classes of finite-difference methods for solving the ill-posed Cauchy problem for the first-order equation with a sectorial operator in a Banach space is analyzed. The necessary and sufficient conditions—close to one another—for the convergence of these methods with a rate polynomial with respect to the regularization parameter or discretization step are obtained in terms of the exponent in the source representability of the solution.
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
Finite-difference frequency-domain study of subwavelength lensing in left-handed materials
Wang, X; Schatz, G C; Gray, Stephen K.; Schatz, George C.
2006-01-01
We show that the finite-difference frequency-domain method is well-suited to study subwavelength lensing effects in left-handed materials (LHM's) and related problems. The method is efficient and works in the frequency domain, eliminating the need for specifying dispersion models for the permeability and permittivity as required by the popular finite-difference time-domain method. We show that "superlensing" in a LHM slab with refractive index n = -1 can be approached by introducing an arbitrarily small loss term. We also study a thin silver slab, which can exhibit subwavelength imaging in the electrostatic limit.
Directory of Open Access Journals (Sweden)
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.
Application of a novel finite difference method to dynamic crack problems
Chen, Y. M.; Wilkins, M. L.
1976-01-01
A versatile finite difference method (HEMP and HEMP 3D computer programs) was developed originally for solving dynamic problems in continuum mechanics. It was extended to analyze the stress field around cracks in a solid with finite geometry subjected to dynamic loads and to simulate numerically the dynamic fracture phenomena with success. This method is an explicit finite difference method applied to the Lagrangian formulation of the equations of continuum mechanics in two and three space dimensions and time. The calculational grid moves with the material and in this way it gives a more detailed description of the physics of the problem than the Eulerian formulation.
DEFF Research Database (Denmark)
Shyroki, Dzmitry; Lægsgaard, Jesper; Bang, Ole; Skorobogatiy, Maksim
As an alternative to the finite-element analysis or subgridding, coordinate transformation is used to “stretch” the fine-structured cladding of a Bragg fiber, and then the fullvector, equidistant-grid finite-difference computations of the modal structure are performed.......As an alternative to the finite-element analysis or subgridding, coordinate transformation is used to “stretch” the fine-structured cladding of a Bragg fiber, and then the fullvector, equidistant-grid finite-difference computations of the modal structure are performed....
Determination of electromagnetic cavity modes using the Finite Difference Frequency-Domain Method
Directory of Open Access Journals (Sweden)
J. Manzanares-Martnez
2010-05-01
Full Text Available In this communication we propose a numerical determination of the electromagnetic modes in a cavity by using the Finite Difference Frequency-Domain Method. We first derive the analytical solution of the system and subsequently we introduce the numerical approximation. The cavity modes are obtained by solving an eigenvalue equation where the eigenvectors describe the eigenfunctions on the real space. It is found that this finite difference method can efficiently and accurately determine the resonance modes of the cavity with a small amount of numerical calculation.
Xie, Shengbai; Ghaisas, Niranjan; Archer, Cristina L.
2015-12-01
The neutral atmospheric boundary layer (ABL) is simulated by finite-difference large-eddy simulations (LES) with various dynamic subgrid-scale (SGS) models. The goal is to understand the sensitivity of the results to several aspects of the simulation set-up: SGS model, numerical scheme for the convective term, resolution, and filter type. Three dynamic SGS models are tested: two scale-invariant models and the Lagrangian-averaged scale-dependent (LASD) model. The results show that the LASD model has the best performance in capturing the law-of-the-wall, because the scale invariance hypothesis is violated in finite-difference LES. Two forms of the convective term are tested, the skew-symmetric and the divergence forms. The choice of the convective term is more important when the LASD model is used and the skew-symmetric scheme leads to better simulations in general. However, at fine resolutions both in space and time, the sensitivity to the convective scheme is reduced. Increasing the resolution improves the performance in general, but does not better capture the law of the wall. The box and Gaussian filters are tested and it is found that, combined with the LASD model, the Gaussian filter is not sufficient to dissipate the small numerical noises, which in turn affects the large-scale motions. In conclusion, to get the most benefits of the LASD model within the finite-difference framework, the simulations need to be set up properly by choosing the right combination of numerical scheme, resolution, and filter type.
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.
Vertical finite-element scheme for the hydrostatic primitive equations on a cubed-sphere
Park, J. R.; Yi, T. H.
2014-12-01
A vertical finite-element (VFE) scheme of three-dimensional hydrostatic primitive equations is adopted for the numerical weather prediction system, which is horizontally discretized with spectral elements on a cubed-sphere. The hybrid pressure-based vertical coordinate is employed to discretize a vertical grid, in which only the full levels of the coordinate are used in the VFE. Vertical integrals and derivatives in the hydrostatic equations are derived based on Galerkin-based finite elements with b-spline functions. These basis functions and their first-order derivatives are constructed using the Cox-de Boor algorithm. The computation of vertical integrals, derivatives and advections in the hydrostatic equations are easily done in physical space by matrix multiplication with the corresponding vertical operators. The VFE discretization scheme implemented into the global three-dimensional hydrostatic model on the cubed-sphere is evaluated by performing ideal test cases including the steady-state, baroclinic wave, 3D Rossby-Haurwitz wave, and mountain-induced Rossby wave train test cases. The two types of the VFE scheme are compared to the vertical finite difference scheme.
Principle of detailed balance and the finite-difference stochastic equation in field theory
International Nuclear Information System (INIS)
The principle of detailed balance for the Markov chain is used to obtain a finite-difference equation which generalizes the Langevin equation in field theory. The advantages of using this approach compared to the conventional Parisi-Wu method are demonstrated for the examples of an exactly solvable problem in zero-dimensional quantum theory and a simple numerical simulation
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
On the spectrum of relativistic Schrödinger equation in finite differences
Berezin, V A; Neronov, Andrii Yu
1999-01-01
We develop a method for constructing asymptotic solutions of finite-difference equations and implement it to a relativistic Schroedinger equation which describes motion of a selfgravitating spherically symmetric dust shell. Exact mass spectrum of black hole formed due to the collapse of the shell is determined from the analysis of asymptotic solutions of the equation.
Dispersion properties of nonradiating configurations: finite-difference time-domain modeling.
Boardman, A D; Marinov, K; Zheludev, N; Fedotov, V A
2005-09-01
A finite-difference time-domain (FDTD) numerical analysis is used to demonstrate that a toroidal solenoid, coaxial with an electric dipole, is a remarkable nonradiating configuration. It can be used to measure the dielectric permittivity of any ambient matter. It becomes a directional radiator at an interface between two dielectric media, depositing energy in the material with the highest polarizability. PMID:16241588
Comparison of finite differences and finite elements in the case of large fast power reactor
International Nuclear Information System (INIS)
A large number of test calculations in two-dimension hexagonal geometry for different configurations of SUPER-PHENIX 1 type Fast Reactor have been performed to compare finite differences theory versus finite elements theory performances. At present, no definitive advantages were found for the application of the finite elements method to two dimensional hexagonal design calculations
A Coupled Finite Difference and Moving Least Squares Simulation of Violent Breaking Wave Impact
DEFF Research Database (Denmark)
Lindberg, Ole; Bingham, Harry B.; Engsig-Karup, Allan Peter
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 specia...
International Nuclear Information System (INIS)
The need to smoothly cover a computational domain of interest generically requires the adoption of several grids. To solve a given problem under this grid structure, one must ensure the suitable transfer of information among the different grids involved. In this work, we discuss a technique that allows one to construct finite-difference schemes of arbitrary high order which are guaranteed to satisfy linear numerical and strict stability. The method relies on the use of difference operators satisfying summation by parts and penalty terms to transfer information between the grids. This allows the derivation of semi-discrete energy estimates for problems admitting such estimates at the continuum. We analyse several aspects of this technique when used in conjunction with high-order schemes and illustrate its use in one-, two- and three-dimensional numerical relativity model problems with non-trivial topologies, including truly spherical black hole excision
Camporeale, E; Bergen, B K; Moulton, J D
2013-01-01
We describe a spectral method for the numerical solution of the Vlasov-Poisson system where the velocity space is decomposed by means of an Hermite basis, and the configuration space is discretized via a Fourier decomposition. The novelty of our approach is an implicit time discretization that allows exact conservation of charge, momentum and energy. The computational efficiency and the cost-effectiveness of this method are compared to the fully-implicit PIC method recently introduced by Markidis and Lapenta (2011) and Chen et al. (2011). The following examples are discussed: Langmuir wave, Landau damping, ion-acoustic wave, two-stream instability. The Fourier-Hermite spectral method can achieve solutions that are several orders of magnitude more accurate at a fraction of the cost with respect to PIC. This paper concludes the study presented in Camporeale et al. (2013) where the same method has been described for a semi-implicit time discretization, and was compared against an explicit PIC.
International Nuclear Information System (INIS)
Coarse-mesh numerical methods are very efficient in the sense that they generate accurate results in short computational time, as the number of floating point operations generally decrease, as a result of the reduced number of mesh points. On the other hand, they generate numerical solutions that do not give detailed information on the problem solution profile, as the grid points can be located considerably away from each other. In this paper we describe two steps for the analytical reconstruction of the coarse-mesh solution generated by the spectral nodal method for neutral particle discrete ordinates (SN) transport model in slab geometry. The first step of the algorithm is based on the analytical reconstruction of the coarse-mesh solution within each discretization cell of the grid set up on the spatial domain. The second step is based on the angular reconstruction of the discrete ordinates solution between two contiguous ordinates of the angular quadrature set used in the SN model. Numerical results are given so we can illustrate the accuracy of the two reconstruction techniques, as described in this paper.
Energy Technology Data Exchange (ETDEWEB)
Barros, Ricardo C., E-mail: rcbarros@pq.cnpq.b [Programa de Pos-graduacao em Modelagem Computacional, Instituto Politecnico (IPRJ/UERJ), Rua Alberto Rangel s/n, 28630-050 Nova Friburgo, RJ (Brazil); Filho, Hermes Alves, E-mail: halves@iprj.uerj.b [Programa de Pos-graduacao em Modelagem Computacional, Instituto Politecnico (IPRJ/UERJ), Rua Alberto Rangel s/n, 28630-050 Nova Friburgo, RJ (Brazil); Platt, Gustavo M., E-mail: gmplatt@iprj.uerj.b [Programa de Pos-graduacao em Modelagem Computacional, Instituto Politecnico (IPRJ/UERJ), Rua Alberto Rangel s/n, 28630-050 Nova Friburgo, RJ (Brazil); Oliveira, Francisco Bruno S., E-mail: fbrunoso@uol.com.b [Programa de Pos-graduacao em Modelagem Computacional, Instituto Politecnico (IPRJ/UERJ), Rua Alberto Rangel s/n, 28630-050 Nova Friburgo, RJ (Brazil); Militao, Damiano S., E-mail: mestredam@yahoo.com.b [Programa de Pos-graduacao em Modelagem Computacional, Instituto Politecnico (IPRJ/UERJ), Rua Alberto Rangel s/n, 28630-050 Nova Friburgo, RJ (Brazil)
2010-11-15
Coarse-mesh numerical methods are very efficient in the sense that they generate accurate results in short computational time, as the number of floating point operations generally decrease, as a result of the reduced number of mesh points. On the other hand, they generate numerical solutions that do not give detailed information on the problem solution profile, as the grid points can be located considerably away from each other. In this paper we describe two steps for the analytical reconstruction of the coarse-mesh solution generated by the spectral nodal method for neutral particle discrete ordinates (S{sub N}) transport model in slab geometry. The first step of the algorithm is based on the analytical reconstruction of the coarse-mesh solution within each discretization cell of the grid set up on the spatial domain. The second step is based on the angular reconstruction of the discrete ordinates solution between two contiguous ordinates of the angular quadrature set used in the S{sub N} model. Numerical results are given so we can illustrate the accuracy of the two reconstruction techniques, as described in this paper.
Directory of Open Access Journals (Sweden)
Ren Roland Colditz
2015-07-01
Full Text Available Land cover mapping for large regions often employs satellite images of medium to coarse spatial resolution, which complicates mapping of discrete classes. Class memberships, which estimate the proportion of each class for every pixel, have been suggested as an alternative. This paper compares different strategies of training data allocation for discrete and continuous land cover mapping using classification and regression tree algorithms. In addition to measures of discrete and continuous map accuracy the correct estimation of the area is another important criteria. A subset of the 30 m national land cover dataset of 2006 (NLCD2006 of the United States was used as reference set to classify NADIR BRDF-adjusted surface reflectance time series of MODIS at 900 m spatial resolution. Results show that sampling of heterogeneous pixels and sample allocation according to the expected area of each class is best for classification trees. Regression trees for continuous land cover mapping should be trained with random allocation, and predictions should be normalized with a linear scaling function to correctly estimate the total area. From the tested algorithms random forest classification yields lower errors than boosted trees of C5.0, and Cubist shows higher accuracies than random forest regression.
International Nuclear Information System (INIS)
Highlights: ► The hybrid SD–SGF–CN spectral nodal method. ► Accurate coarse-mesh solution to SN eigenvalue problems in X,Y geometry. ► Analytical spatial reconstruction scheme within each discretization node. ► Accurate flux profile generated from the spatial reconstruction scheme. - Abstract: Nodal methods are widely regarded as forming an accurate class of coarse-mesh methods for neutron transport problems in the discrete ordinates (SN) formulation. They are also viewed as efficient methods, as the number of floating point operations generally decrease, as a result of the reduced number of mesh points; therefore they generate accurate results in shorter running time. However, the coarse-mesh numerical solutions do not yield detailed information on the solution profile, as the grid points can be located considerably apart from each other. In this paper, we describe an analytical spatial reconstruction of coarse-mesh solutions of the SN transverse integrated nodal equations with constant approximations for the transverse leakage terms, as generated by the hybrid spectral diamond–spectral Green’s function–constant nodal (SD–SGF–CN) method for monoenergetic SN eigenvalue problems in X,Y geometry for neutron multiplying systems. Numerical results for typical model problems are given and we close with general concluding remarks and suggestions for future work
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
DEFF Research Database (Denmark)
Bieniasz, Leslaw K.; Østerby, Ole; Britz, Dieter
1995-01-01
unconditional stability of the Saul'yev algorithms, reported in the literature, the left-right variant of the Saul'yev algorithm becomes unstable for large values of the dimensionless diffusion parameter λ = δt/h2, under mixed boundary conditions. This limitation is not, however, severe for most practical......The stepwise numerical stability of the Saul'yev finite difference discretization of an example diffusional initial boundary value problem from electrochemical kinetics has been investigated using the matrix method of stability analysis. Special attention has been paid to the effect of the...... discretization of the mixed, linear boundary condition on stability, assuming the two-point, forward-difference approximation for the gradient at the left boundary (electrode). Criteria regulating the error growth in time have been identified. In particular it has been shown that, in contrast to the claims of...
Scientific Electronic Library Online (English)
Jorge Mauricio, Ruiz Vera; Ignacio, Mantilla Prada.
2013-06-01
Full Text Available La ecuacin de Derrida-Lebowitz-Speer-Spohn (DLSS) es una ecuacin de evolucin no lineal de cuarto orden. Esta aparece en el estudio de las fluctuaciones de interface de sistemas de espn y en la modelacin de semicoductores cunticos. En este artculo, se presenta una discretizacin por elementos [...] finitos para una formulacin exponencial de la ecuacin DLSS abordada como un sistema acoplado de ecuaciones. Usando la informacin disponible acerca del fenmeno fsico, se establecen las condiciones de contorno para el sistema acoplado. Se demuestra la existencia de la solucin discreta global en el tiempo via un argumento de punto fijo. Los resultados numricos ilustran el carcter cuntico de la ecuacin. Finalmente se presenta un test del orden de convergencia de la discretizacin 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.
An improvement of the douglas scheme for the Black-Scholes equation
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FARES AL-AZEMI
2015-09-01
Full Text Available A well-known finite difference scheme for the valuation of options from the Black-Sholes equation is the Crank-Nicolson scheme. However, in the case of non-smoothpayoffs, the Crank-Nicolson scheme is known to produce unwanted oscillations forthe computed solution. As an alternative, Douglas scheme is generally recommendedfor better resolution of option price because it has fourth order accuracy in assetderivative. However, as noted by Shaw in his book, both these methods show“potentially nasty behavior when applied to simple option pricing”. We note that boththe Crank-Nicolson scheme and the Douglas scheme use a trapezoidal formula for timeintegration which is known to produce unwanted oscillations in the computed solution.This works since the trapezoidal formula is only A-stable and not L-stable. Chawlaand Evans proposed a new L-stable Simpson rule. We investigate the application ofthis L-stable third order rule for the time integration in the Black-Sholes equationafter it has been semi-discretized in the asset derivative by Numerov discretisation.By numerical experimentation with real option valuation problems, we compare theperformance of this new improved version of Douglas with both Crank-Nicolson andDouglas schemes. We also study the performance of this scheme for the valuation ofthe Greeks.
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.
Bohlen, Thomas; Wittkamp, Florian
2016-03-01
We analyse the performance of a higher order accurate staggered viscoelastic time-domain finite-difference method, in which the staggered Adams-Bashforth (ABS) third-order and fourth-order accurate time integrators are used for temporal discretization. ABS is a multistep method that uses previously calculated wavefields to increase the order of accuracy in time. The analysis shows that the numerical dispersion is much lower than that of the widely used second-order leapfrog method. Numerical dissipation is introduced by the ABS method which is significantly smaller for fourth-order than third-order accuracy. In 1-D and 3-D simulation experiments, we verify the convincing improvements of simulation accuracy of the fourth-order ABS method. In a realistic elastic 3-D scenario, the computing time reduces by a factor of approximately 2.4, whereas the memory requirements increase by approximately a factor of 2.2. The ABS method thus provides an alternative strategy to increase the simulation accuracy in time by investing computer memory instead of computing time.
Analysis of a fourth-order compact scheme for convection-diffusion
International Nuclear Information System (INIS)
In, 1984 Gupta et al. introduced a compact fourth-order finite-difference convection-diffusion operator with some very favorable properties. In particular, this scheme does not seem to suffer excessively from spurious oscillatory behavior, and it converges with standard methods such as Gauss Seidel or SOR (hence, multigrid) regardless of the diffusion. This scheme has been rederived, developed (including some variations), and applied in both convection-diffusion and Navier-Stokes equations by several authors. Accurate solutions to high Reynolds-number flow problems at relatively coarse resolutions have been reported. These solutions were often compared to those obtained by lower order discretizations, such as second-order central differences and first-order upstream discretizations. The latter, it was stated, achieved far less accurate results due to the artificial viscosity, which the compact scheme did not include. We show here that, while the compact scheme indeed does not suffer from a cross-stream artificial viscosity (as does the first-order upstream scheme when the characteristic direction is not aligned with the grid), it does include a streamwise artificial viscosity that is inversely proportional to the natural viscosity. This term is not always benign. 7 refs., 1 fig., 1 tab
On One-Dimensional Stretching Functions for Finite-Difference Calculations
Vinokur, M.
1980-01-01
The class of one dimensional stretching function used in finite difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One is an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two sided stretching function, for which the arbitrary slopes at the two ends of the one dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent. The general two sided function has many applications in the construction of finite difference grids.
Implementation of compact finite-difference method to parabolized Navier-Stokes equations
International Nuclear Information System (INIS)
The numerical simulation of the Parabolized Navier-Stokes (PNS) equations for supersonic/hypersonic flow field is obtained by using the fourth-order compact finite-difference method. The PNS equations in the general curvilinear coordinates are solved by using the implicit finite-difference algorithm of Beam and Warming. A shock fitting procedure is utilized to obtain the accurate solution in the vicinity of the shock. The computations are performed for hypersonic axisymmetric flow over a blunt cone. The present results for the flow field along with those of the second-order method are presented and accuracy analysis is performed to insure the fourth-order accuracy of the method. (author)
Finite difference solution of the flux reconstruction problem in nodal reactor analysis
International Nuclear Information System (INIS)
The reconstruction of the fine-mesh neutron flux distribution using coarse-mesh computational results is an important step in a nodal reactor analysis procedure. The problem is divided in a one-dimensional interpolation along the boundaries of the fuel assembly followed by a direct solution of the associated Dirichlet problem. Finite difference techniques are employed for the calculation of the necessary flux corner values. Apart from the fact that this approach can be used for an arbitrary number of energy groups an additional advantage is that it yields an estimate of the order of accuracy. Given these corner values and the known average surface fluxes continuous one-dimensional boundary conditions can be easily constructed by a variety of methods. The determination of the interior solution is based on a direct higher order finite difference technique. The high computational accuracy and efficiency of the proposed method is demonstrated by some results for hexagonal assemblies. (orig.)
International Nuclear Information System (INIS)
The basic principles of the finite-difference approximation applied to the solution of electrostatic field distributions in gaseous proportional counters are given. Using this method, complicated two-dimensional electrostatic problems may be solved, taking into account any number of anodes, each with its own radius, and any cathode shape. A general formula for introducing the anode radii into the calculations is derived and a method of obtaining extremely accurate (up to 0.1%) solutions is developed. Several examples of potential and absolute field distributions for single rectangular and multiwire proportional counters are calculated and compared with exact results according to Tomitani, in order to discuss in detail errors of the finite-difference approximation. (author)
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.
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
Sun, Guangfu
1993-01-01
This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convectio...
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)
Virta, Kristoffer.; Juhlin, Christopher; Kreiss, Gunilla
2015-01-01
We compare two high order finite-difference methods that solve the elastic wave equation in two dimensional domains with curved boundaries and material discontinuities. Two numerical experiments are designed with focus on wave boundary interaction, the response of a pressure wave impinging on a circular cavity and the wave field generated by an explosive impact on the wall an underground tunnel. Qualitative comparisons of the results are made where similarities and differences are pointed out.
Simulation of realistic rotor blade-vortex interactions using a finite-difference technique
Hassan, Ahmed A.; Charles, Bruce D.
1989-01-01
A numerical finite-difference code has been used to predict helicopter blade loads during realistic self-generated three-dimensional blade-vortex interactions. The velocity field is determined via a nonlinear superposition of the rotor flowfield. Data obtained from a lifting-line helicopter/rotor trim code are used to determine the instantaneous position of the interaction vortex elements with respect to the blade. Data obtained for three rotor advance ratios show a reasonable correlation with wind tunnel data.
Numerical techniques in linear duct acoustics. [finite difference and finite element analyses
Baumeister, K. J.
1980-01-01
Both finite difference and finite element analyses of small amplitude (linear) sound propagation in straight and variable area ducts with flow, as might be found in a typical turboject engine duct, muffler, or industrial ventilation system, are reviewed. Both steady state and transient theories are discussed. Emphasis is placed on the advantages and limitations associated with the various numerical techniques. Examples of practical problems are given for which the numerical techniques have been applied.
On supraconvergence phenomenon for second order centered finite differences on non-uniform grids
Khakimzyanov, Gayaz
2015-01-01
In the present note we consider an example of a boundary value problem for a simple second order ordinary differential equation, which may exhibit a boundary layer phenomenon. We show that usual central finite differences, which are second order accurate on a uniform grid, can be substantially upgraded to the fourth order by a suitable choice of the underlying non-uniform grid. This example is quite pedagogical and may give some ideas for more complex problems.
Comparing finite difference forward models using free energy based on multiple sparse priors
Strobbe, Gregor; Lopez, Jose; Montes Restrepo, Victoria Eugenia; van Mierlo, Pieter; Hallez, Hans; Vandenberghe, Stefaan
2012-01-01
Due to the ill-posed nature of the EEG source localization problem, the spatial resolution of the reconstructed activity is limited to several centimeters (Baillet, 2001). Advanced forward modeling of the head can contribute to improve the spatial resolution (Hallez, 2007). The boundary element method or BEM is commonly used due to its computation speed. More advanced volume modeling methods, such as finite difference methods or FDM, are computationally more intensive but allow estimating sou...
Solution of nondeterministic finite element and finite difference heat conduction simulations
Energy Technology Data Exchange (ETDEWEB)
Padovan, J.; Guo, Y.H. (Akron Univ., OH (USA))
1989-01-01
A formal solution methodology is developed for nondeterministic finite element and finite difference simulations of heat conduction. The generality of the procedure is such that statistical variations in both system and environmental characteristics can be handled. This includes the capability to analyze essentially any level of probabilistic variation. For simplification of use, a priori convergence and truncation criteria for the procedure are developed. These enable problem-dependent resource requirements to be predetermined. 14 refs.
A simple finite-difference grid with non-constant intervals
SUNDQVIST, HILDING; Veronis, George
2011-01-01
A finite difference network with non-uniform grid intervals, hi, such that hi = hi-1 + O(h2i-1) is presented. The use of the grid network for two specific choices of hi - hi-1 for a linear boundary layer problem shows that it has distinct advantages in computational efficiency and in accuracy of representation of the analytical solution.DOI: 10.1111/j.2153-3490.1970.tb01933.x
TRUMP3-JR: a finite difference computer program for nonlinear heat conduction problems
International Nuclear Information System (INIS)
Computer program TRUMP3-JR is a revised version of TRUMP3 which is a finite difference computer program used for the solution of multi-dimensional nonlinear heat conduction problems. Pre- and post-processings for input data generation and graphical representations of calculation results of TRUMP3 are avaiable in TRUMP3-JR. The calculation equations, program descriptions and user's instruction are presented. A sample problem is described to demonstrate the use of the program. (author)
Kamiński M.; Supeł Ł.
2016-01-01
It is widely known that lateral-torsional buckling of a member under bending and warping restraints of its cross-sections in the steel structures are crucial for estimation of their safety and durability. Although engineering codes for steel and aluminum structures support the designer with the additional analytical expressions depending even on the boundary conditions and internal forces diagrams, one may apply alternatively the traditional Finite Element or Finite Difference Methods (FEM, F...
A finite difference based tool for analysing ground source heat pump system
Bag, Ramakrishna; Sedighi, Majid; Manju; Rhys Thomas, Hywel
2015-01-01
The ground heat exchangers (GHE) consist of pipes buried in the soil and are used for transferring heat between the soil and the heat exchanger pipes of the ground source heat pump (GSHP). This paper presents the development of a numerical tool for anlysing the behaviour of horizontal ground source heat pump system. The tool was developed in Visual C++ environment. Impicilit finite difference heat conduction method was employed. The numerical solution was obtained by LU factori...
Modeling and Analysis of Printed Antenna Using Finite Difference Time Domain Algorithm
Directory of Open Access Journals (Sweden)
Dr.T.SHANMUGANANTHAM,
2010-12-01
Full Text Available 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 communications and RFID applications.
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...
Modeling and Simulation of Hamburger Cooking Process Using Finite Difference and CFD Methods
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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.
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-β 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 provide a reference for further research into the electrical simulation of PSPSs
Ren, B. Y.; Wang, L.; Tzou, H. S.; Yue, H. H.
2010-08-01
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-β 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 provide a reference for further research into the electrical simulation of PSPSs.
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 lquation de Boltzmann pour le transport des neutrinos dans les modles de supernovae effondrement de cur ainsi quune description dtaille de lIsotropic Diffusion Source Approximation (IDSA dveloppe dans [6]. De plus, nous prsentons le traitement numrique dun modle de Boltzmann simplifi bas sur une dcomposition en temps de loprateur et sur un algorithme de volumes finis ainsi que ladaptation de la discrtisation de lIDSA de [6] notre modle. Les tudes de lerreur de discrtisation faites sur le modle de Boltzmann simplifi et sur lIDSA montrent que les erreurs sont dordre un dans les deux cas. A laide dun exemple numrique, nous comparons et interprtons en dtail les deux modles. Pour cet exemple, lerreur de modlisation de lIDSA par rapport au modle de Boltzmann simplifi est dtermine numriquement et localise.
Zhang, Hong
2016-01-01
Saturation overshoot and pressure overshoot are studied by incorporating dynamic capillary pressure, capillary pressure hysteresis and hysteretic dynamic coefficient with a traditional fractional flow equation. Using the method of lines, the discretizations are constructed by applying Castillo-Grone's mimetic operators in the space direction and explicit trapezoidal integrator in the time direction. Convergence tests and conservation property of the schemes are presented. Computed profiles capture both the saturation overshoot and pressure overshoot phenomena. Comparisons between numerical results and experiments illustrate the effectiveness and different features of the models.
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)
S. S. Das, M. Mohanty, R. K. Padhy, M. Sahu
2012-01-01
Full Text Available This paper considers the magnetohydrodynamic flow and heat transfer in a viscous incompressible fluid between two parallel porous plates experiencing a discontinuous change in wall temperature. An explicit finite difference scheme has been employed to solve the coupled non-linear equations governing the flow. The flow phenomenon has been characterized by Hartmann number, suction Reynolds number, channel Reynolds number and Prandtl number. The effects of these parameters on the velocity and temperature distribution have been analyzed and the results are presented with the aid of figures. It is observed that a growing suction parameter R retards the velocity of the flow field both in MHD as well as non-MHD flow. The effect of increasing Hartmann number is to decrease the transverse component of velocity for both suction and injection and in absence of suction and injection, while it decreases the axial component of velocity up to the middle of the channel and beyond this the effect reverses. There is a sharp fluctuation in temperature near the walls and at the middle of the channel which may be attributed to the discontinuous change in wall temperature. For fluids having low Prandtl number such as air, the temperature assumes negative values.
Lipp, V P; Garcia, M E; Ivanov, D S
2015-01-01
We present a finite-difference integration algorithm for solution of a system of differential equations containing a diffusion equation with nonlinear terms. The approach is based on Crank-Nicolson method with predictor-corrector algorithm and provides high stability and precision. Using a specific example of short-pulse laser interaction with semiconductors, we give a detailed description of the method and apply it for the solution of the corresponding system of differential equations, one of which is a nonlinear diffusion equation. The calculated dynamics of the energy density and the number density of photoexcited free carriers upon the absorption of laser energy are presented for the irradiated thin silicon film. The energy conservation within 0.2% has been achieved for the time step $10^4$ times larger than that in case of the explicit scheme, for the chosen numerical setup. We also present a few examples of successful application of the method demonstrating its benefits for the theoretical studies of la...
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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.
Ren, Zhiming; Liu, Yang; Zhang, Qunshan
2014-05-01
Full waveform inversion (FWI) has the potential to provide preferable subsurface model parameters. The main barrier of its applications to real seismic data is heavy computational amount. Numerical modelling methods are involved in both forward modelling and backpropagation of wavefield residuals, which spend most of computational time in FWI. We develop a time-space domain finite-difference (FD) method and adaptive variable-length spatial operator scheme in numerical simulation of viscoacoustic equation and extend them into the viscoacoustic FWI. Compared with conventional FD methods, different operator lengths are adopted for different velocities and quality factors, which can reduce the amount of computation without reducing accuracy. Inversion algorithms also play a significant role in FWI. In conventional single-scale methods, it is likely to converge to local minimums especially when the initial model is far from the real model. To tackle the problem, we introduce the second generation wavelet transform to implement the multiscale FWI. Compared to other multiscale methods, our method has advantages of ease of implementation and better time-frequency local analysis ability. The L2 norm is widely used in FWI and gives invalid model estimates when the data is contaminated with strong non-uniform noises. We apply the L1-norm and the Huber-norm criteria in the time-domain FWI to improve its antinoise ability. Our strategies have been successfully applied in synthetic experiments to both onshore and offshore reflection seismic data. The results of the viscoacoustic Marmousi example indicate that our new FWI scheme consumes smaller computer resources. In addition, the viscoacoustic Overthrust example shows its better convergence and more reasonable velocity and quality factor structures. All these results demonstrate that our method can improve inversion accuracy and computational efficiency of FWI.