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1

On second-order mimetic and conservative finite-difference discretization schemes

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 ...

Rojas, S.; Guevara-jordan, J. M.

2008-01-01

2

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 Lejago 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)

3

On second-order mimetic and conservative finite-difference discretization schemes

Directory of Open Access Journals (Sweden)

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

S Rojas

2008-12-01

4

On second-order mimetic and conservative finite-difference discretization schemes

Scientific Electronic Library Online (English)

Full Text Available SciELO Mexico | Language: English Abstract in spanish Aunque la derivación del esquema se puede realizar usando la reciente metodología de discretización numérica conocida como Diferencias Finitas Miméticas, estaremos presentando la derivación de un esquema de discretización mimético en diferencias finitas de segundo orden en una forma mas intuitiva, m [...] ediante el uso de expansiones de Taylor. Considerando que los estudiantes se familiarizan con expansiones de Taylor en los primeros cursos de cálculo y métodos matemáticos para físicos, pensamos que la presente alternativa de presentar este nuevo esquema de discretización es más favorable de ser asimilada en cursos de computación numérica tanto de pregrado como de postgrado. La robusticidad del esquema será ilustrada encontrando la solución numérica de un problema unidimensional del tipo capa límite difícil de resolver en forma numérica y que se basa en la ecuación de difusión estacionaria. Más aun, dado que el esquema de discretización alcanza segundo orden de precisión en todo el dominio computacional (incluyendo las fronteras), como ejercicio comparativo el mismo puede ser rápidamente aplicado para resolver ejemplos comúnmente encontrados en textos sobre métodos numéricos aplicados y que se resuelven usando otras metodologías numéricas (incluyendo algunos esquemas de discretización en diferencias finitas) Abstract in english Although the scheme could be derived on the grounds of a relatively new numerical discretization methodology known as Mimetic Finite-Difference Approach, the derivation of a second-order mimetic finite difference discretization scheme will be presented in a more intuitive way, using Taylor expansion [...] s. Since students become familiar with Taylor expansions in earlier calculus and mathematical methods for physicist courses, one finds this approach of presenting this new discretization scheme to be more easily handled in courses on numerical computations of both undergraduate and graduated programs. The robustness of the resulting discretized equations will be illustrated by finding the numerical solution of an essentially hard-to-solve, one-dimensional, boundary-layer-like problem, based on the steady diffusion equation. Moreover, given that the presented mimetic discretization scheme attains second-order accuracy in the entire computational domain (including the boundaries), as a comparative exercise the discretized equations can be readily applied in solving examples commonly found in texbooks on applied numerical methods and solved numerically via other discretization schemes (including some of the standard finite-diffence discretization schemes)

S, Rojas; J.M, Guevara-Jordan.

2008-12-01

5

International Nuclear Information System (INIS)

The three-dimensional, time-dependent convection-diffusion equation (CDE) is considered. An exponential transformation is used to collectively transform the CDE. The idea of global discretization is used, where attention is paid to the whole transformed CDE, but not to the individual spatial and temporal derivatives in the equation. Four finite difference schemes for both CDE and transformed CDE are established. The modified partial differential equations of these schemes are obtained, which indicate that the trunction errors of the schemes can be of second and fourth order, depending on the prescription of the time step length. Some characteristic physical parameters, i.e., local Reynolds number, local Strouhal number, and viscous diffusive length, are introduced into the schemes and the viscous diffusive length is found to be a significant parameter in relating temporal discretisation with spatial discretisation. A series of benchmark analytical solutions of Navier-Stokes and Burgers equations, as well s the numerical solutions using the well-known discretisation schemes, are used to investigate the properties of the derived schemes. The high-order schemes achieve higher resolutions over the conventional schemes without decreasing much the sparsity of the matrix structures. Grid refinement studies reveal that the inverse exponential transformation of the finite difference schemes tends to destroy some resolution of the schemes, especially for large local Reynolds nues, especially for large local Reynolds number. 18 refs., 9 figs., 4 tabs

6

Conservative properties of finite difference schemes for incompressible flow

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.

Morinishi, Youhei

1995-01-01

7

Superconvergent finite difference discretization for reactor calculations

International Nuclear Information System (INIS)

Mesh centered and mesh finite difference discretizations can be derived formally from a primal and dual variational principle, using Gauss-Lobatto quadratures. We show that Gauss-Legendre quadratures can also be applied to the same primal and dual functionals in order to obtain a more accurate discretization: the superconvergent finite difference method. An efficient ADI (Alternating Direction Implicit) numerical technique with a supervectorization procedure was set up to solve the resulting matrix system. Validation results are given for the IAEA 2-D, Biblis and IAEA 3-D benchmarks and for a typical full-core 3-d representation of a pressurized water reactor at the beginning of the second cycle. 13 refs., 6 tabs

8

Applications of nonstandard finite difference schemes

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

Mickens, Ronald E

2000-01-01

9

An optimized finite-difference scheme for wave propagation problems

Two fully-discrete finite-difference schemes for wave propagation problems are presented, a maximum-order scheme and an optimized (or spectral-like) scheme. Both combine a seven-point spatial operator and an explicit six-stage time-march method. The maximum-order operator is fifth-order in space and is sixth-order in time for a linear problem with periodic boundary conditions. The phase and amplitude errors of the schemes obtained using Fourier analysis are given and compared with a second-order and a fourth-order method. Numerical experiments are presented which demonstrate the usefulness of the schemes for a range of problems. For some problems, the optimized scheme leads to a reduction in global error compared to the maximum-order scheme with no additional computational expense.

Zingg, D. W.; Lomax, H.; Jurgens, H.

1993-01-01

10

An optimized finite-difference scheme for wave propagation problems

International Nuclear Information System (INIS)

Two fully-discrete finite-difference schemes for wave propagation problems are presented, a maximum-order scheme and an optimized (or spectral-like) scheme. Both combine a seven-point spatial operator and an explicit six-stage time-march method. The maximum-order operator is fifth-order in space and is sixth-order in time for a linear problem with periodic boundary conditions. The phase and amplitude errors of the schemes obtained using Fourier analysis are given and compared with a second-order and a fourth-order method. Numerical experiments are presented which demonstrate the usefulness of the schemes for a range of problems. For some problems, the optimized scheme leads to a reduction in global error compared to the maximum-order scheme with no additional computational expense. 16 refs

11

Finite-difference schemes for anisotropic diffusion

In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 1012 times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretization schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.

van Es, Bram; Koren, Barry; de Blank, Hugo J.

2014-09-01

12

Optimizations on Designing High-Resolution Finite-Difference Schemes

We describe a general optimization procedure for both maximizing the resolution characteristics of existing finite differencing schemes as well as designing finite difference schemes that will meet the error tolerance requirements of numerical solutions. The procedure is based on an optimization process. This is a generalization of the compact scheme introduced by Lele in which the resolution is improved for single, one-dimensional spatial derivative, whereas in the present approach the complete scheme, after spatial and temporal discretizations, is optimized on a range of parameters of the scheme and the governing equations. The approach is to linearize and Fourier analyze the discretized equations to check the resolving power of the scheme for various wave number ranges in the solution and optimize the resolution to satisfy the requirements of the problem. This represents a constrained nonlinear optimization problem which can be solved to obtain the nodal weights of discretization. An objective function is defined in the parametric space of wave numbers, Courant number, Mach number and other quantities of interest. Typical criterion for defining the objective function include the maximization of the resolution of high wave numbers for acoustic and electromagnetic wave propagations and turbulence calculations. The procedure is being tested on off-design conditions of non-uniform mesh, non-periodic boundary conditions, and non-constant wave speeds for scalar and system of equations. This includes the solution of wave equations and Euler equations using a conventional scheme with and without optimization and the design of an optimum scheme for the specified error tolerance.

Liu, Yen; Koomullil, George; Kwak, Dochan (Technical Monitor)

1994-01-01

13

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

14

Dispersion-dissipation condition for finite difference schemes

In this short note, by analyzing the dispersion and dissipation of explicit finite difference scheme, a dispersion-dissipation condition is derived to determine the minimum dissipation required to damp the artificial high-wavenumber waves in the solution. The example application to our previous developed WENO-CU6-M2 scheme suggests that this condition can be used as an general guidance on optimizing the dissipation of a numerical scheme.

Hu, X Y

2012-01-01

15

Explicit and implicit finite difference schemes for fractional Cattaneo equation

In this paper, the numerical solution of fractional (non-integer)-order Cattaneo equation for describing anomalous diffusion has been investigated. Two finite difference schemes namely an explicit predictor-corrector and totally implicit schemes have been developed. In developing each scheme, a separate formulation approach for the governing equations has been considered. The explicit predictor-corrector scheme is the fractional generalization of well-known MacCormack scheme and has been called Generalized MacCormack scheme. This scheme solves two coupled low-order equations and simultaneously computes the flux term with the main variable. Fully implicit scheme however solves a single high-order undecomposed equation. For Generalized MacCormack scheme, stability analysis has been studied through Fourier method. Through a numerical test, the experimental order of convergency of both schemes has been found. Then, the domain of applicability and some numerical properties of each scheme have been discussed.

Ghazizadeh, H. R.; Maerefat, M.; Azimi, A.

2010-09-01

16

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.

17

An explicit finite difference scheme for the Camassa-Holm equation

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general $H^1$ initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of Camassa-Holm equation.

Coclite, Giuseppe Maria; Risebro, Nils Henrik

2008-01-01

18

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

19

Generalized finite-difference time-domain schemes for solving nonlinear Schrodinger equations

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.

Moxley, Frederick Ira, III

20

A Pseudo-compact Conservative Average Finite Difference Scheme for Dissipation SRLW Eqation

Directory of Open Access Journals (Sweden)

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

ZHENG Mao-bo

2014-01-01

21

In this paper, we utilize the maximum-principle-preserving flux limiting technique, originally designed for high order weighted essentially non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to develop a class of high order positivity-preserving finite difference WENO methods for the ideal magnetohydrodynamic (MHD) equations. Our schemes, under the constrained transport (CT) framework, can achieve high order accuracy, a discrete divergence-free condition...

Christlieb, Andrew J.; Liu, Yuan; Tang, Qi; Xu, Zhengfu

2014-01-01

22

In this work we develop a class of high-order finite difference weighted essentially non-oscillatory (FD-WENO) schemes for solving the ideal magnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this work is to use efficient high-order WENO spatial discretizations with high-order strong stability-preserving Runge-Kutta (SSP-RK) time-stepping schemes. Numerical results have shown that with such methods we are able to resolve solution structures that are only vis...

Christlieb, Andrew J.; Rossmanith, James A.; Tang, Qi

2013-01-01

23

Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.

Fournie?, Michel; Du?ring, Bertram; Ju?ngel, Ansgar

2004-01-01

24

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

25

Single-cone real-space finite difference scheme for the time-dependent Dirac equation

A finite difference scheme for the numerical treatment of the (3+1)D Dirac equation is presented. Its staggered-grid intertwined discretization treats space and time coordinates on equal footing, thereby avoiding the notorious fermion doubling problem. This explicit scheme operates entirely in real space and leads to optimal linear scaling behavior for the computational effort per space-time grid-point. It allows for an easy and efficient parallelization. A functional for a norm on the grid is identified. It can be interpreted as probability density and is proved to be conserved by the scheme. The single-cone dispersion relation is shown and exact stability conditions are derived. Finally, a single-cone scheme and its properties are presented for the two-component (2+1)D Dirac equation.

Hammer, René; Arnold, Anton

2013-01-01

26

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.

Hong, Qianying; Lai, Ming-jun; Wang, Jingyue

2013-01-01

27

High-Order Finite Difference GLM-MHD Schemes for Cell-Centered MHD

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate in smooth regions of the flow, avoid loss of accuracy in proximity of smooth extrema and provide sharp non-oscillatory transitions at discontinuities. We suggest a numerical formulation based on a cell-centered approach where all of the primary flow variables are discretized at the zone center. The divergence-free condition is enforced by augmenting the MHD equations with a generalized Lagrange multiplier yielding a mixed hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175 (2002) 645-673). The resulting...

Mignone, A; Bodo, G

2010-01-01

28

In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent and discuss the applicability of them to nonlinear PDE's as well as to the case of variable coefficients. Moreover, we systematically use another symbolic technique, namely the cylindrical algebraic decomposition, in order to derive the conditions on the von Neumann stability of a difference s cheme for a linear PDE with constant coefficients. For stable schemes we demonst rate algorithmic and symbolic approach to handle both continuous and discrete di spersion. We present an implementation of tools for generation of schemes, which rely on Gr\\"obner basis, in the system SINGULAR and present numerous e xamples, computed with our implementation. In the stability analysis, we use the system MATHEMATICA for cylindrical algebraic decomposition.

Levandovskyy, Viktor

2010-01-01

29

In this paper, a class of finite difference schemes which achieves low dispersion and controllable dissipation in smooth region and robust shock-capturing capabilities in the vicinity of discontinuities is presented. Firstly, a sufficient condition for semi-discrete finite difference schemes to have independent dispersion and dissipation is derived. This condition enables a novel approach to separately optimize the dissipation and dispersion properties of finite difference schemes and a class of schemes with minimized dispersion and controllable dissipation is thus obtained. Secondly, for the purpose of shock-capturing, one of these schemes is used as the linear part of the WENO scheme with symmetrical stencils to constructed an improved WENO scheme. At last, the improved WENO scheme is blended with its linear counterpart to form a new hybrid scheme for practical applications. The proposed scheme is accurate, flexible and robust. The accuracy and resolution of the proposed scheme are tested by the solutions of several benchmark test cases. The performance of this scheme is further demonstrated by its application in the direct numerical simulation of compressible turbulent channel flow between isothermal walls.

Sun, Zhen-Sheng; Ren, Yu-Xin; Larricq, Cédric; Zhang, Shi-ying; Yang, Yue-cheng

2011-06-01

30

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)

31

Stable higher order finite-difference schemes for stellar pulsation calculations

Context: Calculating stellar pulsations requires a sufficient accuracy to match the quality of the observations. Many current pulsation codes apply a second order finite-difference scheme, combined with Richardson extrapolation to reach fourth order accuracy on eigenfunctions. Although this is a simple and robust approach, a number of drawbacks exist thus making fourth order schemes desirable. A robust and simple finite-difference scheme, which can easily be implemented in e...

Reese, D. R.

2013-01-01

32

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.

Abramopoulos, Frank

1988-01-01

33

In this work we develop a class of high-order finite difference weighted essentially non-oscillatory (FD-WENO) schemes for solving the ideal magnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this work is to use efficient high-order WENO spatial discretizations with high-order strong stability-preserving Runge-Kutta (SSP-RK) time-stepping schemes. Numerical results have shown that with such methods we are able to resolve solution structures that are only visible at much higher grid resolutions with lower-order schemes. The key challenge in applying such methods to ideal MHD is to control divergence errors in the magnetic field. We achieve this by augmenting the base scheme with a novel high-order constrained transport approach that updates the magnetic vector potential. The predicted magnetic field from the base scheme is replaced by a divergence-free magnetic field that is obtained from the curl of this magnetic potential. The non-conservative weakly hyperbolic system that the magnetic vecto...

Christlieb, Andrew J; Tang, Qi

2013-01-01

34

A High Order Finite Difference Scheme with Sharp Shock Resolution for the Euler Equations

We derive a high-order finite difference scheme for the Euler equations that satisfies a semi-discrete energy estimate, and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution. The formulation of the semi-discrete energy estimate is based on a symmetrization of the Euler equations that preserves the homogeneity of the flux vector, 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. Around discontinuities or sharp gradients, refined grids are created on which the discrete equations are solved after adding a newly constructed artificial viscosity. The positioning of the sub-grids and computation of the viscosity are aided by a detection algorithm which is based on a multi-scale 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.

Gerritsen, Margot; Olsson, Pelle

1996-01-01

35

High-Order Entropy Stable Finite Difference Schemes for Nonlinear Conservation Laws: Finite Domains

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.

Fisher, Travis C.; Carpenter, Mark H.

2013-01-01

36

Particle methods revisited: a class of high-order finite-difference schemes

We propose a new analysis of particle method with remeshing. We derive a class of high-order finite difference methods. Our analysis is completed by numerical comparisons with Lax-Wendroff schemes for the Burger equation.

Cottet, Georges-henri; Weynans, Lisl

2006-01-01

37

Finite-difference scheme for the numerical solution of the Schroedinger equation

A finite-difference scheme for numerical integration of the Schroedinger equation is constructed. Asymptotically (r goes to infinity), the method gives the exact solution correct to terms of order r exp -2.

Mickens, Ronald E.; Ramadhani, Issa

1992-01-01

38

In this paper, we propose the parametrized maximum principle preserving (MPP) flux limiter, originally developed in [Z. Xu, Math. Comp., (2013), in press], to the semi- Lagrangian finite difference weighted essentially non-oscillatory scheme for solving the Vlasov equation. The MPP flux limiter is proved to maintain up to fourth order accuracy for the semi-Lagrangian finite difference scheme without any time step restriction. Numerical studies on the Vlasov-Poisson system de...

Xiong, Tao; Qiu, Jing-mei; Xu, Zhengfu; Christlieb, Andrew

2013-01-01

39

A fourth-order finite difference scheme for the numerical solution of 1D linear hyperbolic equation

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 imp...

Akbar Mohebbi

2013-01-01

40

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

41

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)

42

High-order flux correction/finite difference schemes for strand grids

A novel high-order method combining unstructured flux correction along body surfaces and high-order finite differences normal to surfaces is formulated for unsteady viscous flows on strand grids. The flux correction algorithm is applied in each unstructured layer of the strand grid, and the layers are then coupled together via a source term containing derivatives in the strand direction. Strand-direction derivatives are approximated to high-order via summation-by-parts operators for first derivatives and second derivatives with variable coefficients. We show how this procedure allows for the proper truncation error canceling properties required for the flux correction scheme. The resulting scheme possesses third-order design accuracy, but often exhibits fourth-order accuracy when higher-order derivatives are employed in the strand direction, especially for highly viscous flows. We prove discrete conservation for the new scheme and time stability in the absence of the flux correction terms. Results in two dimensions are presented that demonstrate improvements in accuracy with minimal computational and algorithmic overhead over traditional second-order algorithms.

Katz, Aaron; Work, Dalon

2015-02-01

43

2-STAGE, 2-LEVEL FINITE-DIFFERENCE SCHEMES FOR NON-LINEAR PARABOLIC EQUATIONS

In this paper we are concerned with four, two-stage, two-level finite difference schemes which are easy to implement and apply to non-linear parabolic equations. We show that these schemes are better (with respect to accuracy and stability) than a method of lines package and some recently published second-order, three-level schemes. A convergence proof for one of the schemes is given, using energy estimates. © 1982 Academic Press Inc. (London) Limited.

Meek, P.; Norbury, J.

1982-01-01

44

Nonstandard finite difference schemes for Michaelis-Menten type reaction-diffusion equations

We compare and investigate the performance of the exact scheme of the Michaelis-Menten (M-M) ordinary differential equation with several new non-standard finite difference (NSFD) schemes that we construct by using Mickens’ rules. Furthermore, the exact scheme of the M-M equation is used to design several dynamically consistent NSFD schemes for related reactiondiffusion equations, advection-reaction equations and advection-reaction-diffusion equations. Numerical simulations th...

Chapwanya, Michael; Lubuma, Jean M. -s; Mickens, Ronald E.

2013-01-01

45

In this work we develop a class of high-order finite difference weighted essentially non-oscillatory (FD-WENO) schemes for solving the ideal magnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this work is to use efficient high-order WENO spatial discretizations with high-order strong stability-preserving Runge-Kutta (SSP-RK) time-stepping schemes. Numerical results have shown that with such methods we are able to resolve solution structures that are only visible at much higher grid resolutions with lower-order schemes. The key challenge in applying such methods to ideal MHD is to control divergence errors in the magnetic field. We achieve this by augmenting the base scheme with a novel high-order constrained transport approach that updates the magnetic vector potential. The predicted magnetic field from the base scheme is replaced by a divergence-free magnetic field that is obtained from the curl of this magnetic potential. The non-conservative weakly hyperbolic system that the magnetic vector potential satisfies is solved using a version of FD-WENO developed for Hamilton-Jacobi equations. The resulting numerical method is endowed with several important properties: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as point values on the same mesh (i.e., there is no mesh staggering); (2) both the spatial and temporal orders of accuracy are fourth-order; (3) no spatial integration or multidimensional reconstructions are needed in any step; and (4) special limiters in the magnetic vector potential update are used to control unphysical oscillations in the magnetic field. Several 2D and 3D numerical examples are presented to verify the order of accuracy on smooth test problems and to show high-resolution on test problems that involve shocks.

Christlieb, Andrew J.; Rossmanith, James A.; Tang, Qi

2014-07-01

46

Stable higher order finite-difference schemes for stellar pulsation calculations

Context: Calculating stellar pulsations requires a sufficient accuracy to match the quality of the observations. Many current pulsation codes apply a second order finite-difference scheme, combined with Richardson extrapolation to reach fourth order accuracy on eigenfunctions. Although this is a simple and robust approach, a number of drawbacks exist thus making fourth order schemes desirable. A robust and simple finite-difference scheme, which can easily be implemented in either 1D or 2D stellar pulsation codes is therefore required. Aims: One of the difficulties in setting up higher order finite-difference schemes for stellar pulsations is the so-called mesh-drift instability. Current ways of dealing with this defect include introducing artificial viscosity or applying a staggered grids approach. However these remedies are not well-suited to eigenvalue problems, especially those involving non-dissipative systems, because they unduly change the spectrum of the operator, introduce supplementary free parameter...

Reese, D R

2013-01-01

47

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 ...

48

International Nuclear Information System (INIS)

We successfully apply fourth-order accurate finite difference methods with nonuniform scheme to analysis the symmetric slot waveguides. The results of numerical simulations show that the present nonuniform formula offers the results more accurate than the previously presented second order schemes

49

The metrics and Jacobian in the fluid motion governing equations under curvilinear coordinate system have a variety of equivalent differential forms, which may have different discretization errors with the same difference scheme. The discretization errors of metrics and Jacobian may cause serious computational instability and inaccuracy in numerical results, especially for high-order finite difference schemes. It has been demonstrated by many researchers that the Geometric Conservation Law (GCL) is very important for high-order Finite Difference Methods (FDMs), and a proper form of metrics and Jacobian, which can satisfy the GCL, can considerably reduce discretization errors and computational instability. In order to satisfy the GCL for FDM, we have previously developed a Conservative Metric Method (CMM) to calculate the metrics [1] and the difference scheme ?3 in the CMM is determined with the suggestion ?3=?2. In this paper, a Symmetrical Conservative Metric Method (SCMM) is newly proposed based on the discussions of the metrics and Jacobian in FDM from geometry viewpoint by following the concept of vectorized surface and cell volume in Finite Volume Methods (FVMs). Interestingly, the expressions of metrics and Jacobian obtained by using the SCMM with second-order central finite difference scheme are equivalent to the vectorized surfaces and cell volumes, respectively. The main advantage of SCMM is that it makes the calculations based on high-order WCNS schemes aroud complex geometry flows possible and somewhat easy. Numerical tests on linear and nonlinear problems indicate that the quality of numerical results may be largely enhanced by utilizing the SCMM, and the advantage of the SCMM over other forms of metrics and Jacobian may be more evident on highly nonuniform grids.

Deng, Xiaogang; Min, Yaobing; Mao, Meiliang; Liu, Huayong; Tu, Guohua; Zhang, Hanxin

2013-04-01

50

The dispersion and dissipation properties of a scheme are of great importance for the simulation of flow fields which involve a broad range of length scales. In order to improve the spectral properties of the finite difference scheme, the authors have previously proposed the idea of optimizing the dispersion and dissipation properties separately and a fourth order scheme based on the minimized dispersion and controllable dissipation (MDCD) technique is thus constructed [29]. In the present paper, we further investigate this technique and extend it to a sixth order finite difference scheme to solve the Euler and Navier-Stokes equations. The dispersion properties of the scheme is firstly optimized by minimizing an elaborately designed integrated error function. Then the dispersion-dissipation condition which is newly derived by Hu and Adams [30] is introduced to supply sufficient dissipation to damp the unresolved wavenumbers. Furthermore, the optimized scheme is blended with an optimized Weighted Essentially Non-Oscillation (WENO) scheme to make it possible for the discontinuity-capturing. In this process, the approximation-dispersion-relation (ADR) approach is employed to optimize the spectral properties of the nonlinear scheme to yield the true wave propagation behavior of the finite difference scheme. Several benchmark test problems, which include broadband fluctuations and strong shock waves, are solved to validate the high-resolution, the good discontinuity-capturing capability and the high-efficiency of the proposed scheme.

Sun, Zhen-sheng; Luo, Lei; Ren, Yu-xin; Zhang, Shi-ying

2014-08-01

51

High-order conservative finite difference GLM–MHD schemes forcell-centered MHD

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate i...

Tzeferacos, Petros; Mignone, Andrea

2010-01-01

52

A stable and conservative method for locally adapting the design order of finite difference schemes

A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing Summation-By-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable. Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the developed switching procedure is combined with the MUSCL technique.

Eriksson, Sofia; Abbas, Qaisar; Nordstro?m, Jan

2011-01-01

53

Error analysis of finite difference schemes applied to hyperbolic initial boundary value problems

Finite difference methods for the numerical solution of mixed initial boundary value problems for hyperbolic equations are studied. The reported investigation has the objective to develop a technique for the total error analysis of a finite difference scheme, taking into account initial approximations, boundary conditions, and interior approximation. Attention is given to the Cauchy problem and the initial approximation, the homogeneous problem in an infinite strip with inhomogeneous boundary data, the reflection of errors in the boundaries, and two different boundary approximations for the leapfrog scheme with a fourth order accurate difference operator in space.

Skollermo, G.

1979-01-01

54

In this paper, we propose the parametrized maximum principle preserving (MPP) flux limiter, originally developed in [37], to the semi-Lagrangian finite difference weighted essentially non-oscillatory scheme for solving the Vlasov equation. The MPP flux limiter is proved to maintain up to fourth order accuracy for the semi-Lagrangian finite difference scheme without any time step restriction. Numerical studies on the Vlasov-Poisson system demonstrate the performance of the proposed method and its ability in preserving the positivity of the probability distribution function while maintaining the high order accuracy.

Xiong, Tao; Qiu, Jing-Mei; Xu, Zhengfu; Christlieb, Andrew

2014-09-01

55

A 2-STAGE, 2-LEVEL FINITE-DIFFERENCE SCHEME FOR MOVING BOUNDARY-PROBLEMS

A two-stage, two-level finite difference scheme is devised which, after applying a coordinate transformation, requires only a single iteration of a modified Newton method to produce second-order approximations to the solution of nonlinear parabolic moving boundary problems. Numerical evidence of unconditional stability and second-order convergence (in space and time) to both the solution and the moving boundary is presented for two particular problems. The proposed scheme is expected to be co...

Meek, P.; Norbury, J.

1984-01-01

56

International Nuclear Information System (INIS)

The goal of this work is to determine classes of traveling solitary wave solutions for a differential approximation of a finite-difference scheme by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of non-linear wave equation. The occurrence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to has a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solution of the original continuous equations.

57

Invariant meshless discretization schemes

International Nuclear Information System (INIS)

A method is introduced for the construction of meshless discretization schemes which preserve Lie symmetries of the differential equations that these schemes approximate. The method exploits the fact that equivariant moving frames provide a way of associating invariant functions with non-invariant functions. An invariant meshless approximation of a nonlinear diffusion equation is constructed. Comparative numerical tests with a non-invariant meshless scheme are presented. These tests yield that invariant meshless schemes can lead to substantially improved numerical solutions compared to numerical solutions generated by non-invariant meshless schemes. (fast track communication)

58

Convergence of a nonlinear finite difference scheme for the Kuramoto-Tsuzuki equation

A nonlinear finite difference scheme is studied for solving the Kuramoto-Tsuzuki equation. Because the maximum estimate of the numerical solution can not be obtained directly, it is difficult to prove the stability and convergence of the scheme. In this paper, we introduce the Brouwer-type fixed point theorem and induction argument to prove the unique existence and convergence of the nonlinear scheme. An iterative algorithm is proposed for solving the nonlinear scheme, and its convergence is proved. Based on the iterative algorithm, some linearized schemes are presented. Numerical examples are carried out to verify the correction of the theory analysis. The extrapolation technique is applied to improve the accuracy of the schemes, and some interesting results are obtained.

Wang, Shanshan; Wang, Tingchun; Zhang, Luming; Guo, Boling

2011-06-01

59

Stable higher order finite-difference schemes for stellar pulsation calculations

Context. Calculating stellar pulsations requires high enough accuracy to match the quality of the observations. Many current pulsation codes apply a second-order finite-difference scheme, combined with Richardson extrapolation to reach fourth-order accuracy on eigenfunctions. Although this is a simple and robust approach, a number of drawbacks exist that make fourth-order schemes desirable. A robust and simple finite-difference scheme that can easily be implemented in either 1D or 2D stellar pulsation codes, is therefore required. Aims: One of the difficulties in setting up higher order finite-difference schemes for stellar pulsations is the so-called mesh-drift instability. Current ways of dealing with this defect include introducing artificial viscosity or applying a staggered grid approach; however, these remedies are not well-suited to eigenvalue problems, especially those involving non-dissipative systems, because they unduly change the spectrum of the operator, introduce supplementary free parameters, or lead to complications when applying boundary conditions. Methods: We propose here a new method, inspired from the staggered grid strategy, which removes this instability while bypassing the above difficulties. Furthermore, this approach lends itself to superconvergence, a process in which the accuracy of the finite differences is boosted by one order. Results: This new approach is successfully applied to stellar pulsation calculations, and is shown to be accurate, flexible with respect to the underlying grid, and able to remove mesh drift. Conclusions: Although specifically designed for stellar pulsation calculations, this method can easily be applied to many other physical or mathematical problems.

Reese, D. R.

2013-07-01

60

Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme

In this paper, an implicit exponential finite-difference scheme (Expo FDM) has been proposed for solving two dimensional nonlinear coupled viscous Burgers’ equations (VBEs) with appropriate initial and boundary conditions. The accuracy of the method has been illustrated by taking two numerical examples. Results are compared with exact solution and those already available in the literature by finding the L1, L2, L? and ER errors. Excellent numerical results indicate that the proposed schem...

Srivastava, Vineet K.; Sarita Singh; Awasthi, Mukesh K.

2013-01-01

61

An explicit finite-difference solution of hypersonic flows using rational Runge-Kutta scheme

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).

Satofuka, Nobuyuki; Morinishi, Koji

62

Application of the symplectic finite-difference time-domain scheme to electromagnetic simulation

International Nuclear Information System (INIS)

An explicit fourth-order finite-difference time-domain (FDTD) scheme using the symplectic integrator is applied to electromagnetic simulation. A feasible numerical implementation of the symplectic FDTD (SFDTD) scheme is specified. In particular, new strategies for the air-dielectric interface treatment and the near-to-far-field (NFF) transformation are presented. By using the SFDTD scheme, both the radiation and the scattering of three-dimensional objects are computed. Furthermore, the energy-conserving characteristic hold for the SFDTD scheme is verified under long-term simulation. Numerical results suggest that the SFDTD scheme is more efficient than the traditional FDTD method and other high-order methods, and can save computational resources

63

This paper is devoted to the construction and analysis of finite difference methods for solving a class of time-fractional subdiffusion equations. Based on the certain superconvergence at some particular points of the fractional derivative by the traditional first-order Grünwald-Letnikov formula, some effective finite difference schemes are derived. The obtained schemes can achieve the global second-order numerical accuracy in time, which is independent of the values of anomalous diffusion exponent ? (0 order scheme and the spatial fourth-order compact scheme, respectively, are established for the one-dimensional problem along with the strict analysis on the unconditional stability and convergence of these schemes by the discrete energy method. Furthermore, the extension to the two-dimensional case is also considered. Numerical experiments support the correctness of the theoretical analysis and effectiveness of the new developed difference schemes.

Gao, Guang-Hua; Sun, Hai-Wei; Sun, Zhi-Zhong

2015-01-01

64

2D numerical simulation of the MEP energy-transport model with a finite difference scheme

International Nuclear Information System (INIS)

A finite difference scheme of Scharfetter-Gummel type is used to simulate a consistent energy-transport model for electron transport in semiconductors devices, free of any fitting parameters, formulated on the basis of the maximum entropy principle. Simulations of silicon n+-n-n+ diodes, 2D-MESFET and 2D-MOSFET and comparisons with the results obtained by a direct simulation of the Boltzmann transport equation and with other energy-transport models, known in the literature, show the validity of the model and the robustness of the numerical scheme

65

Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme

Directory of Open Access Journals (Sweden)

Full Text Available In this paper, an implicit exponential finite-difference scheme (Expo FDM has been proposed for solving two dimensional nonlinear coupled viscous Burgers’ equations (VBEs with appropriate initial and boundary conditions. The accuracy of the method has been illustrated by taking two numerical examples. Results are compared with exact solution and those already available in the literature by finding the L1, L2, L? and ER errors. Excellent numerical results indicate that the proposed scheme is efficient, reliable and robust technique for the numerical solutions of Burgers’ equation.

Vineet K. Srivastava

2013-08-01

66

Directory of Open Access Journals (Sweden)

Full Text Available ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????Fractional order differential equations are generalizations of classical differential equations. They are widely used in the fields of diffusive transport, finance, nonlinear dynamics, signal processing and others. In this paper, an implicit finite difference method for a class of initial-boundary value space-time fractional two-sided space partial differential equations with variable coefficients on a finite domain is established. The stability and convergence order are analyzed for the resulted implicit scheme. With mathematical induction skills, the scheme is proved to be unconditionally stable and convergent.

??

2013-05-01

67

We present a new approach for the construction of stochastic subgrid scale parameterizations. Starting from a high-resolution finite-difference discretization of some model equations, the new approach is based on splitting the model variables into fast, small-scale and slow, large-scale modes by averaging the model discretization over neighboring grid cells. After that, the fast modes are eliminated by applying a stochastic mode reduction procedure. This procedure is a generalization of the mode reduction strategy proposed by Majda, Timofeyev & Vanden-Eijnden, in that it allows for oscillations in the closure assumption. The new parameterization is applied to the forced Burgers equation and is compared with a Smagorinsky-type subgrid scale closure.

Dolaptchiev, S. I.; Achatz, U.; Timofeyev, I.

2013-06-01

68

In finite-difference (FD) acoustic forward modelling, the parameter settings for the perfectly matched layer (PML) are case-dependent. There is no explicit PML formula that can be applied for most acoustic models without tuning, especially, for the fourth-order FD scheme. In this paper, we propose an explicit PML formula for the acoustic frequency-domain FD with second-order and fourth-order accuracies, respectively. The fourth-order FD scheme uses a special treatment for the boundary. The number of points in the PML is fixed to be 10 and 15 for the second-order and fourth-order FD schemes, respectively. The maximum artificial attenuation parameter associated with the PML formula is automatically calculated based on the FD grid size and the value of the compressional velocity of the boundary cells of the interior domain. Numerical tests confirm that this empirical formula achieves the desired accuracy for 2-D and 3-D media for grid sizes varying from 1 to 200 m. For the fourth-order FD scheme, the proposed PML formula works effectively up to 25 points per wavelength for both 2-D and 3-D media. Beyond that, the error of the PML discretization becomes larger than the discretization error in the interior domain. For such cases and to keep a fourth-order accuracy, a larger number of points in the PML (thicker PML region) needs to be employed.

Pan, Guangdong; Abubakar, Aria; Habashy, Tarek M.

2012-01-01

69

A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics

International Nuclear Information System (INIS)

The authors present a high-order accurate weighted essentially non-oscillatory (WENO) finite difference scheme for solving the equations of ideal magnetohydrodynamics (MHD). This scheme is a direct extension of a WENO scheme, which has been successfully applied to hydrodynamic problems. The WENO scheme follows the same idea of an essentially non-oscillatory (ENO) scheme with an advantage of achieving higher-order accuracy with fewer computations. Both ENO and WENO can be easily applied to two and three spatial dimensions by evaluating the fluxes dimension-by-dimension. Details of the WENO scheme as well as the construction of a suitable eigen-system, which can properly decompose various families of MHD waves and handle the degenerate situations, are presented. Numerical results are shown to perform well for the one-dimensional Brio-Wu Riemann problems, the two-dimensional Kelvin-Helmholtz instability problems, and the two-dimensional Orszag-Tang MHD vortex system. They also demonstrate the importance of maintaining the divergence free condition for the magnetic field in achieving numerical stability. The tests also show the advantages of using the higher-order scheme. The new 5th-order WENO MHD code can attain an accuracy comparable with that of the second-order schemes with many fewer grid points

70

A fourth order accurate finite difference scheme for the computation of elastic waves

A finite difference for elastic waves is introduced. The model is based on the first order system of equations for the velocities and stresses. The differencing is fourth order accurate on the spatial derivatives and second order accurate in time. The model is tested on a series of examples including the Lamb problem, scattering from plane interf aces and scattering from a fluid-elastic interface. The scheme is shown to be effective for these problems. The accuracy and stability is insensitive to the Poisson ratio. For the class of problems considered here it is found that the fourth order scheme requires for two-thirds to one-half the resolution of a typical second order scheme to give comparable accuracy.

Bayliss, A.; Jordan, K. E.; Lemesurier, B. J.; Turkel, E.

1986-01-01

71

A general strategy exists for constructing Energy Stable Weighted Essentially Non Oscillatory (ESWENO) finite difference schemes up to eighth-order on periodic domains. These ESWENO schemes satisfy an energy norm stability proof for both continuous and discontinuous solutions of systems of linear hyperbolic equations. Herein, boundary closures are developed for the fourth-order ESWENO scheme that maintain wherever possible the WENO stencil biasing properties, while satisfying the summation-by-parts (SBP) operator convention, thereby ensuring stability in an L2 norm. Second-order, and third-order boundary closures are developed that achieve stability in diagonal and block norms, respectively. The global accuracy for the second-order closures is three, and for the third-order closures is four. A novel set of non-uniform flux interpolation points is necessary near the boundaries to simultaneously achieve 1) accuracy, 2) the SBP convention, and 3) WENO stencil biasing mechanics.

Fisher, Travis C.; Carpenter, Mark H.; Yamaleev, Nail K.; Frankel, Steven H.

2009-01-01

72

a Two-Level Second-Order Finite Difference Scheme for the Single Term Structure Equation

In the paper [6] the classical single factor term structure equation for models that predict non-negative interest rates is numerically studied. For these models the authors proposed a second order accurate three-level finite difference scheme (FDs) using the appropriate boundary conditions at zero. For the same problem we propose a two-level second-order accurate FDs. We also propose an effective algorithm for solving the difference schemes, for which also follows the positivity of the numerical solution. The flexibility of our FDs makes it easy to change the drift and diffusion terms in the model. The numerical experiments confirm the second-order of accuracy of the scheme and the positivity-convexity property.

Chernogorova, T.; Valkov, R.

2011-12-01

73

Heat capacity estimators for random series path-integral methods by finite-difference schemes

Previous heat capacity estimators used in path integral simulations either have large variances that grow to infinity with the number of path variables or require the evaluation of first and second order derivatives of the potential. In the present paper, we show that the evaluation of the total energy by the T-method estimator and of the heat capacity by the TT-method estimator can be implemented by a finite difference scheme in a stable fashion. As such, the variances of t...

Predescu, Cristian; Sabo, Dubravko; Doll, J. D.; Freeman, David L.

2003-01-01

74

The role of a geometric conservation law (GCL) on a finite-difference scheme is revisited for conservation laws, and the conservative forms of coordinate-transformation metrics are introduced in general dimensions. The sufficient condition of a linear high-order finite-difference scheme is arranged in detail, for which the discretized conservative coordinate-transformation metrics and Jacobian satisfy the GCL identities on three-dimensional moving and deforming grids. Subsequently, the geometric interpretation of the metrics and Jacobian discretized by a linear high-order finite-difference scheme is discussed, and only the symmetric conservative forms of the discretized metrics and Jacobian are shown to have the appropriate geometric structures. The symmetric and asymmetric conservative forms of the metrics and Jacobian are examined by the computation of an inviscid compressible fluid on highly-skewed stationary and deforming grids using sixth-order compact and fourth-order explicit central-difference schemes, respectively. The resolution of the isentropic vortex and the robustness of the computation are improved by employing symmetric conservative forms on the coordinate-transformation metrics and Jacobian that have an appropriate geometry background. An integrated conservation of conservative quantities is also attained on the deforming grid when symmetric conservative forms are adopted to the time metrics and Jacobian.

Abe, Yoshiaki; Nonomura, Taku; Iizuka, Nobuyuki; Fujii, Kozo

2014-03-01

75

A fourth-order finite difference scheme for the numerical solution of 1D linear hyperbolic equation

Directory of Open Access Journals (Sweden)

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.

Akbar Mohebbi

2013-10-01

76

In this article, we present a simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations by optimally selecting the time step used to advance the numerical solution and adding defect correction terms in a non-iterative manner. The power of the technique is its ability to extract as much accuracy as possible from existing finite difference schemes with minimal additional effort. Through straightforward nu...

Chu, Kevin T.

2008-01-01

77

Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes

In the reconstruction step of (2r-1) order weighted essentially non-oscillatory conservative finite difference schemes (WENO) for solving hyperbolic conservation laws, nonlinear weights ?k and ?k, such as the WENO-JS weights by Jiang et al. and the WENO-Z weights by Borges et al., are designed to recover the formal (2r-1) order (optimal order) of the upwinded central finite difference scheme when the solution is sufficiently smooth. The smoothness of the solution is determined by the lower order local smoothness indicators ?k in each substencil. These nonlinear weight formulations share two important free parameters in common: the power p, which controls the amount of numerical dissipation, and the sensitivity ?, which is added to ?k to avoid a division by zero in the denominator of ?k. However, ? also plays a role affecting the order of accuracy of WENO schemes, especially in the presence of critical points. It was recently shown that, for any design order (2r-1), ? should be of ?(?x2) (?(?xm) means that ??C?xm for some C independent of ?x, as ?x?0) for the WENO-JS scheme to achieve the optimal order, regardless of critical points. In this paper, we derive an alternative proof of the sufficient condition using special properties of ?k. Moreover, it is unknown if the WENO-Z scheme should obey the same condition on ?. Here, using same special properties of ?k, we prove that in fact the optimal order of the WENO-Z scheme can be guaranteed with a much weaker condition ?=?(?xm), where m(r,p)?2 is the optimal sensitivity order, regardless of critical points. Both theoretical results are confirmed numerically on smooth functions with arbitrary order of critical points. This is a highly desirable feature, as illustrated with the Lax problem and the Mach 3 shock-density wave interaction of one dimensional Euler equations, for a smaller ? allows a better essentially non-oscillatory shock capturing as it does not over-dominate over the size of ?k. We also show that numerical oscillations can be further attenuated by increasing the power parameter 2?p?r-1, at the cost of increased numerical dissipation. Compact formulas of ?k for WENO schemes are also presented.

Don, Wai-Sun; Borges, Rafael

2013-10-01

78

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...

Bommaraju, C.; Marklein, R.; Chinta, P. K.

2005-01-01

79

This paper reports a two-dimensional time-domain inverse scattering algorithm based upon the finite-difference time domain method (FDTD) for determining the shape of a perfectly conducting cylinder. FDTD is used to solve the scattering electromagnetic wave of a perfectly conducting cylinder. The inverse problem is resolved by an optimization approach and the global searching scheme asynchronous particle swarm optimization is then employed to search the parameter space. By properly processing the scattered field, some electromagnetic properties can be reconstructed. A set of representative numerical results is presented to demonstrate that the proposed approach is able to efficiently reconstruct the electromagnetic properties of metallic scatterer even when the initial guess is far away from the exact one. In addition, the effects of Gaussian noises on imaging reconstruction are also investigated.

Chen, Chien-Hung; Chiu, Chien-Ching; Sun, Chi-Hsien; Chang, Wan-Ling

2011-01-01

80

International Nuclear Information System (INIS)

An entirely new discrete levels segment has been created by the Budapest group according to the recommended principles, using the Evaluated Nuclear Structure Data File, ENSDF as a source. The resulting segment contains 96,834 levels and 105,423 gamma rays for 2,585 nuclei, with their characteristics such as energy, spin, parity, half-life as well as gamma-ray energy and branching percentage. Isomer flags for half-lives longer than 1 s have been introduced. For those 1,277 nuclei having at least ten known levels the cutoff level numbers Nm have been determined from fits to the cumulative number of levels. The level numbers Nc associated with the cutoff energies Uc, corresponding to the upper energy limit of levels with unique spin and parity, have been included for each nuclide. The segment has the form of an ASCII file which follows the extended ENEA Bologna convention. For the RIPL Starter File the new Budapest file is recommended as a Discrete Level Schemes Segment because it is most complete, up-to-date, and also well documented. Moreover, the cutoff energies have been determined in a consistent way, giving also hints about basic level density parameters. The recommended files are budapest-levels.dat and budapest-cumulative.dat. As alternative choices, the libraries from Beijing, Bologna, JAERI, Obninsk and Livermore may also be used for special applications. (author)

81

We propose highly accurate finite-difference schemes for simulating wave propagation problems described by linear second-order hyperbolic equations. The schemes are based on the summation by parts (SBP) approach modified for applications with violation of input data smoothness. In particular, we derive and implement stable schemes for solving elastodynamic anisotropic problems described by the Navier wave equation in complex geometry. To enhance potential of the method, we u...

Dovgilovich, Leonid; Sofronov, Ivan

2014-01-01

82

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...

Hammer, René; Arnold, Anton

2013-01-01

83

We consider a divergence-free non-dissipative fourth-order explicit staggered finite difference scheme for the hyperbolic Maxwell's equations. Special one-sided difference operators are derived in order to implement the scheme near metal boundaries and dielectric interfaces. Numerical results show the scheme is long-time stable, and is fourth-order convergent over complex domains that include dielectric interfaces and perfectly conducting surfaces. We also examine the scheme's behavior near metal surfaces that are not aligned with the grid axes, and compare its accuracy to that obtained by the Yee scheme.

Yefet, Amir; Petropoulos, Peter G.

1999-01-01

84

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.

Mickens, Ronald E.

1989-01-01

85

Directory of Open Access Journals (Sweden)

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.

M.M. Arabshahi

2012-08-01

86

The fourth-order accurate, three-point finite-difference Numerov spatial discretization provides accurate and efficient solutions to the time-dependent governing differential equations of electrochemical kinetics in one-dimensional space geometry, when the equations contain first time derivatives of the solution, second spatial derivatives, and homogeneous reaction terms only. However, the original Numerov discretization is not applicable when the governing equations involve first spatial derivative terms. To overcome this limitation, an appropriately extended Numerov discretization is required. We examine the utility of one of such extensions, first described by Chawla. Relevant discrete formulae are outlined for systems of linear governing equations involving first derivative terms, and applied to five representative example models of electrochemical transient experiments. The extended Numerov discretization proves to have an accuracy and efficiency comparable to the original Numerov scheme, and its accuracy is typically up to four orders of magnitude higher, compared to the conventional, second-order accurate spatial discretization, commonly used in electrochemistry. This results in a considerable improvement of efficiency. Therefore, the application of the extended Numerov discretization to the electrochemical kinetic simulations can be fully recommended. PMID:15067683

Bieniasz, Les?aw K

2004-06-01

87

We develop a second-order high-resolution finite difference scheme to approximate the solution of a mathematical model describing the within-host dynamics of malaria infection. The model consists of two nonlinear partial differential equations coupled with three nonlinear ordinary differential equations. Convergence of the numerical method to the unique weak solution with bounded total variation is proved. Numerical simulations demonstrating the achievement of the designed accuracy are presented. PMID:23541675

Ackleh, Azmy S; Ma, Baoling; Thibodeaux, Jeremy J

2013-09-01

88

In this paper, we develop parametrized positivity satisfying flux limiters for the high order finite difference Runge-Kutta weighted essentially non-oscillatory (WENO) scheme solving compressible Euler equations to maintain positive density and pressure. Negative density and pressure, which often leads to simulation blow-ups or nonphysical solutions, emerges from many high resolution computations in some extreme cases. The methodology we propose in this paper is a nontrivial...

Xiong, Tao; Qiu, Jing-mei; Xu, Zhengfu

2014-01-01

89

Comment on the finite difference schemes for the time-dependent schroedinger equation

International Nuclear Information System (INIS)

The method of finite differences is applied to the solution of the two-dimensional time-dependent Schroedinger equation. The equidistant mesh is used and the accuracy of the formulae approximating the kinetic energy term by polynomials of the 2nd, 4th and 6th order in the mesh size is discussed. The numerical results are compared to the exact solutions corresponding to the square well potential. (orig.)

90

Computational Aero-Acoustic Using High-order Finite-Difference Schemes

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 differ...

Zhu, Wei Jun; Shen, Wen Zhong; Sørensen, Jens Nørkær

2010-01-01

91

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.

Harte, Philip T.

1994-01-01

92

A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coecient 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 ...

Nikkar, Samira; Nordstro?m, Jan

2014-01-01

93

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.

Xiong, Tao; Qiu, Jing-Mei; Xu, Zhengfu

2013-11-01

94

A few families of counterexamples are provided to "A proof that the discrete singular convolution (DSC)/Lagrange-distributed approximation function (LDAF) method is inferior to high order finite differences", Journal of Computational Physics, 214, 538-549 (2006).

Wei, G W; Zhao, Shan

2006-01-01

95

Fourth-order finite difference scheme for the incompressible Navier-Stokes equations in a disk

We develop an efficient fourth-order finite difference method for solving the incompressible Navier-Stokes equations in the vorticity-stream function formulation on a disk. We use the fourth-order Runge-Kutta method for the time integration and treat both the convection and diffusion terms explicitly. Using a uniform grid with shifting a half mesh away from the origin, we avoid placing the grid point directly at the origin; thus, no pole approximation is needed. Besides, on such grid, a fourth-order fast direct method is used to solve the Poisson equation of the stream function. By Fourier filtering the vorticity in the azimuthal direction at each time stage, we are able to increase the time step to a reasonable size. The numerical results of the accuracy test and the simulation of a vortex dipole colliding with circular wall are presented.

Lai, Ming-Chih

2003-07-01

96

Perfect plane-wave source for a high-order symplectic finite-difference time-domain scheme

International Nuclear Information System (INIS)

The method of splitting a plane-wave finite-difference time-domain (SP-FDTD) algorithm is presented for the initiation of plane-wave source in the total-field / scattered-field (TF/SF) formulation of high-order symplectic finite-difference time-domain (SFDTD) scheme for the first time. By splitting the fields on one-dimensional grid and using the nature of numerical plane-wave in finite-difference time-domain (FDTD), the identical dispersion relation can be obtained and proved between the one-dimensional and three-dimensional grids. An efficient plane-wave source is simulated on one-dimensional grid and a perfect match can be achieved for a plane-wave propagating at any angle forming an integer grid cell ratio. Numerical simulations show that the method is valid for SFDTD and the residual field in SF region is shrinked down to ?300 dB. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)

97

International Nuclear Information System (INIS)

The problem of laminar flow and heat transfer in the presence of a uniform transverse magnetic and electric field between two parallel porous plates experiencing discontinuous change in wall temperature has been considered. An explicit finite difference scheme has been employed to find the solution of the coupled non linear equations. The flow phenomenon has been characterised by Hartmann number (M), suction Reynolds number (R), Channel Reynolds number (R*) and Prandtl number (P). The effect of these parameters on the velocity distribution and the temperature distribution have been discussed and some interesting results are presented. (author)

98

A spectral finite difference method is applied to analysis on magnetic levitation as a major unsteady-state problem in magnetohydrodynamics. Vorticity-stream function formulation is introduced in conjunction with Maxwell's equations, and the non-linear term of Ohm's law for a liquid metal is included. For the purpose of analysis treated is a liquid metal occupying a volume such that no shear stresses and no normal velocity components on the free surface are used as dynamic boundary conditions. Externally applied electromagnetic fields consist of no electromagnetic field at infinity and fields produced by circular coils placed horizontally near the liquid metal. Presented are lift force, magnetic fields and flow fields for several parameters. Numerical data for high viscosity on dimensionless force with the dimensionless vertical coil position are qualitatively in good agreement with experimental data for a solid metal [J. Appl. Phys. 23 (1952) 545]. The effects of the Reynolds number, the Strouhal number and the number of the external coil(s) on levitation force, the magnetic field and the flow field are clarified.

Im, Kichang; Mochimaru, Yoshihiro

2005-02-01

99

An implicit finite difference method of fourth order accuracy in space and time is introduced for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The basic form of the method is a two-level scheme which is unconditionally stable and nondissipative. The scheme uses only three mesh points at level t and three mesh points at level t + delta t. The dissipative version of the basic method given is conditionally stable under the CFL (Courant-Friedrichs-Lewy) condition. This version is particularly useful for the numerical solution of problems with strong but nonstiff dynamic features, where the CFL restriction is reasonable on accuracy grounds. Numerical results are provided to illustrate properties of the proposed method.

Harten, A.; Tal-Ezer, H.

1981-01-01

100

In this paper, we propose a new WENO finite difference procedure for nonlinear degenerate parabolic equations which may contain discontinuous solutions. Our scheme is based on the method of lines, with a high-order accurate conservative approximation to each of the diffusion terms based on an idea that has been recently presented by Liu et al. [Y. Liu, C.-W. Shu, M. Zhang, High order finite difference WENO schemes for non-linear degenerate parabolic equations, SIAM J. Sci. Comput. 33 (2011) 939-965]. Our scheme tries to circumvent the negative ideal weights that appear when applying the standard WENO idea, as is done in Liu et al. (2011) [13]. In one-dimensional case, first we obtain an optimum polynomial on a six-points stencil. This optimum polynomial is sixth-order accurate in regions of smoothness. Then, we consider this optimum polynomial as a symmetric and convex combination of four polynomials with ideal weights. Following the methodology of the classic WENO procedure, then we calculate the non-oscillatory weights with the ideal weights. Numerical examples are provided to demonstrate the resolution power and accuracy of the scheme. Finally, the new method is extended to multi-dimensional problems by dimension-by-dimension approach. More examples of multi-dimension problems are presented to show that our method remains non-oscillatory while giving good resolution of discontinuities. Finally, we would like to mention that this paper combines and extends the techniques proposed in [13] and Levy et al. (2000) [24].

Abedian, Rooholah; Adibi, Hojatollah; Dehghan, Mehdi

2013-08-01

101

Scientific Electronic Library Online (English)

Full Text Available SciELO Brazil | Language: English Abstract in english We present a computational study of the solution of the Falkner-Skan equation (a thirdorder boundary value problem arising in boundary-layer theory) using high-order and high-order-compact finite differences schemes. There are a number of previously reported solution approaches that adopt a reduced- [...] order system of equations, and numerical methods such as: shooting, Taylor series, Runge-Kutta and other semi-analytic methods. Interestingly, though, methods that solve the original non-reduced third-order equation directly are absent from the literature. Two high-order schemes are presented using both explicit (third-order) and implicit compact-difference (fourth-order) formulations on a semi-infinite domain; to our knowledge this is the first time that high-order finite difference schemes are presented to find numerical solutions to the non-reduced-order Falkner-Skan equation directly. This approach maintains the simplicity of Taylor-series coefficient matching methods, avoiding complicated numerical algorithms, and in turn presents valuable information about the numerical behaviour of the equation. The accuracy and effectiveness of this approach is established by comparison with published data for accelerating, constant and decelerating flows; excellent agreement is observed. In general, the numerical behaviour of formulations that seek an optimum physical domain size (for a given computational grid) is discussed. Based on new insight into such methods, an alternative optimisation procedure is proposed that should increase the range of initial seed points for which convergence can be achieved.

Carlos, Duque-Daza; Duncan, Lockerby; Carlos, Galeano.

2011-12-01

102

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.

Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul

1994-01-01

103

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.

Harten, A.; Tal-Ezer, H.

1981-01-01

104

Total-energy minimization of few-body electron systems in the real-space finite-difference scheme

International Nuclear Information System (INIS)

A practical and high-accuracy computation method to search for ground states of few-electron systems is presented on the basis of the real-space finite-difference scheme. A linear combination of Slater determinants is employed as a many-electron wavefunction, and the total-energy functional is described in terms of overlap integrals of one-electron orbitals without the constraints of orthogonality and normalization. In order to execute a direct energy minimization process of the energy functional, the steepest-descent method is used. For accurate descriptions of integrals which include bare-Coulomb potentials of ions, the time-saving double-grid technique is introduced. As an example of the present method, calculations for the ground state of the hydrogen molecule are demonstrated. An adiabatic potential curve is illustrated, and the accessibility and accuracy of the present method are discussed.

105

Locally exact modifications of discrete gradient schemes

International Nuclear Information System (INIS)

Locally exact integrators preserve linearization of the original system at every point. We construct energy-preserving locally exact discrete gradient schemes for arbitrary multidimensional canonical Hamiltonian systems by modifying classical discrete gradient schemes. Modifications of this kind are found for any discrete gradient.

106

Locally exact modifications of discrete gradient schemes

Locally exact integrators preserve linearization of the original system at every point. We construct energy-preserving locally exact discrete gradient schemes for arbitrary multidimensional canonical Hamiltonian systems by modifying classical discrete gradient schemes. Modifications of this kind are found for any discrete gradient.

Cie?li?ski, Jan L

2013-01-01

107

The development of numerically efficient models for atmospheric flow is a topic of ongoing research. Need for such models arises wherever the interest is predominantly in the dynamics of a coarse-grained flow which is, however, interacting with small-scale structures not to be resolved explicitly. Typical applications are, e.g., the parameterization of meso-scale or synoptic-scale eddies in large-scale climate models or the treatment of the impact of small-scale turbulence on the larger turbulent structures in large-eddy simulations. We present a systematic framework for the development of a stochastic closure for local averages of various atmospheric quantities. The parameterization is derived from the finite difference discretization of the full model equations by utilizing stochastic mode reduction techniques [1]. It includes linear and nonlinear corrections, as well as additive and multiplicative noise terms. The new parameterization is implemented for the Burgers equation, where we consider a stochastically forced case as well as the inviscid case. In order to assess the performance of the closure, it is compared with two benchmark parameterizations: a Smagorinsky sub-grid scale model and a purely empirical linear stochastic parameterization. The new parameterization improves the representation of the inertial energy range (forced case) and higher order statistical moments (inviscid case).

Dolaptchiev, S.; Achatz, U.; Timofeyev, I.

2012-04-01

108

A New Scheme for Discrete HJB Equations

Directory of Open Access Journals (Sweden)

Full Text Available In this paper we propose a relaxation scheme for solving discrete HJB equations based on scheme II [1] of Lions and Mercier. The convergence of the new scheme has been established. Numerical example shows that the scheme is efficient.

Zhanyong Zou

2014-10-01

109

Accuracy in finite difference (FD) solutions to spontaneous rupture problems is controlled principally by the scheme used to represent the fault discontinuity, and not by the grid geometry used to represent the continuum. We have numerically tested three fault representation methods, the Thick Fault (TF) proposed by Madariaga et al (1998), the Stress Glut (SG) described by Andrews (1999), and the Staggered-Grid Split-Node (SGSN) methods proposed by Dalguer and Day (2006), each implemented in a the fourth-order velocity-stress staggered-grid (VSSG) FD scheme. The TF and the SG methods approximate the discontinuity through inelastic increments to stress components ("inelastic-zone" schemes) at a set of stress grid points taken to lie on the fault plane. With this type of scheme, the fault surface is indistinguishable from an inelastic zone with a thickness given by a spatial step dx for the SG, and 2dx for the TF model. The SGSN method uses the traction-at-split-node (TSN) approach adapted to the VSSG FD. This method represents the fault discontinuity by explicitly incorporating discontinuity terms at velocity nodes in the grid, with interactions between the "split nodes" occurring exclusively through the tractions (frictional resistance) acting between them. These tractions in turn are controlled by the jump conditions and a friction law. Our 3D tests problem solutions show that the inelastic-zone TF and SG methods show much poorer performance than does the SGSN formulation. The SG inelastic-zone method achieved solutions that are qualitatively meaningful and quantitatively reliable to within a few percent. The TF inelastic-zone method did not achieve qualitatively agreement with the reference solutions to the 3D test problem, and proved to be sufficiently computationally inefficient that it was not feasible to explore convergence quantitatively. The SGSN method gives very accurate solutions, and is also very efficient. Reliable solution of the rupture time is reached with a median resolution of the cohesive zone of only ~2 grid points, and efficiency is competitive with the Boundary Integral (BI) method. The results presented here demonstrate that appropriate fault representation in a numerical scheme is crucial to reduce uncertainties in numerical simulations of earthquake source dynamics and ground motion, and therefore important to improving our understanding of earthquake physics in general.

Dalguer, L. A.; Day, S. M.

2006-12-01

110

A high-order compact finite-difference scheme for large-eddy simulation of active flow control

The purpose of this article is to summarize a computational approach, which developed and matured over an extended period of time, and has been shown to be useful for performing large-eddy simulation (LES) of flows with active control. Because of the nature of active flow control, simulation of this class of problems typically cannot be carried out accurately by methods less sophisticated than LES. Active control flowfields are highly unsteady, and can be characterized by small-scale fluid structures which are produced by the control process, but may also be inherent in the original uncontrolled situation. The numerical scheme is predicated upon an implicit time-marching algorithm, and utilizes a high-order compact finite-difference approximation to represent spatial derivatives. Robustness of the scheme is maintained by employing a low-pass Pade-type nondispersive spatial filter, which also accounts for the fine-scale turbulent dissipation that otherwise is traditionally provided by an explicitly added subgrid-scale (SGS) stress model. Geometrically complex applications are accommodated by an overset grid technique, where spatial accuracy is preserved through use of high-order interpolation. Utility of the method is illustrated by specific computational examples, including suppression of acoustic resonance in supersonic cavity flow, leading-edge vortex control of a delta wing, efficiency enhancement of a transitional highly loaded low-pressure turbine blade, and separation control of a wall-mounted hump model. Control techniques represented in these examples are comprised of both steady and pulsed mass injection or removal, as well as plasma-based actuation. For each case, features of the flowfield are elucidated and the solutions are compared to the baseline situation where no control was enforced. Where available, comparisons are also made with experimental data.

Rizzetta, Donald P.; Visbal, Miguel R.; Morgan, Philip E.

2008-08-01

111

We analyse 13 3-D numerical time-domain explicit schemes for modelling seismic wave propagation and earthquake motion for their behaviour with a varying P-wave to S-wave speed ratio (VP/VS). The second-order schemes include three finite-difference, three finite-element and one discontinuous-Galerkin schemes. The fourth-order schemes include three finite-difference and two spectral-element schemes. All schemes are second-order in time. We assume a uniform cubic grid/mesh and present all schemes in a unified form. We assume plane S-wave propagation in an unbounded homogeneous isotropic elastic medium. We define relative local errors of the schemes in amplitude and the vector difference in one time step and normalize them for a unit time. We also define the equivalent spatial sampling ratio as a ratio at which the maximum relative error is equal to the reference maximum error. We present results of the extensive numerical analysis. We theoretically (i) show how a numerical scheme sees the P and S waves if the VP/VS ratio increases, (ii) show the structure of the errors in amplitude and the vector difference and (iii) compare the schemes in terms of the truncation errors of the discrete approximations to the second mixed and non-mixed spatial derivatives. We find that four of the tested schemes have errors in amplitude almost independent on the VP/VS ratio. The homogeneity of the approximations to the second mixed and non-mixed spatial derivatives in terms of the coefficients of the leading terms of their truncation errors as well as the absolute values of the coefficients are key factors for the behaviour of the schemes with increasing VP/VS ratio. The dependence of the errors in the vector difference on the VP/VS ratio should be accounted for by a proper (sufficiently dense) spatial sampling.

Moczo, Peter; Kristek, Jozef; Galis, Martin; Chaljub, Emmanuel; Etienne, Vincent

2011-12-01

112

International Nuclear Information System (INIS)

The results of numerical simulation of fluid flow and heat transfer in the rod bundle with geometrical disturbance are presented. The geometry of the rod bundle was chosen according to the benchmark problem for 9th IAHR Working Group Meeting (April 7-9, 1998, Grenoble, France). For such a case, experimental data for local velocity and wall shear stress distributions were obtained by group of F. Mantlic at NRI (Czech Republic). Another series of the experiments which provide a data on the wall temperature profiles had been done at the IPPE (Russia). Both experiments provide complete set of data for comparison with the results of numerical simulation. Reynolds equation for axial velocity component has been simulated in two dimensions. Turbulent shear stresses have been simulated by turbulent eddy viscosity with anisotropy defined for radial and azimuth components. Secondary flows have not been taken into consideration. The averaged energy conservation equation closed with anisotropic turbulent conductivity coefficients was simulated. Reynolds and energy conservation equations have been discretized by the Efficient Finite-Difference (EFD) scheme based on the 'locally exact' analytical solution. The comparison of the accuracy of the EFD method and traditional central-difference scheme has been performed. The benchmark problem has been simulated using components of the Computational Object-Oriented Library for Fluid Dynamics (COOLFD) which is a new-generation programming tool aimed to improve the development of the CFD application for complex calculation areas such as rod bundle of nuclear reactor. Comparison of calculated results and experimental data is presented for the local shear stress, axial velocity and the wall temperature distributions in the 'geometrically disturbed' region around dislocated rod. (author)

113

International Nuclear Information System (INIS)

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

114

Numerical simulation of seismic waves using a discrete particle scheme

A particle-based model for the simulation of wave propagation is presented. The model is based on solid-state physics principles and considers a piece of rock to be a Hookean material composed of discrete particles representing fundamental intact rock units. These particles interact at their contact points and experience reversible elastic forces proportional to their displacement from equilibrium. Particles are followed through space by numerically solving their equations of motion. We demonstrate that a numerical implementation of this scheme is capable of modelling the propagation of elastic waves through heterogeneous isotropic media. The results obtained are compared with a high-order finite difference solution to the wave equation. The method is found to be accurate, and thus offers an alternative to traditional continuum-based wave simulators.

Toomey, Aoife; Bean, Christopher J.

2000-06-01

115

A mesh system composed of multiple overset body-conforming grids is described for adapting finite-difference procedures to complex aircraft configurations. In this so-called 'chimera mesh,' a major grid is generated about a main component of the configuration and overset minor grids are used to resolve all other features. Methods for connecting overset multiple grids and modifications of flow-simulation algorithms are discussed. Computational tests in two dimensions indicate that the use of multiple overset grids can simplify the task of grid generation without an adverse effect on flow-field algorithms and computer code complexity.

Steger, J. L.; Dougherty, F. C.; Benek, J. A.

1983-01-01

116

A steady-state axisymmetric flow field of a liquid metal in a coreless induction furnace under an axisymmetric magnetic field is analyzed numerically, using a spectral finite difference method. Vorticity-stream function formulation is used in conjunction with Maxwell's equations, in a boundary-fitted coordinate system. For boundary conditions, both no-slip on the wall and no shear stress tensor on the free surface are used as dynamic conditions, and a field equivalent to the magnetic field induced by external coils is adopted as an electromagnetic field condition. Presented are streamlines, magnetic streamlines, and radial profiles of the axial velocity component at two Reynolds numbers for various parameters. It is found that the flow field varies remarkably according to the Reynolds number, the dimensionless height of the liquid metal, and the dimensionless height of external coils.

Im, Kichang; Mochimaru, Yoshihiro

117

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.

Wieselquist, William A.; Anistratov, Dmitriy Y.; Morel, Jim E.

2014-09-01

118

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.

Caplan, Ronald M

2011-01-01

119

Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations

Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in...

Amirali, I.; Amiraliyev, G. M.; Cakir, M.; Cimen, E.

2014-01-01

120

Discrete schemes for Gaussian curvature and their convergence

In this paper, several discrete schemes for Gaussian curvature are surveyed. The convergence property of a modified discrete scheme for the Gaussian curvature is considered. Furthermore, a new discrete scheme for Gaussian curvature is resented. We prove that the new scheme converges at the regular vertex with valence not less than 5. By constructing a counterexample, we also show that it is impossible for building a discrete scheme for Gaussian curvature which converges over the regular vertex with valence 4. Finally, asymptotic errors of several discrete scheme for Gaussian curvature are compared.

Zhiqiang Xu

2008-01-01

121

An over-relaxation procedure is applied to the MacCormack finite-difference scheme in order to reduce the computation time required to obtain a steady-state solution. The implementation of this acceleration procedure to an existing computer program using the regular MacCormack method is extremely simple and does not require additional storage. The over-relaxation procedure does not alter the steady-state solution, which is second-order accurate. The method is first applied to Burgers' equation. A stability condition and an expression for the increase in the rate of convergence are derived. The method is then applied to the calculation of the hypersonic viscous flow over a flat plate, using the complete Navier-Stokes equations, and the inviscid flow over a wedge. Reductions in computing time by factors of 3 and 1.5, respectively, are obtained by over-relaxation.

Desideri, J.-A.; Tannehill, J. C.

1977-01-01

122

Mimetic finite difference method

The mimetic finite difference (MFD) method mimics fundamental properties of mathematical and physical systems including conservation laws, symmetry and positivity of solutions, duality and self-adjointness of differential operators, and exact mathematical identities of the vector and tensor calculus. This article is the first comprehensive review of the 50-year long history of the mimetic methodology and describes in a systematic way the major mimetic ideas and their relevance to academic and real-life problems. The supporting applications include diffusion, electromagnetics, fluid flow, and Lagrangian hydrodynamics problems. The article provides enough details to build various discrete operators on unstructured polygonal and polyhedral meshes and summarizes the major convergence results for the mimetic approximations. Most of these theoretical results, which are presented here as lemmas, propositions and theorems, are either original or an extension of existing results to a more general formulation using polyhedral meshes. Finally, flexibility and extensibility of the mimetic methodology are shown by deriving higher-order approximations, enforcing discrete maximum principles for diffusion problems, and ensuring the numerical stability for saddle-point systems.

Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail

2014-01-01

123

Explicit finite difference methods for the delay pseudoparabolic equations.

Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown. PMID:24688392

Amirali, I; Amiraliyev, G M; Cakir, M; Cimen, E

2014-01-01

124

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

125

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.

Alizadeh Nomeli, M.; Riaz, A.

2012-12-01

126

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

127

Discretized Newton-relaxation solution of high resolution flux-difference split schemes

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.

Whitfield, David L.; Taylor, Lafe K.

1991-01-01

128

A Two-Timescale Discretization Scheme for Collocation

The development of a two-timescale discretization scheme for collocation is presented. This scheme allows a larger discretization to be utilized for smoothly varying state variables and a second finer discretization to be utilized for state variables having higher frequency dynamics. As such. the discretization scheme can be tailored to the dynamics of the particular state variables. In so doing. the size of the overall Nonlinear Programming (NLP) problem can be reduced significantly. Two two-timescale discretization architecture schemes are described. Comparison of results between the two-timescale method and conventional collocation show very good agreement. Differences of less than 0.5 percent are observed. Consequently. a significant reduction (by two-thirds) in the number of NLP parameters and iterations required for convergence can be achieved without sacrificing solution accuracy.

Desai, Prasun; Conway, Bruce A.

2004-01-01

129

We investigate the connections between several recent methods for the discretization of ani\\-so\\-tropic 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...

Droniou, Jerome; Eymard, Robert; Galloue?t, Thierry; Herbin, Raphaele

2008-01-01

130

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.

Droniou, Jerome; Gallouët, Thierry; Herbin, Raphaele

2008-01-01

131

The accuracy of finite difference methods is related to the mesh choice and cell size. Concerning the micromagnetism of nano-objects, we show here that discretization issues can drastically affect the symmetry of the problem and therefore the resulting computed properties of lattices of interacting curved nanomagnets. In this paper, we detail these effects for the multi-axis kagome lattice. Using the OOMMF finite difference method, we propose an alternative way of discretizing the nanomagnet shape via a variable moment per cell scheme. This method is shown to be efficient in reducing discretization effects.

Riahi, H.; Montaigne, F.; Rougemaille, N.; Canals, B.; Lacour, D.; Hehn, M.

2013-07-01

132

Optimized Discretization Schemes For Brain Images

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 TV...

USHA RANI.N,; Subbaiah, Dr P. V.; Venkata Rao, Dr D.

2011-01-01

133

Optimized Discretization Schemes For Brain Images

Directory of Open Access Journals (Sweden)

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.

USHA RANI.N,

2011-02-01

134

Mimetic finite difference methods in image processing

Directory of Open Access Journals (Sweden)

Full Text Available We introduce the use of mimetic methods to the imaging community, for the solution of the initial-value problems ubiquitous in the machine vision and image processing and analysis fields. PDE-based image processing and analysis techniques comprise a host of applications such as noise removal and restoration, deblurring and enhancement, segmentation, edge detection, inpainting, registration, motion analysis, etc. Because of their favorable stability and efficiency properties, semi-implicit finite difference and finite element schemes have been the methods of choice (in that order of preference. We propose a new approach for the numerical solution of these problems based on mimetic methods. The mimetic discretization scheme preserves the continuum properties of the mathematical operators often encountered in the image processing and analysis equations. This is the main contributing factor to the improved performance of the mimetic method approach, as compared to both of the aforementioned popular numerical solution techniques. To assess the performance of the proposed approach, we employ the Catté-Lions-Morel-Coll model to restore noisy images, by solving the PDE with the three numerical solution schemes. For all of the benchmark images employed in our experiments, and for every level of noise applied, we observe that the best image restored by using the mimetic method is closer to the noise-free image than the best images restored by the other two methods tested. These results motivate further studies of the application of the mimetic methods to other imaging problems. Mathematical subject classification: Primary: 68U10; Secondary: 65L12.

C. Bazan

2011-01-01

135

Mimetic finite difference methods in image processing

Scientific Electronic Library Online (English)

Full Text Available SciELO Brazil | Language: English Abstract in english We introduce the use of mimetic methods to the imaging community, for the solution of the initial-value problems ubiquitous in the machine vision and image processing and analysis fields. PDE-based image processing and analysis techniques comprise a host of applications such as noise removal and res [...] toration, deblurring and enhancement, segmentation, edge detection, inpainting, registration, motion analysis, etc. Because of their favorable stability and efficiency properties, semi-implicit finite difference and finite element schemes have been the methods of choice (in that order of preference). We propose a new approach for the numerical solution of these problems based on mimetic methods. The mimetic discretization scheme preserves the continuum properties of the mathematical operators often encountered in the image processing and analysis equations. This is the main contributing factor to the improved performance of the mimetic method approach, as compared to both of the aforementioned popular numerical solution techniques. To assess the performance of the proposed approach, we employ the Catté-Lions-Morel-Coll model to restore noisy images, by solving the PDE with the three numerical solution schemes. For all of the benchmark images employed in our experiments, and for every level of noise applied, we observe that the best image restored by using the mimetic method is closer to the noise-free image than the best images restored by the other two methods tested. These results motivate further studies of the application of the mimetic methods to other imaging problems. Mathematical subject classification: Primary: 68U10; Secondary: 65L12.

C., Bazan; M., Abouali; J., Castillo; P., Blomgren.

136

One of the ongoing issues with fractional-order diffusion models is the design of efficient numerical schemes for the space and time discretizations. Until now, most models have relied on a low-order finite difference (FD) method to discretize both the fractional-order space and time derivatives. Some numerical schemes using low-order finite elements (FE) have also been proposed. Both the FD and FE methods have long been used to solve integer-order partial differential equations. These low-or...

Hanert, Emmanuel; Piret, Ce?cile; International Symposium On Fractional Pdes Theory, Numerics And Applications

2013-01-01

137

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

138

The coupled double diffusive natural convection and radiation in a tilted and differentially heated square cavity containing a non-gray air-CO2 (or air-H2O) mixtures was numerically investigated. The horizontal walls are insulated and impermeable and the vertical walls are maintained at different temperatures and concentrations. The hybrid lattice Boltzmann method with the multiple-relaxation time model is used to compute the hydrodynamics and the finite difference method to determine temperatures and concentrations. The discrete ordinates method combined to the spectral line-based weighted sum of gray gases model is used to compute the radiative term and its spectral aspect. The effects of the inclination angle on the flow, thermal and concentration fields are analyzed for both aiding and opposing cases. It was found that radiation gas modifies the structure of the velocity and thermal fields by generating inclined stratifications and promoting the instabilities in opposing flows.

Moufekkir, Fayçal; Moussaoui, Mohammed Amine; Mezrhab, Ahmed; Naji, Hassan

2014-09-01

139

Applications of an exponential finite difference technique

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.

Handschuh, Robert F.; Keith, Theo G., Jr.

1988-01-01

140

Finite-difference models of ordinary differential equations - Influence of denominator functions

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.

Mickens, Ronald E.; Smith, Arthur

1990-01-01

141

Stable and Accurate Interpolation Operators for High-Order Multi-Block Finite-Difference Methods

Block-to-block interface interpolation operators are constructed for several common high-order finite difference discretizations. In contrast to conventional interpolation operators, these new interpolation operators maintain the strict stability, accuracy and conservation of the base scheme even when nonconforming grids or dissimilar operators are used in adjoining blocks. The stability properties of the new operators are verified using eigenvalue analysis, and the accuracy...

Mattsson, K.; Carpenter, Mark H.

2009-01-01

142

A free energy satisfying finite difference method for Poisson--Nernst--Planck equations

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 probabi...

Liu, Hailiang; Wang, Zhongming

2013-01-01

143

We consider an initial-boundary value problem for a generalized 2D time-dependent Schr\\"odinger equation on a semi-infinite strip. For the Crank-Nicolson finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniform in time $L^2$-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the related practical error analysis.

Ducomet, Bernard; Zlotnik, Ilya

2013-01-01

144

Suboptimal greedy power allocation schemes for discrete bit loading.

We consider low cost discrete bit loading based on greedy power allocation (GPA) under the constraints of total transmit power budget, target BER, and maximum permissible QAM modulation order. Compared to the standard GPA, which is optimal in terms of maximising the data throughput, three suboptimal schemes are proposed, which perform GPA on subsets of subchannels only. These subsets are created by considering the minimum SNR boundaries of QAM levels for a given target BER. We demonstrate how these schemes can significantly reduce the computational complexity required for power allocation, particularly in the case of a large number of subchannels. Two of the proposed algorithms can achieve near optimal performance including a transfer of residual power between subsets at the expense of a very small extra cost. By simulations, we show that the two near optimal schemes, while greatly reducing complexity, perform best in two separate and distinct SNR regions. PMID:24501578

Al-Hanafy, Waleed; Weiss, Stephan

2013-01-01

145

Finite difference methods for the time fractional diffusion equation on non-uniform meshes

Since fractional derivatives are integrals with weakly singular kernel, the discretization on the uniform mesh may lead to poor accuracy. The finite difference approximation of Caputo derivative on non-uniform meshes is investigated in this paper. The method is applied to solve the fractional diffusion equation and a semi-discrete scheme is obtained. The unconditional stability and H1 norm convergence are proved. A fully discrete difference scheme is constructed with space discretization by compact difference method. The error estimates are established for two kinds of nonuniform meshes. Numerical tests are carried out to support the theoretical results and comparing with the method on uniform grid shows the efficiency of our methods. Moreover, a moving local refinement technique is introduced to improve the temporal accuracy of numerical solution.

Zhang, Ya-nan; Sun, Zhi-zhong; Liao, Hong-lin

2014-05-01

146

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.)

147

Finite Difference Method of Modelling Groundwater Flow

Directory of Open Access Journals (Sweden)

Full Text Available In this study, finite difference method is used to solve the equations that govern groundwater flow to obtain flow rates, flow direction and hydraulic heads through an aquifer. The aim therefore is to discuss the principles of Finite Difference Method and its applications in groundwater modelling. To achieve this, a rectangular grid is overlain an aquifer in order to obtain an exact solution. Initial and boundary conditions are then determined. By discretizing the system into grids and cells that are small compared to the entire aquifer, exact solutions are obtained. A flow chart of the computational algorithm for particle tracking is also developed. Results show that under a steady-state flow with no recharge, pathlines coincide with streamlines. It is also found that the accuracy of the numerical solution by Finite Difference Method is largely dependent on initial particle distribution and number of particles assigned to a cell. It is therefore concluded that Finite Difference Method can be used to predict the future direction of flow and particle location within a simulation domain.

Magnus.U. Igboekwe

2011-03-01

148

Accurate finite difference methods for time-harmonic wave propagation

Finite difference methods for solving problems of time-harmonic acoustics are developed and analyzed. Multidimensional inhomogeneous problems with variable, possibly discontinuous, coefficients are considered, accounting for the effects of employing nonuniform grids. A weighted-average representation is less sensitive to transition in wave resolution (due to variable wave numbers or nonuniform grids) than the standard pointwise representation. Further enhancement in method performance is obtained by basing the stencils on generalizations of Pade approximation, or generalized definitions of the derivative, reducing spurious dispersion, anisotropy and reflection, and by improving the representation of source terms. The resulting schemes have fourth-order accurate local truncation error on uniform grids and third order in the nonuniform case. Guidelines for discretization pertaining to grid orientation and resolution are presented.

Harari, Isaac; Turkel, Eli

1994-01-01

149

High-order Finite Difference Solution of Euler Equations for Nonlinear Water Waves

DEFF Research Database (Denmark)

The incompressible Euler equations are solved with a free surface, the position of which is captured by applying an Eulerian kinematic boundary condition. The solution strategy follows that of [1, 2], applying a coordinate-transformation to obtain a time-constant spatial computational domain which is discretized using arbitrary-order finite difference schemes on a staggered grid with one optional stretching in each coordinate direction. The momentum equations and kinematic free surface condition are integrated in time using the classic fourth-order Runge-Kutta scheme. Mass conservation is satisfied implicitly, at the end of each time stage, by constructing the pressure from a discrete Poisson equation, derived from the discrete continuity and momentum equations and taking the time-dependent physical domain into account. An efficient preconditionedDefect Correction (DC) solution of the discrete Poisson equation for the pressure is presented, in which the preconditioning step is based on an order-multigrid formulation with a direct solution on the lowest order-level. This ensures fast convergence of the DC method with a computational effort which scales linearly with the problem size. Results obtained with a two-dimensional implementation of the model are compared with highly accurate stream function solutions to the nonlinear wave problem, which show the approximately expected convergence rates and a clear advantage of using high-order finite difference schemes in combination with the Euler equations.

Christiansen, Torben Robert Bilgrav; Bingham, Harry B.

2012-01-01

150

Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation

In this paper we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two dimensional heat equation. We employ respectively a second-order and a fourth-order schemes for the spatial derivatives and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is non-singular. Numerical experiments carried out on serial computers, show the unconditional stability of the proposed method and the high accuracy achieved by the fourth-order scheme.

Kouatchou, Jules

1999-01-01

151

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

152

Three discrete integrable coupling schemes associated with relativistic Toda lattice equation

In this paper, three discrete integrable coupling schemes from three different semi-direct sums of Lie algebras are illustrated. By the discrete variational identity, three different bi-Hamiltonian structures are proposed about discrete relativistic Toda lattice hierarchies, which are derived from three different types of discrete coupling spectral problems and the three hierarchies are all reduced to the discrete relativistic Toda lattice equations. In this way, many other multiform types and classifications of other integrable couplings can also be deduced.

Zhao, Qiu-Lan; Li, Xin-Yue; Liu, Fa-Sheng

2013-10-01

153

Convergence of discrete schemes for the Perona-Malik equation

We prove the convergence, up to a subsequence, of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation u=(?())x, ?(p):=1/2 >log(1+p), when the initial datum u¯ is 1-Lipschitz out of a finite number of jump points, and we characterize the problem satisfied by the limit solution. In the more difficult case when u¯ has a whole interval where ?(u) is negative, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points. The limit solution u we obtain is the same as the one obtained by replacing ?(?) with the truncated function min(?(?),1), and it turns out that u solves a free boundary problem. The free boundary consists of the points dividing the region where |u|>1 from the region where |u|?1. Finally, we consider the full space-time discretization (implicit in time) of the Perona-Malik equation, and we show that, if the time step is small with respect to the spatial grid h, then the limit is the same as the one obtained with the spatial semidiscrete scheme. On the other hand, if the time step is large with respect to h, then the limit solution equals u¯, i.e., the standing solution of the convexified problem.

Bellettini, G.; Novaga, M.; Paolini, M.; Tornese, C.

154

Exponential Finite-Difference Technique

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.

Handschuh, Robert F.

1989-01-01

155

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 a...

Caserta, A.

1998-01-01

156

A one dimensional fractional diffusion model with the Riemann-Liouville fractional derivative is studied. First, a second order discretization for this derivative is presented and then an unconditionally stable weighted average finite difference method is derived. The stability of this scheme is established by von Neumann analysis. Some numerical results are shown, which demonstrate the efficiency and convergence of the method. Additionally, some physical properties of this fractional diffusion system are simulated, which further confirm the effectiveness of our method.

Sousa, Ercília

2011-01-01

157

To stably employ multiband k.p model for analyzing the band structure in semiconductor heterostructures without spurious solutions (SSs), the Hermitian forward and backward difference (HFBD) scheme for finite difference method (FDM) is presented. The HFBD is the discretization scheme that eliminates the difference instability and employs the Burt-Foreman Hermitian operator ordering without geometric asymmetry. The difference instability arises from employing Foreman's strategy (FS). FS removes SSs caused by unphysical bowing in bulk dispersion curve meanwhile the HFBD is the only difference scheme that can accurately adapt for it. In comparison with other recent strategies, the proposed method in this paper is as accurate and reliable as FS, along with preserving the rapidness and simplicity of FDM. This difference scheme shows stable convergence without any SSs under variable grid size. Therefore, a wide range of experiment-determined band parameters can be applied to large-scale stable simulation with this method regardless of the SSs they originally generate.

Jiang, Yu; Ma, Xunpeng; Xu, Yun; Song, Guofeng

2014-11-01

158

We consider semi-discrete first-order finite difference schemes for a nonlinear degenerate convection-diffusion equations in one space dimension, and prove an L1 error estimate. Precisely, we show that the L1 loc difference between the approximate solution and the unique entropy solution converges at a rate O(\\Deltax 1/11), where \\Deltax is the spatial mesh size. If the diffusion is linear, we get the convergence rate O(\\Deltax 1/2), the point being that the O is independent of the size of the diffusion

Karlsen, K H; Risebro, And N H

2011-01-01

159

We consider semi-discrete first-order finite difference schemes for a nonlinear degenerate convection-diffusion equations in one space dimension, and prove an L1 error estimate. Precisely, we show that the L1 loc difference between the approximate solution and the unique entropy solution converges at a rate O(\\Deltax 1/11), where \\Deltax is the spatial mesh size. If the diffusion is linear, we get the convergence rate O(\\Deltax 1/2), the point being that the O is independent...

Karlsen, K. H.; Koley, U.; Risebro, N. H.

2011-01-01

160

A semi-discrete scheme for the stochastic Landau-Lifshitz equation

We propose a new convergent time semi-discrete scheme for the stochastic Landau-Lifshitz-Gilbert equation. The scheme is only linearly implicit and does not require the resolution of a nonlinear problem at each time step. Using a martingale approach, we prove the convergence in law of the scheme up to a subsequence.

Alouges, Franc?ois; Bouard, Anne; Hocquet, Antoine

2014-01-01

161

Nonlinear Conservative Difference Scheme for the Rosenau-RLW Equation

Numerical solution for the Rosenau-RLW equation is studied by a new nonlinear conservative finite difference scheme with parameter ? . Conservations of discrete mass and discrete energy are discussed. Second order convergence and unconditional stability of the scheme are also derived using prior estimate and energy method. Numerical results show that it can achieve better accuracy under adjusting the value of parameter ? .

Zheng, Kelong; Zhou, Guangya

2014-01-01

162

Finite difference neuroelectric modeling software.

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

Dang, Hung V; Ng, Kwong T

2011-06-15

163

A GOST-like Blind Signature Scheme Based on Elliptic Curve Discrete Logarithm Problem

In this paper, we propose a blind signature scheme and three practical educed schemes based on elliptic curve discrete logarithm problem. The proposed schemes impart the GOST signature structure and utilize the inherent advantage of elliptic curve cryptosystems in terms of smaller key size and lower computational overhead to its counterpart public key cryptosystems such as RSA and ElGamal. The proposed schemes are proved to be secure and have less time complexity in comparis...

Hosseini, Hossein; Bahrak, Behnam; Hessar, Farzad

2013-01-01

164

A New Signature Scheme Based on Factoring and Discrete Logarithm Problems

Directory of Open Access Journals (Sweden)

Full Text Available In 1994, He and Kiesler proposed a digital signature scheme which was based on the factoring and the discrete logarithm problem both. Same year, Shimin-Wei modi?ed the He-Kiesler signature scheme. In this paper, we propose an improvement of Shimin-Wei signature scheme based on factorization and discrete logarithm problem both with di?erent parameters and using a collision-free one-way hash function. In our opinion, our scheme is more secure than the earlier one.

Swati Verma

2012-07-01

165

We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omn\\`es \\cite{DomOmnes}) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" $L^\\infty$ weak-$\\star$ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, ...). Our results co...

Andreianov, Boris; Karlsen, Kenneth H

2009-01-01

166

On a space discretization scheme for the Fractional Stochastic Heat Equations

In this work, we introduce a new discretization to the fractional Laplacian and use it to elaborate an approximation scheme for fractional heat equations perturbed by a multiplicative cylindrical white noise. In particular, we estimate the rate of convergence.

Debbi, Latifa

2011-01-01

167

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.

Ducomet, Bernard; Romanova, Alla

2013-01-01

168

On the Total Variation of High-Order Semi-Discrete Central Schemes for Conservation Laws

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.

Bryson, Steve; Levy, Doron

2004-01-01

169

Convergence of discrete duality finite volume schemes for the cardiac bidomain model

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented.

Andreianov, Boris; Karlsen, Kenneth H; Pierre, Charles

2010-01-01

170

Implicit time-dependent finite different algorithm for quench simulation

Energy Technology Data Exchange (ETDEWEB)

A magnet in a fusion machine has many difficulties in its application because of requirement of a large operating current, high operating field and high breakdown voltage. A cable-in-conduit (CIC) conductor is the best candidate to overcome these difficulties. However, there remained uncertainty in a quench event in the cable-in-conduit conductor because of a difficulty to analyze a fluid dynamics equation. Several scientists, then, developed the numerical code for the quench simulation. However, most of them were based on an explicit time-dependent finite difference scheme. In this scheme, a discrete time increment is strictly restricted by CFL (Courant-Friedrichs-Lewy) condition. Therefore, long CPU time was consumed for the quench simulation. Authors, then, developed a new quench simulation code, POCHI1, which is based on an implicit time dependent scheme. In POCHI1, the fluid dynamics equation is linearlized according to a procedure applied by Beam and Warming and then, a tridiagonal system can be offered. Therefore, no iteration is necessary to solve the fluid dynamics equation. This leads great reduction of the CPU time. Also, POCHI1 can cope with non-linear boundary condition. In this study, comparison with experimental results was carried out. The normal zone propagation behavior was investigated in two samples of CIC conductors which had different hydraulic diameters. The measured and simulated normal zone propagation length showed relatively good agreement. However, the behavior of the normal voltage shows a little disagreement. These results indicate necessity to improve the treatment of the heat transfer coefficient in the turbulent flow region and the electric resistivity of the copper stabilizer in high temperature and high field region. (author).

Koizumi, Norikiyo; Takahashi, Yoshikazu; Tsuji, Hiroshi [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment

1994-12-01

171

Implicit time-dependent finite different algorithm for quench simulation

International Nuclear Information System (INIS)

A magnet in a fusion machine has many difficulties in its application because of requirement of a large operating current, high operating field and high breakdown voltage. A cable-in-conduit (CIC) conductor is the best candidate to overcome these difficulties. However, there remained uncertainty in a quench event in the cable-in-conduit conductor because of a difficulty to analyze a fluid dynamics equation. Several scientists, then, developed the numerical code for the quench simulation. However, most of them were based on an explicit time-dependent finite difference scheme. In this scheme, a discrete time increment is strictly restricted by CFL (Courant-Friedrichs-Lewy) condition. Therefore, long CPU time was consumed for the quench simulation. Authors, then, developed a new quench simulation code, POCHI1, which is based on an implicit time dependent scheme. In POCHI1, the fluid dynamics equation is linearlized according to a procedure applied by Beam and Warming and then, a tridiagonal system can be offered. Therefore, no iteration is necessary to solve the fluid dynamics equation. This leads great reduction of the CPU time. Also, POCHI1 can cope with non-linear boundary condition. In this study, comparison with experimental results was carried out. The normal zone propagation behavior was investigated in two samples of CIC conductors which had different hydraulic diameters. The measured and simulated normal zone propagation length showed relatively good agreement. However, the behavior of the normal voltage shows a little disagreement. These results indicate necessity to improve the treatment of the heat transfer coefficient in the turbulent flow region and the electric resistivity of the copper stabilizer in high temperature and high field region. (author)

172

On some fundamental finite difference inequalities

The main object of this paper is to establish some new finite difference inequalities which can be used as tools in the study of various problems in the theory of certain classes of finite difference and sum-difference equations.

Pachpatte, B. G.

2001-01-01

173

This paper is concerned with a few novel RBF-based numerical schemes discretizing partial differential equations. For boundary-type methods, we derive the indirect and direct symmetric boundary knot methods (BKM). The resulting interpolation matrix of both is always symmetric irrespective of boundary geometry and conditions. In particular, the direct BKM applies the practical physical variables rather than expansion coefficients and becomes very competitive to the boundary element method. On the other hand, based on the multiple reciprocity principle, we invent the RBF-based boundary particle method (BPM) for general inhomogeneous problems without a need using inner nodes. The direct and symmetric BPM schemes are also developed. For domain-type RBF discretization schemes, by using the Green integral we develop a new Hermite RBF scheme called as the modified Kansa method (MKM), which differs from the symmetric Hermite RBF scheme in that the MKM discretizes both governing equation and boundary conditions on the...

Chen, W

2001-01-01

174

This article analyzes the changes in the number of cases of various clients of the pyramid and the establishment of the basic rules of the pyramid schemes based on discrete models. The article is also a continuation of previous work [1], which had formulas to simulate the amount collected by the pyramid scheme

Kovalenko A. V.; Urtenov M. K.; Chagarov R. H.

2012-01-01

175

Discrete unified gas kinetic scheme with force term for incompressible fluid flows

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...

Wu, Chen; Chai, Zhenhua; Wang, Peng

2014-01-01

176

A Review of High-Order and Optimized Finite-Difference Methods for Simulating Linear Wave Phenomena

This paper presents a review of high-order and optimized finite-difference methods for numerically simulating the propagation and scattering of linear waves, such as electromagnetic, acoustic, or elastic waves. The spatial operators reviewed include compact schemes, non-compact schemes, schemes on staggered grids, and schemes which are optimized to produce specific characteristics. The time-marching methods discussed include Runge-Kutta methods, Adams-Bashforth methods, and the leapfrog method. In addition, the following fourth-order fully-discrete finite-difference methods are considered: a one-step implicit scheme with a three-point spatial stencil, a one-step explicit scheme with a five-point spatial stencil, and a two-step explicit scheme with a five-point spatial stencil. For each method studied, the number of grid points per wavelength required for accurate simulation of wave propagation over large distances is presented. Recommendations are made with respect to the suitability of the methods for specific problems and practical aspects of their use, such as appropriate Courant numbers and grid densities. Avenues for future research are suggested.

Zingg, David W.

1996-01-01

177

An energy conserving finite-difference model of Maxwell's equations for soliton propagation

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.

Bachiri, H; Vázquez, L

1997-01-01

178

A free energy satisfying finite difference method for Poisson-Nernst-Planck equations

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.

Liu, Hailiang; Wang, Zhongming

2014-07-01

179

Original Signer's Forgery Attacks on Discrete Logarithm Based Proxy Signature Schemes

Directory of Open Access Journals (Sweden)

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.

Tianjie Cao

2007-05-01

180

Novel Two-Scale Discretization Schemes for Lagrangian Hydrodynamics

Energy Technology Data Exchange (ETDEWEB)

In this report we propose novel higher order conservative schemes of discontinuous Galerkin (or DG) type for the equations of gas dynamics in Lagrangian coordinates suitable for general unstructured finite element meshes. The novelty of our approach is in the formulation of two-scale non-oscillatory function recovery procedures utilizing integral moments of the quantities of interest (pressure and velocity). The integral moments are computed on a primary mesh (cells or zones) which defines our original scale that governs the accuracy of the schemes. In the non-oscillatory smooth function recovery procedures, we introduce a finer mesh which defines the second scale. Mathematically, the recovery can be formulated as nonlinear energy functional minimization subject to equality and nonlinear inequality constraints. The schemes are highly accurate due to both the embedded (local) mesh refinement features as well as the ability to utilize higher order integral moments. The new DG schemes seem to offer an alternative to currently used artificial viscosity techniques and limiters since the two-scale recovery procedures aim at resolving these issues. We report on some preliminary tests for the lowest order case, and outline some possible future research directions.

Vassilevski, P

2008-05-29

181

The mimetic finite difference method for elliptic problems

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.

Veiga, Lourenço Beirão; Manzini, Gianmarco

2014-01-01

182

An Efficient Elliptic Curve Discrete Logarithm based Trapdoor Hash Scheme without Key Exposure

Directory of Open Access Journals (Sweden)

Full Text Available The trapdoor hash function plays essential role in constructing certain secure digital signature, and signature scheme that composed by trapdoor hash function is widely applied in different fields. However, the key exposure problem of trapdoor hash scheme has brought great distress. In this paper, an efficient trapdoor hash scheme without key exposure based on elliptic curve discrete logarithm is put forward and its security is analyzed, the scheme satisfies the five properties of trapdoor hash functions: effective calculation, trapdoor collision, collision resistance, key exposure resistance and semantic security. Through comparing and analyzing with the existing schemes, it shows that the proposed scheme, which has only multiplicative complexity and removes the operations of computing finite field element inverse, is more advantage in terms of safety and efficiency. Moreover, the scheme supports batch computation that it can greatly improve the efficiency of verification.

Yi Sun

2013-11-01

183

Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows

A phase-field-based hybrid model that combines the lattice Boltzmann method with the finite difference method is proposed for simulating immiscible thermocapillary flows with variable fluid-property ratios. Using a phase field methodology, an interfacial force formula is analytically derived to model the interfacial tension force and the Marangoni stress. We present an improved lattice Boltzmann equation (LBE) method to capture the interface between different phases and solve the pressure and velocity fields, which can recover the correct Cahn-Hilliard equation (CHE) and Navier-Stokes equations. The LBE method allows not only use of variable mobility in the CHE, but also simulation of multiphase flows with high density ratio because a stable discretization scheme is used for calculating the derivative terms in forcing terms. An additional convection-diffusion equation is solved by the finite difference method for spatial discretization and the Runge-Kutta method for time marching to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. The model is first validated against analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of a three-dimensional deformable droplet and bubble at various Marangoni numbers and density ratios, and satisfactory agreement is obtained between numerical results and theoretical predictions.

Liu, Haihu; Valocchi, Albert J.; Zhang, Yonghao; Kang, Qinjun

2013-01-01

184

Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows.

A phase-field-based hybrid model that combines the lattice Boltzmann method with the finite difference method is proposed for simulating immiscible thermocapillary flows with variable fluid-property ratios. Using a phase field methodology, an interfacial force formula is analytically derived to model the interfacial tension force and the Marangoni stress. We present an improved lattice Boltzmann equation (LBE) method to capture the interface between different phases and solve the pressure and velocity fields, which can recover the correct Cahn-Hilliard equation (CHE) and Navier-Stokes equations. The LBE method allows not only use of variable mobility in the CHE, but also simulation of multiphase flows with high density ratio because a stable discretization scheme is used for calculating the derivative terms in forcing terms. An additional convection-diffusion equation is solved by the finite difference method for spatial discretization and the Runge-Kutta method for time marching to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. The model is first validated against analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of a three-dimensional deformable droplet and bubble at various Marangoni numbers and density ratios, and satisfactory agreement is obtained between numerical results and theoretical predictions. PMID:23410429

Liu, Haihu; Valocchi, Albert J; Zhang, Yonghao; Kang, Qinjun

2013-01-01

185

Energy Technology Data Exchange (ETDEWEB)

We present a comprehensive error estimation of four spatial discretization schemes of the two-dimensional Discrete Ordinates (SN) equations on Cartesian grids utilizing a Method of Manufactured Solution (MMS) benchmark suite based on variants of Larsen’s benchmark featuring different orders of smoothness of the underlying exact solution. The considered spatial discretization schemes include the arbitrarily high order transport methods of the nodal (AHOTN) and characteristic (AHOTC) types, the discontinuous Galerkin Finite Element method (DGFEM) and the recently proposed higher order diamond difference method (HODD) of spatial expansion orders 0 through 3. While AHOTN and AHOTC rely on approximate analytical solutions of the transport equation within a mesh cell, DGFEM and HODD utilize a polynomial expansion to mimick the angular flux profile across each mesh cell. Intuitively, due to the higher degree of analyticity, we expect AHOTN and AHOTC to feature superior accuracy compared with DGFEM and HODD, but at the price of potentially longer grind times and numerical instabilities. The latter disadvantages can result from the presence of exponential terms evaluated at the cell optical thickness that arise from the semianalytical solution process. This work quantifies the order of accuracy and the magnitude of the error of all four discretization methods for different optical thicknesses, scattering ratios and degrees of smoothness of the underlying exact solutions in order to verify or contradict the aforementioned intuitive expectation.

Sebastian Schunert; Yousry Y. Azmy; Damien Fournier

2011-05-01

186

Accuracy Analysis for Finite-Volume Discretization Schemes on Irregular Grids

A new computational analysis tool, downscaling test, is introduced and applied for studying the convergence rates of truncation and discretization errors of nite-volume discretization schemes on general irregular (e.g., unstructured) grids. The study shows that the design-order convergence of discretization errors can be achieved even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all. The downscaling test is a general, efficient, accurate, and practical tool, enabling straightforward extension of verification and validation to general unstructured grid formulations. It also allows separate analysis of the interior, boundaries, and singularities that could be useful even in structured-grid settings. There are several new findings arising from the use of the downscaling test analysis. It is shown that the discretization accuracy of a common node-centered nite-volume scheme, known to be second-order accurate for inviscid equations on triangular grids, degenerates to first order for mixed grids. Alternative node-centered schemes are presented and demonstrated to provide second and third order accuracies on general mixed grids. The local accuracy deterioration at intersections of tangency and in flow/outflow boundaries is demonstrated using the DS tests tailored to examining the local behavior of the boundary conditions. The discretization-error order reduction within inviscid stagnation regions is demonstrated. The accuracy deterioration is local, affecting mainly the velocity components, but applies to any order scheme.

Diskin, Boris; Thomas, James L.

2010-01-01

187

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 Comer 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.

Thompson, Kelly Glen

188

An exponential integrator scheme for time discretization of nonlinear stochastic wave equation

This work is devoted to convergence analysis of an exponential integrator scheme for semi-discretization in time of nonlinear stochastic wave equation (SWE). A unified framework is set forth, which covers important cases of multiplicative and additive noise. Within this framework, the scheme is shown to converge uniformly in the strong $L^p$-sense with strong convergence rates given.The abstract results are then applied to several concrete examples, including SWE with genera...

Wang, Xiaojie

2013-01-01

189

Discrete ordinate method with a new and a simple quadrature scheme

International Nuclear Information System (INIS)

Evaluation of the radiative component in heat-transfer problems is often difficult and expensive. To address this problem, in the recent past, attention has been focused on improving the performance of various approximate methods. Computational efficiency of any method depends to a great extent on the quadrature schemes that are used to compute the source term and heat flux. The discrete ordinate method (DOM) is one of the oldest and still the most widely used methods. To make this method computationally more attractive, various types of quadrature schemes have been suggested over the years. In the present work, a new quadrature scheme has been suggested. The new scheme is a simple one and does not involve complicated mathematics for determination of direction cosines and weights. It satisfies all the required moments. To test the suitability of the new scheme, four benchmark problems were considered. In all cases, the proposed quadrature scheme was found to give accurate results

190

A parallel adaptive finite difference algorithm for petroleum reservoir simulation

Energy Technology Data Exchange (ETDEWEB)

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)

Hoang, Hai Minh

2005-07-01

191

In this paper we present a numerical method for solving elliptic equations in an arbitrary domain (described by a level-set function) with general boundary conditions (Dirichlet, Neumann, Robin, etc.) on Cartesian grids, using finite difference discretization and non-eliminated ghost values. A system of Ni+Ng equations in Ni+Ng unknowns is obtained by finite difference discretization on the Ni internal grid points, and second order interpolation to define the conditions for the Ng ghost values. The resulting large sparse linear system is then solved by a multigrid technique. The novelty of the papers can be summarized as follows: general strategy to discretize the boundary condition to second order both in the solution and its gradient; a relaxation of inner equations and boundary conditions by a fictitious time method, inspired by the stability conditions related to the associated time dependent problem (with a convergence proof for the first order scheme); an effective geometric multigrid, which maintains the structure of the discrete system at all grid levels. It is shown that by increasing the relaxation step of the equations associated to the boundary conditions, a convergence factor close to the optimal one is obtained. Several numerical tests, including variable coefficients, anisotropic elliptic equations, and domains with kinks, show the robustness, efficiency and accuracy of the approach.

Coco, Armando; Russo, Giovanni

2013-05-01

192

Effects of Mesh Irregularities on Accuracy of Finite-Volume Discretization Schemes

The effects of mesh irregularities on accuracy of unstructured node-centered finite-volume discretizations are considered. The focus is on an edge-based approach that uses unweighted least-squares gradient reconstruction with a quadratic fit. For inviscid fluxes, the discretization is nominally third order accurate on general triangular meshes. For viscous fluxes, the scheme is an average-least-squares formulation that is nominally second order accurate and contrasted with a common Green-Gauss discretization scheme. Gradient errors, truncation errors, and discretization errors are separately studied according to a previously introduced comprehensive methodology. The methodology considers three classes of grids: isotropic grids in a rectangular geometry, anisotropic grids typical of adapted grids, and anisotropic grids over a curved surface typical of advancing layer grids. The meshes within the classes range from regular to extremely irregular including meshes with random perturbation of nodes. Recommendations are made concerning the discretization schemes that are expected to be least sensitive to mesh irregularities in applications to turbulent flows in complex geometries.

Diskin, Boris; Thomas, James L.

2012-01-01

193

A unified formalism for spatial discretization schemes of transport equations in slab geometry

International Nuclear Information System (INIS)

It is shown that most of the spatial discretization schemes of transport equations in slab geometry which have been developed recently are particular applications of a general finite element oriented formalism developed by this author and his collaborators for the numerical integration of systems of stiff ordinary differential equations. (author)

194

An Efficient Signcryption Scheme based on The Elliptic Curve Discrete Logarithm Problem

Directory of Open Access Journals (Sweden)

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.

Fatima Amounas

2013-02-01

195

A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs

We consider issues of stability of time-discretization schemes with exacttreatment of the linear part(ELP schemes) for solving nonlinear PDEs. A distinctive feature of ELP schemes is the exact evaluation of the contribution of the linear term, that is if the nonlinear term of the equation is zero, then the scheme reduces to the evaluation of the exponential function of the operator representing the linear term. Computing and applying the exponential or other functions of operators with variable coefficients in the usual manner requires evaluating dense matrices and is highly inefficient. It turns out that computing the exponential of strictly elliptic operators in the wavelet system of coordinates yields sparse matrices (for a finite but arbitrary accuracy). This observation makes our approach practical in a number of applications. In particular, we consider applications of ELP schemes to advection-diffusion equations. We study the stability of these schemes and show that both explicit and implicit ELP schemes have distinctly different stability properties if compared with known implicit-explicit schemes. For example, we describe explicit schemes with stability regions similar to those of typical implicit schemes used for solving advection-diffusion equations.

Beylkin, Gregory; Keiser, James M.; Vozovoi, Lev

1998-12-01

196

International Nuclear Information System (INIS)

The Coarse Mesh Finite Difference (CMFD) acceleration was devised to enhance the computational acceleration for the high order diffusion calculation such as nodal diffusion. And then it could be successfully applied to the acceleration of the transport eigenvalue calculations using Method of Characteristics (MOC). This method is quite effective for the fission source iteration by conserving the reaction rates inside each coarse mesh through the non-linear updating of the interface net currents from the high order transport equation. Fourier analysis for the fixed source and eigenvalue problems showed that the coupling of the high order transport and the low order CMFD calculations is not unconditionally stable. The partial current based CMFD (pCMFD) was devised to consider the interface partial currents for the better performance. Fourier analysis for the eigenvalue problems showed that the coupling of the high order SC (Step Characteristics) transport and the low order pCMFD calculations is unconditionally stable. The NEWT code is a multi-group discrete ordinate neutron transport code with flexible meshing capabilities. This code adopts the Extended Step Characteristic (ESC) approach for the arbitrary polygon meshes. In NEWT an acceleration scheme for the fission source iteration has been available only for the rectangular domain boundaries by using coarse mesh finite difference acceleration method only with rectangular coarse meshes. Therefore no acceleration schemee meshes. Therefore no acceleration scheme could be applied to the wedge, triangle, hexagon and their symmetric domain boundaries. The conventional and the partial current based unstructured CMFD acceleration schemes (uCMFD and upCMFD) with the unstructured coarse meshes were implemented to be used for any domain boundaries

197

Weighted Average Finite Difference Methods for Fractional Reaction-Subdiffusion Equation

Directory of Open Access Journals (Sweden)

Full Text Available In this article, a numerical study for fractional reaction-subdiffusion equations is introduced using a class of finite difference methods. These methods are extensions of the weighted average methods for ordinary (non-fractional reaction-subdiffusion equations. A stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. Simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, are given and checked numerically. Numerical test examples, figures, and comparisons have been presented for clarity.doi:10.14456/WJST.2014.50

Nasser Hassen SWEILAM

2014-04-01

198

Notes on Accuracy of Finite-Volume Discretization Schemes on Irregular Grids

Truncation-error analysis is a reliable tool in predicting convergence rates of discretization errors on regular smooth grids. However, it is often misleading in application to finite-volume discretization schemes on irregular (e.g., unstructured) grids. Convergence of truncation errors severely degrades on general irregular grids; a design-order convergence can be achieved only on grids with a certain degree of geometric regularity. Such degradation of truncation-error convergence does not necessarily imply a lower-order convergence of discretization errors. In these notes, irregular-grid computations demonstrate that the design-order discretization-error convergence can be achieved even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all.

Diskin, Boris; Thomas, James L.

2011-01-01

199

Finite Difference Migration Imaging of Magnetotellurics

we put forward a new migration imaging technique of Magnetotellurics (MT) data based on improved finite difference method, which increased the accuracy of difference equation and imaging resolution greatly. We also discussed the determination of background resistivity and reimaging. The processing results of theoretical model and case study indicated that this method was a more practical and effective for MT imaging. Finally the characteristics of finite difference migration imaging were summ...

Runlin Du; Zhan Liu

2013-01-01

200

In this manuscript, we present a computational model to approximate the solutions of a partial differential equation which describes the growth dynamics of microbial films. The numerical technique reported in this work is an explicit, nonlinear finite-difference methodology which is computationally implemented using Newton's method. Our scheme is compared numerically against an implicit, linear finite-difference discretization of the same partial differential equation, whose computer coding requires an implementation of the stabilized bi-conjugate gradient method. Our numerical results evince that the nonlinear approach results in a more efficient approximation to the solutions of the biofilm model considered, and demands less computer memory. Moreover, the positivity of initial profiles is preserved in the practice by the nonlinear scheme proposed. PMID:23850847

Macías-Díaz, J E; Macías, Siegfried; Medina-Ramírez, I E

2013-12-01

201

A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in R3 is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a surface to a sequence of BIEs defined on a generating curve for the surface. It can handle loads that are not necessarily rotationally symmetric. Nyström discretization is used to discretize the BIEs on the generating curve. The quadrature is a high-order Gaussian rule that is modified near the diagonal to retain high-order accuracy for singular kernels. The reduction in dimensionality, along with the use of high-order accurate quadratures, leads to small linear systems that can be inverted directly via, e.g., Gaussian elimination. This makes the scheme particularly fast in environments involving multiple right hand sides. It is demonstrated that for BIEs associated with the Laplace and Helmholtz equations, the kernel in the reduced equations can be evaluated very rapidly by exploiting recursion relations for Legendre functions. Numerical examples illustrate the performance of the scheme; in particular, it is demonstrated that for a BIE associated with Laplace's equation on a surface discretized using 320,800 points, the set-up phase of the algorithm takes 1 min on a standard laptop, and then solves can be executed in 0.5 s.

Young, P.; Hao, S.; Martinsson, P. G.

2012-06-01

202

A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation

In this paper we analyze the large-time behavior of the augmented Burgers equation. We first study the well-posedness of the Cauchy problem and obtain $L^1$-$L^p$ decay rates. The asymptotic behavior of the solution is obtained by showing that the convolution term $K*u_{xx}$ behaves as $u_{xx}$ for large times. Then, we propose a semi-discrete numerical scheme that preserves this asymptotic behavior, by introducing two corrector factors in the discretization of the non-local...

Ignat, Liviu I.; Pozo, Alejandro

2014-01-01

203

A Finite Difference-Based Modeling Approach for Prediction of Steel Hardenability

In this research work an independent finite difference-based modeling approach was adopted for determination of the hardenability of steels. In this model, at first, cooling curves were generated by solving transient heat transfer equation through discretization with pure explicit finite difference scheme coupled with MATLAB-based programing in view of variable thermo-physical properties of 1080 steel. The cooling curves were solved against 50% transformation nose of TTT diagram in order to predict hardening behavior of 1080 steel in terms of hardenability parameters (Grossmann critical diameter, D C; and ideal critical diameter, D I) and the variation of the unhardened core diameter ( D u) to diameter of steel bar ( D) ratio with diameter of steel bar ( D). The experiments were also performed to determine actual D C value of 1080 steel for still water quenching. The D C value obtained by the developed model was found to match the experimental D C value with only 6% deviation. Therefore, the model developed in the present work can be used for direct determination of D I, D C, and D u without resorting to any rigorous experimentation.

Sushanthi, Neethi; Maity, Joydeep

2014-06-01

204

International Nuclear Information System (INIS)

We present the sublattice approach, a procedure to generate, from a given integrable lattice, a sublattice which inherits its integrability features. We consider, as illustrative example of this approach, the discrete Moutard 4-point equation and its sublattice, the self-adjoint 5-point scheme on the star of the square lattice, which are relevant in the theory of the integrable discrete geometries and in the theory of discrete holomorphic and harmonic functions (in this last context, the discrete Moutard equation is called discrete Cauchy-Riemann equation). Therefore an integrable, at one energy, discretization of elliptic two-dimensional operators is considered. We use the sublattice point of view to derive, from the Darboux transformations and superposition formulas of the discrete Moutard equation, the Darboux transformations and superposition formulas of the self-adjoint 5-point scheme. We also construct, from algebro-geometric solutions of the discrete Moutard equation, algebro-geometric solutions of the self-adjoint 5-point scheme. In particular, we show that the corresponding restrictions on the finite-gap data are of the same type as those for the fixed energy problem for the two-dimensional Schroedinger operator. We finally use these solutions to construct explicit examples of discrete holomorphic and harmonic functions, as well as examples of quadrilateral surfaces in R3

205

We present the sublattice approach, a procedure to generate, from a given integrable lattice, a sublattice which inherits its integrability features. We consider, as illustrative example of this approach, the discrete Moutard 4-point equation and its sublattice, the self-adjoint 5-point scheme on the star of the square lattice, which are relevant in the theory of the integrable discrete geometries and in the theory of discrete holomorphic and harmonic functions (in this last context, the discrete Moutard equation is called discrete Cauchy-Riemann equation). Therefore an integrable, at one energy, discretization of elliptic two-dimensional operators is considered. We use the sublattice point of view to derive, from the Darboux transformations and superposition formulas of the discrete Moutard equation, the Darboux transformations and superposition formulas of the self-adjoint 5-point scheme. We also construct, from algebro-geometric solutions of the discrete Moutard equation, algebro-geometric solutions of the self-adjoint 5-point scheme. In particular, we show that the corresponding restrictions on the finite-gap data are of the same type as those for the fixed energy problem for the two-dimensional Schrödinger operator. We finally use these solutions to construct explicit examples of discrete holomorphic and harmonic functions, as well as examples of quadrilateral surfaces in R3.

Doliwa, A.; Grinevich, P.; Nieszporski, M.; Santini, P. M.

2007-01-01

206

High Order Finite Difference Methods in Space and Time

In this thesis, high order accurate discretization schemes for partial differential equations are investigated. In the first paper, the linearized two-dimensional Navier-Stokes equations are considered. A special formulation of the boundary conditions is used and estimates for the solution to the continuous problem in terms of the boundary conditions are derived using a normal mode analysis. Similar estimates are achieved for the discretized equations. For the discretization, a second order f...

Kress, Wendy

2003-01-01

207

Error Estimate for a Fully Discrete Spectral Scheme for Korteweg-de Vries-Kawahara Equation

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.

Koley, U

2011-01-01

208

In this paper, for a class of fractional sub-diffusion equations on a space unbounded domain, firstly, exact artificial boundary conditions, which involve the time-fractional derivatives, are derived using the Laplace transform technique. Then the original problem on the space unbounded domain is reduced to the initial-boundary value problem on a space bounded domain. Secondly, an efficient finite difference approximation for the reduced initial-boundary problem on the space bounded domain is constructed. Different from the method of order reduction used in [37] for the fractional sub-diffusion equations on a space half-infinite domain, the presented difference scheme, which is more simple than that in the previous work, is developed using the direct discretization method, i.e. the approximate method of considering the governing equations at mesh points directly. The stability and convergence of the scheme with numerical accuracy O(?+h2) are proved by means of discrete energy method and Sobolev imbedding inequality, where ? is the order of time-fractional derivative in the governing equation, ? and h are the temporal stepsize and spatial stepsize, respectively. Thirdly, a compact difference scheme for the case of ??2/3 is derived with the truncation errors of fourth-order accuracy for interior points and third-order accuracy for boundary points, respectively. Then the global convergence order O(?+h4) of the compact difference scheme is proved. Finally, numerical experiments are used to verify the numerical accuracy and the efficiency of the obtained schemes.

Gao, Guang-hua; Sun, Zhi-zhong

2013-03-01

209

We introduce the sub-lattice approach, a procedure to generate, from a given integrable lattice, a sub-lattice which inherits its integrability features. We consider, as illustrative example of this approach, the discrete Moutard 4-point equation and its sub-lattice, the self-adjoint 5-point scheme on the star of the square lattice, which are relevant in the theory of the integrable Discrete Geometries and in the theory of discrete holomorphic and harmonic functions (in this last context, the discrete Moutard equation is called discrete Cauchy-Riemann equation). We use the sub-lattice point of view to derive, from the Darboux transformations and superposition formulas of the discrete Moutard equation, the Darboux transformations and superposition formulas of the self-adjoint 5-point scheme. We also construct, from algebro-geometric solutions of the discrete Moutard equation, algebro-geometric solutions of the self-adjoint 5-point scheme. We finally use these solutions to construct explicit examples of discret...

Doliwa, A; Nieszporski, M; Santini, P M

2004-01-01

210

Nonlinear Conservative Difference Scheme for the Rosenau-RLW Equation

Directory of Open Access Journals (Sweden)

Full Text Available Numerical solution for the Rosenau-RLW equation is studied by a new nonlinear conservative finite difference scheme with parameter ? . Conservations of discrete mass and discrete energy are discussed. Second order convergence and unconditional stability of the scheme are also derived using prior estimate and energy method. Numerical results show that it can achieve better accuracy under adjusting the value of parameter ? .

ZHENG Kelong

2014-07-01

211

Exact and high order discretization schemes for Wishart processes and their affine extensions

This work deals with the simulation of Wishart processes and affine diffusions on positive semidefinite matrices. To do so, we focus on the splitting of the infinitesimal generator, in order to use composition techniques as Ninomiya and Victoir or Alfonsi. Doing so, we have found a remarkable splitting for Wishart processes that enables us to sample exactly Wishart distributions, without any restriction on the parameters. It is related but extends existing exact simulation methods based on Bartlett's decomposition. Moreover, we can construct high-order discretization schemes for Wishart processes and second-order schemes for general affine diffusions. These schemes are in practice faster than the exact simulation to sample entire paths. Numerical results on their convergence are given.

Ahdida, Abdelkoddousse

2010-01-01

212

A semi-discrete central scheme for magnetohydrodynamics on orthogonal-curvilinear grids

This work describes a novel scheme for the equations of magnetohydrodynamics on orthogonal-curvilinear grids within a finite-volume framework. The scheme is based on a combination of central-upwind techniques for hyperbolic conservation laws and projection-evolution methods originally developed for Hamilton-Jacobi equations. The scheme is derived in semi-discrete form, and a full-fledged version is obtained by applying any stable and accurate solver for integration in time. The divergence-free condition of the magnetic field is a built-in property of the scheme by virtue of a constrained-transport ansatz for the induction equation. From the general formulation second-order accurate schemes for cylindrical grids and spherical grids are introduced in some more detail pointing out their potential importance in many applications. Special emphasis in this context is put to a treatment of the geometric axis implying severe complications because of the presence of coordinate singularities and associated grid degeneracy. An attempt to tackle these problems is presented. Numerical experiments illustrate the overall robustness and performance of the scheme for a small suite of tests.

Ziegler, U.

2011-02-01

213

International Nuclear Information System (INIS)

This paper addresses the problem of the construction of stable approximation schemes for the one-dimensional linear Schroedinger equation set in an unbounded domain. After a study of the initial boundary-value problem in a bounded domain with a transparent boundary condition, some unconditionally stable discretization schemes are developed for this kind of problem. The main difficulty is linked to the involvement of a fractional integral operator defining the transparent operator. The proposed semi-discretization of this operator yields with a very different point of view the one proposed by Yevick, Friese and Schmidt [J. Comput. Phys. 168 (2001) 433]. Two possible choices of transparent boundary conditions based on the Dirichlet-Neumann (DN) and Neumann-Dirichlet (ND) operators are presented. To preserve the stability of the fully discrete scheme, conform Galerkin finite element methods are employed for the spatial discretization. Finally, some numerical tests are performed to study the respective accuracy of the different schemes

214

A finite-difference lattice Boltzmann approach for gas microflows

Finite-difference Lattice Boltzmann (LB) models are proposed for simulating gas flows in devices with microscale geometries. The models employ the roots of half-range Gauss-Hermite polynomials as discrete velocities. Unlike the standard LB velocity-space discretizations based on the roots of full-range Hermite polynomials, using the nodes of a quadrature defined in the half-space permits a consistent treatment of kinetic boundary conditions. The possibilities of the proposed LB models are illustrated by studying the one-dimensional Couette flow and the two-dimensional driven cavity flow. Numerical and analytical results show an improved accuracy in finite Knudsen flows as compared with standard LB models.

Ghiroldi, G P

2013-01-01

215

In this paper is presented a finite volume (DDFV) scheme for solving elliptic equations with heterogeneous anisotropic conductivity tensor. That method is based on the definition of a discrete divergence and a discrete gradient operator. These discrete operators have close relationships with the continuous ones, in particulat they fulfil a duality property related with the Green formula. The operators are defined in dimension 2 and 3, their duality property is stated and used to establish the...

Coudiere, Yves; Pierre, Charles; Rousseau, Olivier; Turpault, Rodolphe

2008-01-01

216

A coupled discrete unified gas-kinetic scheme for Boussinesq flows

Recently, the discrete unified gas-kinetic scheme (DUGKS) [Z. L. Guo \\emph{et al}., Phys. Rev. E ${\\bf 88}$, 033305 (2013)] based on the Boltzmann equation is developed as a new multiscale kinetic method for isothermal flows. In this paper, a thermal and coupled discrete unified gas-kinetic scheme is derived for the Boussinesq flows, where the velocity and temperature fields are described independently. Kinetic boundary conditions for both velocity and temperature fields are also proposed. The proposed model is validated by simulating several canonical test cases, including the porous plate problem, the Rayleigh-b\\'{e}nard convection, and the natural convection with Rayleigh number up to $10^{10}$ in a square cavity. The results show that the coupled DUGKS is of second order accuracy in space and can well describe the convection phenomena from laminar to turbulent flows. Particularly, it is found that this new scheme has better numerical stability in simulating high Rayleigh number flows compared with the pre...

Wang, Peng; Guo, Zhaoli

2014-01-01

217

Accurate Finite Difference Methods for Option Pricing

Stock options are priced numerically using space- and time-adaptive finite difference methods. European options on one and several underlying assets are considered. These are priced with adaptive numerical algorithms including a second order method and a more accurate method. For American options we use the adaptive technique to price options on one stock with and without stochastic volatility. In all these methods emphasis is put on the control of errors to fulfill predefined tolerance level...

Persson, Jonas

2006-01-01

218

The numerical solution of problems in calculus of variation using Chebyshev finite difference method

International Nuclear Information System (INIS)

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

219

The numerical solution of problems in calculus of variation using Chebyshev finite difference method

Energy Technology Data Exchange (ETDEWEB)

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.

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

220

Energy Technology Data Exchange (ETDEWEB)

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.

Kim, S. [Purdue Univ., West Lafayette, IN (United States)

1994-12-31

221

Optimization of Dengue Epidemics: a test case with different discretization schemes

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.

Rodrigues, Helena Sofia; Torres, Delfim F M; 10.1063/1.3241345

2010-01-01

222

Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids

We present here a number of test cases and meshes which were designed to form a benchmark for finite volume schemes and give a summary of some of the results which were presented by the participants to this benchmark. We address a two-dimensional anisotropic diffusion problem, which is discretized on general, possibly non-conforming meshes. In most cases, the diffusion tensor is taken to be anisotropic, and at times heterogeneous and/or discontinuous. The meshes are either triangular or quadr...

Herbin, Raphaele; Hubert, Florence

2008-01-01

223

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy.

Romá, Federico; Cugliandolo, Leticia F.; Lozano, Gustavo S.

2014-08-01

224

Discrete unified gas kinetic scheme for all Knudsen number flows: II. Compressible case

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...

Guo, Zhaoli; Xu, Kun

2014-01-01

225

In this paper we present a multigrid approach to solve the Poisson equation in arbitrary domain (identified by a level set function) and mixed boundary conditions. The discretization is based on finite difference scheme and ghost-cell method. This multigrid strategy can be applied also to more general problems where a non-eliminated boundary condition approach is used. Arbitrary domain make the definition of the restriction operator for boundary conditions hard to find. A suitable restriction operator is provided in this work, together with a proper treatment of the boundary smoothing, in order to avoid degradation of the convergence factor of the multigrid due to boundary effects. Several numerical tests confirm the good convergence property of the new method.

Coco, Armando

2011-01-01

226

In this work, the implementation of a high-order compact finite-difference lattice Boltzmann method (CFDLBM) is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non-uniform grids. The incompressible form of the discrete Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates. Herein, the spatial derivatives in the resulting lattice Boltzmann (LB) equation in the computational plane are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge-Kutta scheme to provide an accurate and efficient incompressible flow solver. A high-order spectral-type low-pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions, flow non-linearities and grid non-uniformity. All boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates. The accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution. Four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are: unsteady Couette flow and steady flow in a 2-D cavity with non-uniform grid and steady and unsteady flows over a circular cylinder and the NACA0012 hydrofoil at different flow conditions. Results obtained for the above test cases are in good agreement with the existing numerical and experimental results. The study shows the present solution methodology based on the implementation of the high-order compact finite-difference Lattice Boltzmann method (CFDLBM) in the generalized curvilinear coordinates is robust, efficient and accurate for solving steady and unsteady incompressible flows over practical geometries.

Hejranfar, Kazem; Ezzatneshan, Eslam

2014-06-01

227

Adaptive High-Order Finite-Difference Method for Nonlinear Wave Problems

We discuss a scheme for the numerical solution of one-dimensional initial value problems exhibiting strongly localized solutions or finite-time singularities. To accurately and efficiently model such phenomena we present a full space-time adaptive scheme, based on a variable order spatial finite-difference scheme and a 4th order temporal integration with adaptively chosen time step. A wavelet analysis is utilized at regular intervals to adaptively select the order and the grid in accordance w...

Fatkullin, I.; Hesthaven, J. S.

2001-01-01

228

In this study, we examine the acoustic simulation method combining the wave equation finite difference time domain (WE FD-TD) method and compact FDs (CPFDs) for the second derivative. The wave equation compact finite difference time domain (WE CPFD-TD) method does not require calculation of the particle velocity; therefore, it can reduce the calculation time and memory used. Furthermore, for acceleration of simulation, we employ the recursive filtering algorithm and graphics processing unit (GPU) computing. As a result, we clarified that the three-dimensional WE CPFD-TD acoustic simulation using GPU is ca. 25-30 times faster than that using CPU with OpenMP.

Kawada, Naoki; Yoda, Takeshi; Tagawa, Norio; Tsuchiya, Takao; Okubo, Kan

2012-07-01

229

A Novel Robust Zero-Watermarking Scheme Based on Discrete Wavelet Transform

Directory of Open Access Journals (Sweden)

Full Text Available In traditional watermarking algorithms, the insertion of watermark into the original signal inevitably introduces some perceptible quality degradation. Another problem is the inherent conflict between imperceptibility and robustness. Zero-watermarking technique can solve these problems successfully. But most existing zero-watermarking algorithm for audio and image cannot resist against some signal processing manipulations or malicious attacks. In the paper, a novel audio zero-watermarking scheme based on discrete wavelet transform (DWT is proposed, which is more efficient and robust. The experiments show that the algorithm is robust against the common audio signal processing operations such as MP3 compression, re-quantization, re-sampling, low-pass filtering, cutting-replacement, additive white Gaussian noise and so on. These results demonstrate that the proposed watermarking method can be a suitable candidate for audio copyright protection.

Yu Yang

2012-08-01

230

TWO STAGE DISCRETE TIME EXTENDED KALMAN FILTER SCHEME FOR MICRO AIR VEHICLE

Directory of Open Access Journals (Sweden)

Full Text Available Navigation of Micro Air Vehicle (MAV is one of the most challenging areas of twenty first century’s research. Micro Air Vehicle (MAV is the miniaturized configuration of aircraft with a size of six inches in length and below the weight of hundred grams, which includes twenty grams of payload as well. Due to its small size, MAV is highly affected by the wind gust and therefore the navigation of Micro Air Vehicle (MAV is very important because precise navigation is a very basic step for the control of the Micro Air Vehicle (MAV. This paper presents two stage cascaded discrete time Extended Kalman Filter while using INS/GPS based navigation. First stage of this scheme estimates the Euler angles of Micro Air Vehicle (MAV whereas the second stage of this scheme estimates the position of Micro Air Vehicle (MAV in terms of height, longitude and latitude. As the system is considered as non-linear, so Extended Kalman Filter is used. On-board sensors in first stage included MEMS Gyro, MEMS Accelerometer, MEMS Magnetometer whereas second stage includes GPS.

Ali Usman

2012-03-01

231

Finite Difference Method of the Study on Radioactivities DispersionModeling in Environment of Ground

International Nuclear Information System (INIS)

It has been resulted the mathematics equation as model of constructingthe computer algorithm deriving from the transport equation having been theform of radionuclides dispersion in the environment of ground as a result ofdiffusion and advection process. The derivation of mathematics equation usedthe finite difference method into three schemes, the explicit scheme,implicit scheme and Crank-Nicholson scheme. The computer algorithm then wouldbe used as the basic of making the software in case of making a monitoringsystem of automatic radionuclides dispersion on the area around the nuclearfacilities. By having the three schemes, so it would be, in choosing thesoftware system, able to choose the more approximate with the fact. (author)

232

Calculating photonic Green's functions using a non-orthogonal finite difference time domain method

In this paper we shall propose a simple scheme for calculating Green's functions for photons propagating in complex structured dielectrics or other photonic systems. The method is based on an extension of the finite difference time domain (FDTD) method, originally proposed by Yee, also known as the Order-N method, which has recently become a popular way of calculating photonic band structures. We give a new, transparent derivation of the Order-N method which, in turn, enables us to give a simple yet rigorous derivation of the criterion for numerical stability as well as statements of charge and energy conservation which are exact even on the discrete lattice. We implement this using a general, non-orthogonal co-ordinate system without incurring the computational overheads normally associated with non-orthogonal FDTD. We present results for local densities of states calculated using this method for a number of systems. Firstly, we consider a simple one dimensional dielectric multilayer, identifying the suppres...

Ward, A J

1998-01-01

233

A novel variable-grid finite-difference method for fiber lasers

A novel finite-difference method with a variable discrete grid is proposed. The algorithm is applied to the traveling wave model for fiber lasers successfully. The simulation for the CW and the actively Q-switched output of Yb-doped fiber laser is compared with the reported results and the results simulated by a traditional constant-grid finite difference method. The comparisons prove that the new algorithm causes negligible adverse impacts, while the efficiency is largely improved. The novel algorithm presented in this paper can be optimized to be a promising method for solving other sorts of partial differential equations.

Liu, Zhen

2014-11-01

234

Two discretizations of a 9-velocity Boltzmann equation with a BGK collision operator are studied. A Chapman-Enskog expansion of the PDE system predicts that the macroscopic behavior corresponds to the incompressible Navier-Stokes equations with additional terms of order Mach number squared. We introduce a fourth-order scheme and compare results with those of the commonly used lattice Boltzmann discretization and with finite-difference schemes applied to the incompressible Na...

Reider, Marc B.; Sterling, James D.

1993-01-01

235

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.)

236

ON FINITE DIFFERENCES ON A STRING PROBLEM

Directory of Open Access Journals (Sweden)

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.

J. M. Mango

2014-01-01

237

High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations

We present high-order semi-discrete central-upwind numerical schemes for approximating solutions of multi-dimensional Hamilton-Jacobi (HJ) equations. This scheme is based on the use of fifth-order central interpolants like those developed in [1], in fluxes presented in [3]. These interpolants use the weighted essentially nonoscillatory (WENO) approach to avoid spurious oscillations near singularities, and become "central-upwind" in the semi-discrete limit. This scheme provides numerical approximations whose error is as much as an order of magnitude smaller than those in previous WENO-based fifth-order methods [2, 1]. Thee results are discussed via examples in one, two and three dimensions. We also pregnant explicit N-dimensional formulas for the fluxes, discuss their monotonicity and tl!e connection between this method and that in [2].

Bryson, Steve; Levy, Doron; Biegel, Bran R. (Technical Monitor)

2002-01-01

238

In this paper, a novel approach is introduced towards an efficient Finite-Difference Time-Domain (FDTD) algorithm by incorporating the Alternating Direction Implicit (ADI) technique to the Nonorthogonal FDTD (NFDTD) method. This scheme can be regarded as an extension of the conventional ADI-FDTD scheme into a generalized curvilinear coordinate system. The improvement on accuracy and the numerical efficiency of the ADI-NFDTD over the conventional nonorthogonal and the ADI-FDTD algorithms is ca...

Yang Hao; Wei Song

2010-01-01

239

A New Conservative Difference Scheme for the General Rosenau-RLW Equation

Directory of Open Access Journals (Sweden)

Full Text Available A new conservative finite difference scheme is presented for an initial-boundary value problem of the general Rosenau-RLW equation. Existence of its difference solutions are proved by Brouwer fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable, and second-order convergent. Numerical examples show the efficiency of the scheme.

Zhang Tian-De

2010-01-01

240

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.

Al-Mansoori, Saeed; Kunhu, Alavi

2013-10-01

241

In this paper, a stereo image watermarking scheme using discrete wavelet transform (DWT) and feature-based window maching algorithm(FMA) is proposed. A watermark data is embedded into the right image of a stereo image pair by using the DWT algorithm and disparity data is extracted from the left and watermarked right images. And then, disparity data and the left image are transmitted to the recipient through the communication channel. At the receiver, the watermarked right image is reconstructed from the received left image and disparity data by employing the FMA. From the difference between the watermarked and original right images, the embedded watermark image can be finally extracted. From experiments using the stereo image pair of 'Friends' and a watermark data of '3DRC', it is found that PSNR of the watermark image extracted from the reconstructed right image through the FMA and DWT algorithms can be increased up to 2.87dB, 2.58dB on average by comparing with those of the FMA and DCT algorithm when the quantizer scale(Q.S) is kept to be 16 and 20, respectively.

Hwang, Dong-Choon; Ko, Jung-Hwan; Park, Jae-Sung; Kim, Eun-Soo

2004-10-01

242

Approximate Lie Group Analysis of Finite-difference Equations

Approximate group analysis technique, that is, the technique combining the methodology of group analysis and theory of small perturbations, is applied to finite-difference equations approximating ordinary differential equations. Finite-difference equations are viewed as a system of algebraic equations with a small parameter, introduced through the definitions of finite-difference derivatives. It is shown that application of the approximate invariance criterion to this algebr...

Latypov, Azat M.

1995-01-01

243

Dynamic Rupture Simulation of Bending Faults With a Finite Difference Approach

Many questions about physical parameters governing the rupture propagation of earthquakes seem to find their answers within realistic dynamic considerations. Sophisticated constitutive relations based in laboratory experiments have lead to a better understanding of rupture evolution from its very beginning to its arrest. On the other hand, large amount of field observations as well as recent numerical simulations have also demonstrated the importance, in rupture growing, of considering more reasonable geological settings (e.g., bending and step-over fault geometries; heterogeneous surrounding media). So far, despite the development of powerful numerical tools, there still exist some numerical considerations that overstep their possibilities. Authors have solved the dynamic problem by applying the boundary integral equations method (BIEM) in order to explore the influence of fault geometry. This can be possible because of the fact that only the rupture path must be discretized, reducing the impact of numerical discretization. However, the BIEM needs the analytical solution of Green functions that can only be computed for a homogeneous space. Up to date, no interaction with heterogeneous structures can be taken in to account. In contrast, finite difference (FD) approaches have been widely used. In this case, due to the specific discretization of the elastodynamic equations through the entire domain, and the azimuthal anisotropy intrinsic to differential operators, only planar faults have been considered and numerical artefacts have to be carefully checked. In this work, we have used a recently proposed four-order staggered grid finite difference scheme to model in-plane (mode II) dynamic shear fracturing propagation with any pre-established geometry. In contrast with the classical 2-D staggered grid elementary cell in which all the elastic fields are defined in different positions (except the normal stresses), the stencil used here consider the velocity and stress fields separately in only two staggered grids. This permit an efficient treatment of boundary conditions to impose the shear stress drop in the nodes where the stresses are located. On the other hand, the stencil allows the four order Cartesian differential operators being decoupled into two different 45 degrees rotated operators. This procedure reduces numerical anisotropy along preferred directions and provides stable solutions for any fault orientation. Numerical solutions of dynamical fracture still exhibit large oscillations coming from local discretization effects and integration procedures. These perturbations can strongly alter the rupture front velocity and the average slip rate behind the crack tip. We controlled this phenomenon by applying a smoothing Laplacian operator to velocity equations. Such a mathematical tool, provided that suitable input parameters are supplied, helps to vanish these oscillations. Specifying a fault thickness in simulations yields similar results as we scale down numerical parameters, if the same fault geometry is kept. A simple definition of the fault is done placing it in the middle of the grid without using any adhoc numerical ghost plane often used in FD approaches. The fault is a sum of source points taken as close as possible to the hypothetical fault line. Simulations of irregular fault geometry (e.g., bending faults) are possible using the superposition technique. Spontaneous and velocity fixed rupture propagation will be presented with abrupt stress drop, as well as with time- and slip- weakening constitutive laws. Analysis of arbitrarily heterogeneous media surrounding the fault region in the dynamics of seismic sources evolution is possible.

Cruz-Atienza, V. M.; Virieux, J.; Operto, S.

2002-12-01

244

Scheme for measuring experimentally the velocity of pilot waves and the discreteness of time

International Nuclear Information System (INIS)

We consider the following two questions. Suppose that a quantum system suffers a change of the boundary condition or the potential at a given space location. Then (1)when will the wavefunction shows a response to this change at another location? And (2)how does the wavefunction changes?The answer to question (1) could reveal how a quantum system gets information on the boundary condition or the potential. Here we show that if the response takes place immediately, then it can allow superluminal signal transfer. Else if the response propagates in space with a finite velocity, then it could give a simple explanation why our world shows classicality on the macroscopic scale. Furthermore, determining the exact value of this velocity can either clarify the doubts on static experiments for testing Bell's inequality, or support the pilot-wave interpretation of quantum mechanics. We propose a feasible experimental scheme for measuring this velocity, which can be implemented with state-of-art technology, e.g., single-electron biprism interferometry.Question (2) is studied with a square-well potential model, and we find a paradox between the impossibility of superluminal signal transfer and the normalization condition of wavefunctions. To solve the paradox, we predict that when a change of the potential occurs at a given space location, the system will show no response to this change at all, until after a certain time interval. Otherwise either special relativity or quantum mech either special relativity or quantum mechanics will be violated. As a consequence, no physical process can actually happen within Planck time. Therefore it gives a simple proof that time is discrete, with Planck time being the smallest unit. Combining with the answer to question (1), systems with a larger size and a slower velocity could have a larger unit of time, making it possible to test the discreteness of time experimentally. Our result also sets a limit on the speed of computers, and gives instruction to the search of quantum gravity theories.

245

Improving sub-grid scale accuracy of boundary features in regional finite-difference models

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.

Panday, Sorab; Langevin, Christian D.

2012-01-01

246

Most of the localization algorithms in past decade are usually based on Monte Carlo, sequential monte carlo and adaptive monte carlo localization method. In this paper we proposed a new scheme called DQMCL which employs the antithetic variance reduction method to improve the localization accuracy. Most existing SMC and AMC based localization algorithm cannot be used in dynamic sensor network but DQMCL can work well even without need of static sensor network with the help of discrete power con...

Vasim Babu, M.; Ramprasad, Dr A. V.

2014-01-01

247

International Nuclear Information System (INIS)

A novel approach is presented in this paper for improving anisotropic diffusion PDE models, based on the Perona–Malik equation. A solution is proposed from an engineering perspective to adaptively estimate the parameters of the regularizing function in this equation. The goal of such a new adaptive diffusion scheme is to better preserve edges when the anisotropic diffusion PDE models are applied to image enhancement tasks. The proposed adaptive parameter estimation in the anisotropic diffusion PDE model involves self-organizing maps and Bayesian inference to define edge probabilities accurately. The proposed modifications attempt to capture not only simple edges but also difficult textural edges and incorporate their probability in the anisotropic diffusion model. In the context of the application of PDE models to image processing such adaptive schemes are closely related to the discrete image representation problem and the investigation of more suitable discretization algorithms using constraints derived from image processing theory. The proposed adaptive anisotropic diffusion model illustrates these concepts when it is numerically approximated by various discretization schemes in a database of magnetic resonance images (MRI), where it is shown to be efficient in image filtering and restoration applications

248

The fourth-order finite-difference scheme with fully implicit time-marching presently used to computationally study the spatial instability of planar Poiseuille flow incorporates a novel treatment for outflow boundary conditions that renders the buffer area as short as one wavelength. A semicoarsening multigrid method accelerates convergence for the implicit scheme at each time step; a line-distributive relaxation is developed as a robust fast solver that is efficient for anisotropic grids. Computational cost is no greater than that of explicit schemes, and excellent agreement with linear theory is obtained.

Liu, C.; Liu, Z.

1993-01-01

249

Approximate solutions to neutral type finite difference equations

Directory of Open Access Journals (Sweden)

Full Text Available In this article, we study the approximate solutions and the dependency of solutions on parameters to a neutral type finite difference equation, under a given initial condition. A fundamental finite difference inequality, with explicit estimate, is used to establish the results.

Deepak B. Pachpatte

2012-10-01

250

Investigation of Calculation Techniques of Finite Difference Method

Directory of Open Access Journals (Sweden)

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

Audrius Krukonis

2011-03-01

251

Directory of Open Access Journals (Sweden)

Full Text Available This paper deals with monotone finite difference iterative algorithms for solving nonlinear singularly perturbed reaction-diffusion problems of elliptic and parabolic types. Monotone domain decomposition algorithms based on a Schwarz alternating method and on box-domain decomposition are constructed. These monotone algorithms solve only linear discrete systems at each iterative step and converge monotonically to the exact solution of the nonlinear discrete problems. The rate of convergence of the monotone domain decomposition algorithms are estimated. Numerical experiments are presented.

Boglaev Igor

2006-01-01

252

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

253

Directory of Open Access Journals (Sweden)

Full Text Available This paper deals with a finite difference method for a wide class of weakly coupled nonlinear second-order partial differential systems with initial condition and weakly coupled nonlinear implicit boundary conditions. One part of each system is of the parabolic type (degenerated parabolic equations and the other of the elliptic type (equations with a parameter in a cube in $R^{1+n}$. A suitable finite difference scheme is constructed. It is proved that the scheme has a unique solution, and the numerical method is consistent, convergent and stable. The error estimate is given. Moreover, by the method, the differential problem has at most one classical solution. The proof is based on the Banach fixed-point theorem, the maximum principle for difference functional systems of the parabolic type and some new difference inequalities. It is a new technique of studying the mixed-type systems. Examples of physical applications and numerical experiments are presented.

Marian Malec

2007-01-01

254

Optimal variable-grid finite-difference modeling for porous media

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.

Liu, Xinxin; Yin, Xingyao; Li, Haishan

2014-12-01

255

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

256

Calculating photonic Green's functions using a non-orthogonal finite difference time domain method

In this paper we shall propose a simple scheme for calculating Green's functions for photons propagating in complex structured dielectrics or other photonic systems. The method is based on an extension of the finite difference time domain (FDTD) method, originally proposed by Yee, also known as the Order-N method, which has recently become a popular way of calculating photonic band structures. We give a new, transparent derivation of the Order-N method which, in turn, enable...

Ward, A. J.; Pendry, J. B.

1998-01-01

257

Explicit and implicit finite-difference methods for the diffusion equation in two dimensions

International Nuclear Information System (INIS)

In this paper we focus our attention on rotationally symmetric problems, where cylinder coordinates are suitable. For Cartesian grid arrangements finite-difference schemes for the diffusion equation in two spatial dimensions are introduced. The temporal evolution is determined by implicit and explicit techniques. In addition to exactly solvable diffusion model problems we present numerical results of simulation experiments of a diamond disc window. (orig.)

258

A MATLAB implementation of upwind finite differences and adaptive grids in the method of lines

In this paper, we report on the development of a MATLAB library for the solution of partial differential equation systems following the method of lines. In particular, we focus attention on upwind finite difference schemes and grid adaptivity, i.e., grid movement or grid refinement. Several algorithms are presented and their performance is demonstrated with illustrative examples including a fixed-bed reactor with periodic flow reversal, a model of flame propagation, and the Korteweg-de Vries equation.

Wouwer, A. Vande; Saucez, P.; Schiesser, W. E.; Thompson, S.

2005-11-01

259

Time-discretization scheme for quasi-static Maxwell's equations with a non-linear boundary condition

We study a time dependent eddy current equation for the magnetic field H accompanied with a non-linear degenerate boundary condition (BC), which is a generalization of the classical Silver-Müller condition for a non-perfect conductor. More exactly, the relation between the normal components of electrical E and magnetic H fields obeys the following power law [nu]×E=[nu]×(H×[nu][alpha]-1H×[nu]) for some [alpha][set membership, variant](0,1]. We establish the existence and uniqueness of a weak solution in a suitable function space under the minimal regularity assumptions on the boundary [Gamma] and the initial data H0. We design a non-linear time discrete approximation scheme based on Rothe's method and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time-discretization.

Slodicka, Marián; Zemanová, Viera

2008-07-01

260

Using the methods of finite difference equations the discrete analogue of the parabolic and catenary cable are analysed. The fibonacci numbers and the golden ratio arise in the treatment of the catenary.

Peters, James V.

2004-01-01

261

Techniques for correcting approximate finite difference solutions. [considering transonic flow

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.

Nixon, D.

1978-01-01

262

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

263

Generalized coarse-mesh finite difference acceleration for the method of characteristics - 068

International Nuclear Information System (INIS)

Based on generalized coarse mesh re-balance (GCMR) method, this paper proposes a new acceleration method for the method of characteristics (MOC) in unstructured meshes: the generalized coarse-mesh finite difference (GCMFD) method. The GCMFD method, which applies equivalent width of coarse mesh to establish the finite difference discretization, can use unstructured coarse meshes composed of adjacent fine meshes to speed up the MOC iteration. The convergence property of the GCMFD method is controlled by width factor. However, the optimal width factor cannot be given a priori. Method of adjusting width factor automatically is proposed in this paper. Finally, the GCMFD method is adopted in the 3-D neutron transport MOC code TCM. Numerical tests show that the GCMFD, using generalized-geometry coarse meshes, can accelerate the MOC iteration with good speedup. (authors)

264

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

265

Discrete Mechanics and Optimal Control: an Analysis

The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationsh...

Ober-Bloebaum, S; Marsden, J E

2008-01-01

266

In the present paper, a method to reduce the computational cost associated with solving a nonlinear transient heat conduction problem is presented. The proposed method combines the ideas of two level discretization and the multilevel time integration schemes with the proper orthogonal decomposition model order reduction technique. The accuracy and the computational efficiency of the proposed methods is discussed. Several numerical examples are presented for validation of the approach. Compared to the full finite element model, the proposed method significantly reduces the computational time while maintaining an acceptable level of accuracy.

Gaonkar, A. K.; Kulkarni, S. S.

2015-01-01

267

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

268

Directory of Open Access Journals (Sweden)

Full Text Available In this paper, a novel approach is introduced towards an efficient Finite-Difference Time-Domain (FDTD algorithm by incorporating the Alternating Direction Implicit (ADI technique to the Nonorthogonal FDTD (NFDTD method. This scheme can be regarded as an extension of the conventional ADI-FDTD scheme into a generalized curvilinear coordinate system. The improvement on accuracy and the numerical efficiency of the ADI-NFDTD over the conventional nonorthogonal and the ADI-FDTD algorithms is carried out by numerical experiments. The application in the modelling of the Electromagnetic Bandgap (EBG structure has further demonstrated the advantage of the proposed method.

Yang Hao

2010-06-01

269

Gain from a mixed finite-difference formulation for three-dimensional diffusion-theory neutronics

International Nuclear Information System (INIS)

The advantage of a mixed differencing scheme for representing the diffusion theory approximation to neutron transport in three-dimensional triangular-Z geometry is demonstrated for a fast reactor. Most of the early codes employed the mesh edge difference formulation as is used in the German D3E code. A mesh centered formulation was chosen for use on a routine basis with mesh points located at the centers of the finite difference elements instead of at the corners where the internal material interfaces intersect, the VENURE code being the latest to use this scheme. Results are presented for a fast reactor core problem modeling hexagonal assemblies

270

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.

Luneva, Maria; Holt, Jason; Harle, James; Liu, Hedong

2013-04-01

271

In this paper, we present a versatile mathematical formulation of a newly developed 3-D locally conformal Finite Difference (FD) thermal algorithm developed specificallyfor coupled electromagnetic (EM) and heat diffusion simulations utilizing Overlapping Grids (OGFD) in the Cartesian and cylindrical coordinate systems. The motivation for this research arises from an attempt to characterize the dominant thermal transport phenomena typically encountered during the process cycle of a high-power, microwave-assisted material processing system employing a geometrically composite cylindrical multimode heating furnace. The cylindrical FD scheme is only applied to the outer shell of the housing cavity whereas the Cartesian FD scheme is used to advance the temperature elsewhere including top and bottom walls, and most of the inner region of the cavity volume. The temperature dependency of the EM constitutive and thermo-physical parameters of the material being processed is readily accommodated into the OGFD update equations. The time increment, which satisfies the stability constraint of the explicit OGFD time-marching scheme, is derived. In a departure from prior work, the salient features of the proposed algorithm are first, the locally conformal discretization scheme accurately describes the diffusion of heat and second, significant heat-loss mechanisms usually encountered in microwave heating problems at the interfacial boundary temperature nodes have been considered. These include convection and radiation between the surface of the workload and air inside the cavity, heat convection and radiation between the inner cavity walls and interior cavity volume, and free cooling of the outermost cavity walls. PMID:16673831

Al-Rizzo, Hussain M; Tranquilla, Jim M; Feng, Ma

2005-01-01

272

Hybrid semi-Lagrangian finite element-finite difference methods for the Vlasov equation

In this paper, we propose a new conservative hybrid finite element-finite difference method for the Vlasov equation. The proposed methodology uses Strang splitting to decouple the nonlinear high dimensional Vlasov equation into two lower dimensional equations, which describe spatial advection and velocity acceleration/deceleration processes respectively. We then propose to use a semi-Lagrangian (SL) discontinuous Galerkin (DG) scheme (or Eulerian Runge-Kutta (RK) DG scheme with local time stepping) for spatial advection, and use a SL finite difference WENO for velocity acceleration/deceleration. Such hybrid method takes the advantage of DG scheme in its compactness and its ability in handling complicated spatial geometry; while takes the advantage of the WENO scheme in its robustness in resolving filamentation solution structures of the Vlasov equation. The proposed highly accurate methodology enjoys great computational efficiency, as it allows one to use relatively coarse phase space mesh due to the high order nature of the scheme. At the same time, the time step can be taken to be extra large in the SL framework. The quality of the proposed method is demonstrated via basic test problems, such as linear advection and rigid body rotation, and classical plasma problems, such as Landau damping and the two stream instability. Although we only tested 1D1V examples, the proposed method has the potential to be extended to problems with high spatial dimensions and complicated geometry. This constitutes our future research work.

Guo, Wei; Qiu, Jing-Mei

2013-02-01

273

In this paper we present a new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation. The scheme uses a symmetrical multi-point difference formula to represent the partial differentials of the two-dimensional variables, which can improve the accuracy of the numerical solutions to the order of ?x when a (2N+1)-point formula is used for any positive integer N with ?x=?y, while N=1 equivalent to the traditional scheme. On the other hand, the new scheme keeps the same form of the traditional matrix equation so that the standard algebraic eigenvalue algorithm with a real, symmetric, large sparse matrix is still applicable. Therefore, for the same dimension, only a little more CPU time than the traditional one should be used for diagonalizing the matrix. The numerical examples of the two-dimensional harmonic oscillator and the two-dimensional Henon-Heiles potential demonstrate that by using the new method, the error in the numerical solutions can be reduced steadily and extensively through the increase of N, which is more efficient than the traditional methods through the decrease of the step size.

Wang, Zhongcheng; Shao, Hezhu

2009-06-01

274

3D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids

We present a number of test cases and meshes that were designed as a benchmark for numerical schemes dedicated to the approximation of three-dimensional anisotropic and heterogeneous diffusion problems. These numerical schemes may be applied to general, possibly non conforming, meshes composed of tetrahedra, hexahedra and quite distorted general polyhedra. A number of methods were tested among which conforming ?nite element methods, discontinuous Galerkin ?nite element methods, cell-cente...

Eymard, Robert; Henry, Ge?rard; Herbin, Rapahele; Hubert, Florence; Klofkorn, Robert; Manzini, Gianmarco

2011-01-01

275

High order Finite difference Constrained Transport Method for Ideal Magnetohydrodynamic Equations

A new algorithm will be used to simulate the ideal MHD equations based on AMR algorithm and a high order finite difference WENO reconstruction method [Shen, C., Qiu, J.M., Christlieb, A. J., Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations, JCP (2011)]. The base framework of the algorithm will be the magnetic potential advection constrained transport method (MPACT), which was originally a 2nd-order finite volume type solver for ideal MHD equations by Rossmanith, J. [An unstaggered, High-Resolution Constrained Transport Method For Magnetohydrodynamic Flows, SIAM (2006)], but the treatment is significantly different. The important feature of the new algorithm will be (1) the method is finite difference type, (2) all quantities are treated as cell-centered, (3) high order (higher than 2nd) in both time and space, (4) all the quantities are non-oscillatory, (5) AMR will be used as the base framework. Convergence study will be done on the smooth problem. More 1D/2D/2.5D benchmark problems such as Brio-Wu shock tube, Rotor problem and Cloud-shock interaction will be simulated. We expect our algorithm is robust, non-oscillatory and good at resolving solution structures, due to the very low numerical diffusion of high order scheme.

Tang, Qi; Christlieb, Andrew; Guclu, Yaman; Rossmanith, James

2012-10-01

276

Improvement of the finite difference lattice Boltzmann method for low mach number flows

International Nuclear Information System (INIS)

This paper presents a numerical method to compute flow-acoustic resonance at low Mach number within a reasonable computing time. Light water reactors have experienced flow-acoustic resonance which is attributed to unsteady compressible flows at low Mach number. This phenomenon is undesirable because the induced sound causes loud noise and vibrations of the mechanical structures. However, a numerical simulation of this flow-acoustic resonance at low Mach number requires a large computing time and a highly accurate method in order to simultaneously compute flows and acoustic waves. The finite difference lattice Boltzmann method which is a powerful tool for obtaining computational fluid dynamics has high accuracy for simultaneous calculation of flows and acoustic waves. It becomes an efficient method to compute low Mach number flow if the computing time is shortened. In this paper, the finite difference lattice Boltzmann method was sped up. Three improvements were proposed: development of a new particle model, modification of the governing equation, and employment of an efficient time marching scheme. The computing time of the proposed finite difference lattice Boltzmann model was compared with the conventional finite difference lattice Boltzmann model for the calculation of the cubic cavity flow. The results showed that the computing time of the proposed model is 30% of the time needed by the conventional finite difference lattice Boltzmann model. The flow-acoustic resoce Boltzmann model. The flow-acoustic resonance at low Mach number at the side branch was calculated using the proposed model. The numerical results showed quantitative agreement with the experimental data. (author)

277

Directory of Open Access Journals (Sweden)

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.

A. Caserta

1998-06-01

278

Finite-difference synthetic seismograms for SH waves

The research performed under this contract, during the period 12 January 1983 through 11 January 1984, can be divided into two major topics: finite-difference synthetic seismograms for SH waves; and array analysis of the ground motions from the 1971 San Fernando Earthquake. In Section 2, the accuracy and ease of application of the finite-difference method for generating synthetic seismograms of SH wave propagation in cylindrically symmetric media is discussed. The finite-difference method was the advantage that arbitrary density and velocity field in the medium may be specified. A point source is generated by a simple transformation of a line source. The accuracy of the finite-difference seismograms in flat layered media is confirmed by comparison with the Cagniard De-Hoop method. The finite-difference seismograms also agree with a previously untried dipping-layer Cagniard method. An earthquake radiation pattern is approximated by introducing a near-field which has permanent displacement near the source.

Vidale, J.; Helmberger, D. V.; Clayton, R.

1984-08-01

279

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 ...

Hai Fang; Quan Zhou; Kaijia Li

2013-01-01

280

On the spectral properties of random finite difference operators

International Nuclear Information System (INIS)

We study a class of random finite difference operators, a typical example of which is the finite difference Schroedinger operator with a random potential which arises in solid state physics in the tight binding approximation. We obtain with probability one, in various situations, the exact location of the spectrum, and criterions for a given part in the spectrum to be pure point or purely continuous, or for the static electric conductivity to vanish. A general formalism is developped which transforms the study of these random operators into that of the asymptotics of a multiple integral constructed from a given recipe. Finally we apply our criterions and formalism to prove that, with probability one, the one-dimensional finite difference Schroedinger operator with a random potential has pure point spectrum and developps no static conductivity. (orig.)

281

Higher-order finite-difference formulation of periodic Orbital-free Density Functional Theory

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...

Ghosh, Swarnava

2014-01-01

282

A high-order finite-difference frequency domain method is proposed for the analysis of the band diagrams of two-dimensional photonic crystals. This improved formulation is based on Taylor series expansion, local coordinate transformation, boundary conditions matching, and the generalized Douglas scheme. The nine-point second-order formulas are extended to fourth-order accuracy. This proposed scheme can deal with piecewise homogeneous structures with curved dielectric interfaces. PMID:19259167

Chiang, Yen-Chung

2009-03-01

283

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. ...

Srivastava, Vineet K.; Tamsir, M.; Awasthi, Mukesh K.; Sarita Singh

2014-01-01

284

An assessment of semi-discrete central schemes for hyperbolic conservation laws

International Nuclear Information System (INIS)

High-resolution finite volume methods for solving systems of conservation laws have been widely embraced in research areas ranging from astrophysics to geophysics and aero-thermodynamics. These methods are typically at least second-order accurate in space and time, deliver non-oscillatory solutions in the presence of near discontinuities, e.g., shocks, and introduce minimal dispersive and diffusive effects. High-resolution methods promise to provide greatly enhanced solution methods for Sandia's mainstream shock hydrodynamics and compressible flow applications, and they admit the possibility of a generalized framework for treating multi-physics problems such as the coupled hydrodynamics, electro-magnetics and radiative transport found in Z pinch physics. In this work, we describe initial efforts to develop a generalized 'black-box' conservation law framework based on modern high-resolution methods and implemented in an object-oriented software framework. The framework is based on the solution of systems of general non-linear hyperbolic conservation laws using Godunov-type central schemes. In our initial efforts, we have focused on central or central-upwind schemes that can be implemented with only a knowledge of the physical flux function and the minimal/maximal eigenvalues of the Jacobian of the flux functions, i.e., they do not rely on extensive Riemann decompositions. Initial experimentation with high-resolution central schemes suggests that contact discontinuitiesemes suggests that contact discontinuities with the concomitant linearly degenerate eigenvalues of the flux Jacobian do not pose algorithmic difficulties. However, central schemes can produce significant smearing of contact discontinuities and excessive dissipation for rotational flows. Comparisons between 'black-box' central schemes and the piecewise parabolic method (PPM), which relies heavily on a Riemann decomposition, shows that roughly equivalent accuracy can be achieved for the same computational cost with both methods. However, PPM clearly outperforms the central schemes in terms of accuracy at a given grid resolution and the cost of additional complexity in the numerical flux functions. Overall we have observed that the finite volume schemes, implemented within a well-designed framework, are extremely efficient with (potentially) very low memory storage. Finally, we have found by computational experiment that second and third-order strong-stability preserving (SSP) time integration methods with the number of stages greater than the order provide a useful enhanced stability region. However, we observe that non-SSP and non-optimal SSP schemes with SSP factors less than one can still be very useful if used with time-steps below the standard CFL limit. The 'well-designed' integration schemes that we have examined appear to perform well in all instances where the time step is maintained below the standard physical CFL limit

285

An assessment of semi-discrete central schemes for hyperbolic conservation laws.

Energy Technology Data Exchange (ETDEWEB)

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.

Christon, Mark Allen; Robinson, Allen Conrad; Ketcheson, David Isaac

2003-09-01

286

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)

287

We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection-diffusion-reaction equations in two or three space dimensions. We discretize the time evolution, convection, reaction, and sources terms on a given grid, which can be nonmatching and can contain nonconvex elements, by means of the cell-centered finite volume method. To discretize the diffusion term, we construct a conforming simplicial mesh with the vertices given by the original grid and use the finite el...

Eymard, R.; Hilhorst, D.; Vohralik, M.

2009-01-01

288

In this paper an improved finite volume scheme to discretize diffusive flux on a non-orthogonal mesh is proposed. This approach, based on an iterative technique initially suggested by Khosla [P.K. Khosla, S.G. Rubin, A diagonally dominant second-order accurate implicit scheme, Computers and Fluids 2 (1974) 207-209] and known as deferred correction, has been intensively utilized by Muzaferija [S. Muzaferija, Adaptative finite volume method for flow prediction using unstructured meshes and multigrid approach, Ph.D. Thesis, Imperial College, 1994] and later Fergizer and Peric [J.H. Fergizer, M. Peric, Computational Methods for Fluid Dynamics, Springer, 2002] to deal with the non-orthogonality of the control volumes. Using a more suitable decomposition of the normal gradient, our scheme gives accurate solutions in geometries where the basic idea of Muzaferija fails. First the performances of both schemes are compared for a Poisson problem solved in quadrangular domains where control volumes are increasingly skewed in order to test their robustness and efficiency. It is shown that convergence properties and the accuracy order of the solution are not degraded even on extremely skewed mesh. Next, the very stable behavior of the method is successfully demonstrated on a randomly distorted grid as well as on an anisotropically distorted one. Finally we compare the solution obtained for quadrilateral control volumes to the ones obtained with a finite element code and with an unstructured version of our finite volume code for triangular control volumes. No differences can be observed between the different solutions, which demonstrates the effectiveness of our approach.

Traoré, Philippe; Ahipo, Yves Marcel; Louste, Christophe

2009-08-01

289

Serpentine: Finite Difference Methods for Wave Propagation in Second Order Formulation

International Nuclear Information System (INIS)

Wave propagation phenomena are important in many DOE applications such as nuclear explosion monitoring, geophysical exploration, estimating ground motion hazards and damage due to earthquakes, non-destructive testing, underground facilities detection, and acoustic noise propagation. There are also future applications that would benefit from simulating wave propagation, such as geothermal energy applications and monitoring sites for carbon storage via seismic reflection techniques. In acoustics and seismology, it is of great interest to increase the frequency bandwidth in simulations. In seismic exploration, greater frequency resolution enables shorter wave lengths to be included in the simulations, allowing for better resolution in the seismic imaging. In nuclear explosion monitoring, higher frequency seismic waves are essential for accurate discrimination between explosions and earthquakes. When simulating earthquake induced motion of large structures, such as nuclear power plants or dams, increased frequency resolution is essential for realistic damage predictions. Another example is simulations of micro-seismic activity near geothermal energy plants. Here, hydro-fracturing induces many small earthquakes and the time scale of each event is proportional to the square root of the moment magnitude. As a result, the motion is dominated by higher frequencies for smaller seismic events. The above wave propagation problems are all governed by systems of hyperbolic partial governed by systems of hyperbolic partial differential equations in second order differential form, i.e., they contain second order partial derivatives of the dependent variables. Our general research theme in this project has been to develop numerical methods that directly discretize the wave equations in second order differential form. The obvious advantage of working with hyperbolic systems in second order differential form, as opposed to rewriting them as first order hyperbolic systems, is that the number of differential equations in the second order system is significantly smaller. Another issue with re-writing a second order system into first order form is that compatibility conditions often must be imposed on the first order form. These (Saint-Venant) conditions ensure that the solution of the first order system also satisfies the original second order system. However, such conditions can be difficult to enforce on the discretized equations, without introducing additional modeling errors. This project has previously developed robust and memory efficient algorithms for wave propagation including effects of curved boundaries, heterogeneous isotropic, and viscoelastic materials. Partially supported by internal funding from Lawrence Livermore National Laboratory, many of these methods have been implemented in the open source software WPP, which is geared towards 3-D seismic wave propagation applications. This code has shown excellent scaling on up to 32,768 processors and has enabled seismic wave calculations with up to 26 Billion grid points. TheWPP calculations have resulted in several publications in the field of computational seismology, e.g.. All of our current methods are second order accurate in both space and time. The benefits of higher order accurate schemes for wave propagation have been known for a long time, but have mostly been developed for first order hyperbolic systems. For second order hyperbolic systems, it has not been known how to make finite difference schemes stable with free surface boundary conditions, heterogeneous material properties, and curvilinear coordinates. The importance of higher order accurate methods is not necessarily to make the numerical solution more accurate, but to reduce the computational cost for obtaining a solution within an acceptable error tolerance. This is because the accuracy in the solution can always be improved by reducing the grid size h. However, in practice, the available computational resources might not be large enough to solve the problem with a low order method.

290

In this study we examined a fast and efficient calculation method for compact finite-differences (FDs). The proposed method employs a recursive filter (RF) method. This method is based on z-transformation and its RF algorithm. Analysis of sound propagation using compact FDs with the RF method was implemented. Using the RF algorithm, we can implement compact finite difference time domain (FD-TD) analysis as a simpler code than a conventional technique using LU factorization, i.e., the Tomas method. The results obtained in this study clarified that the RF method reduces calculation time required for compact FDs.

Yoda, Takeshi; Okubo, Kan; Tagawa, Norio; Tsuchiya, Takao

2011-07-01

291

Finite difference time domain modelling of particle accelerators

International Nuclear Information System (INIS)

Finite Difference Time Domain (FDTD) modelling has been successfully applied to a wide variety of electromagnetic scattering and interaction problems for many years. Here the method is extended to incorporate the modelling of wake fields in particle accelerators. Algorithmic comparisons are made to existing wake field codes, such as MAFIA T3. 9 refs., 7 figs

292

HEATING-7, Multidimensional Finite-Difference Heat Conduction Analysis

International Nuclear Information System (INIS)

1 - Description of program or function: HEATING 7.2i and 7.3 are the most recent developments in a series of heat-transfer codes and obsolete all previous versions distributed by RSICC as SCA-1/HEATING5 and PSR-199/HEATING 6. Note that Unix and PC versions of HEATING7 are available in the CCC-545/SCALE 4.4 package. HEATING can solve steady-state and/or transient heat conduction problems in one-, two-, or three-dimensional Cartesian, cylindrical, or spherical coordinates. A model may include multiple materials, and the thermal conductivity, density, and specific heat of each material may be both time- and temperature-dependent. The thermal conductivity may also be anisotropic. Materials may undergo change of phase. Thermal properties of materials may be input or may be extracted from a material properties library. Heat- generation rates may be dependent on time, temperature, and position, and boundary temperatures may be time- and position-dependent. The boundary conditions, which may be surface-to-environment or surface-to-surface, may be specified temperatures or any combination of prescribed heat flux, forced convection, natural convection, and radiation. The boundary condition parameters may be time- and/or temperature-dependent. General gray body radiation problems may be modeled with user-defined factors for radiant exchange. The mesh spacing may be variable along each axis. HEATING uses a run-time memory allocation scheme to avoid having to recompile to match mee to avoid having to recompile to match memory requirements for each specific problem. HEATING utilizes free-form input. In June 1997 HEATING 7.3 was added to the HEATING 7.2i packages, and the Unix and PC versions of both 7.2i and 7.3 were merged into one package. HEATING 7.3 is being released as a beta-test version; therefore, it does not entirely replace HEATING 7.2i. There is no published documentation for HEATING 7.3; but a listing of input specifications, which reflects changes for 7.3, is included in the PSR-199 documentation. For 3-D problems, surface fluxes may be plotted with H7TECPLOT which requires the proprietary software TECPLOT. HEATING 7.3 runs under Windows95 and WindowsNT on PC's. No future modifications are planned for HEATING7. See README.1ST for more information. 2 - Method of solution: Three steady-state solution techniques are available: point-successive over-relaxation iterative method with extrapolation, direct-solution (for one-dimensional or two-dimensional problems), and conjugate gradient. Transient problems may be solved using any one of several finite-difference schemes: Crank-Nicolson implicit, Classical Implicit Procedure (CIP), Classical Explicit Procedure (CEP), or Levy explicit method (which for some circumstances allows a time step greater than the CEP stability criterion.) The solution of the system of equations arising from the implicit techniques is accomplished by point-successive over-relaxation iteration and includes procedures to estimate the optimum acceleration parameter. 3 - Restrictions on the complexity of the problem: All surfaces in a model must be parallel to one of the coordinate axes which makes modeling complex geometries difficult. Transient change of phase problems can only be solved with one of the explicit techniques - an implicit change-of-phase capability has not been implemented

293

International Nuclear Information System (INIS)

In this paper, a H-infinity fault detection and diagnosis (FDD) scheme for a class of discrete nonlinear system fault using output probability density estimation is presented. Unlike classical FDD problems, the measured output of the system is viewed as a stochastic process and its square root probability density function (PDF) is modeled with B-spline functions, which leads to a deterministic space-time dynamic model including nonlinearities, uncertainties. A weighting mean value is given as an integral function of the square root PDF along space direction, which leads a function only about time and can be used to construct residual signal. Thus, the classical nonlinear filter approach can be used to detect and diagnose the fault in system. A feasible detection criterion is obtained at first, and a new H-infinity adaptive fault diagnosis algorithm is further investigated to estimate the fault. Simulation example is given to demonstrate the effectiveness of the proposed approaches.

294

An exact expression of steady discrete shocks was recently obtained by the author in [9] for a class of residual-based compact schemes (RBC) applied to the inviscid Bürgers equation in a finite domain. Following the same lines, the analysis is extended to an infinite domain for a scalar conservation law with a general convex flux. For the dissipative high-order schemes considered, discrete shocks in infinite domain or with boundary conditions at short distance (Rankine-Hugoniot relations) are found to be very close. Besides, the present analytical description of shock capturing in infinite domain is explicit and so simple that it could lead to a new approach for correcting parasitic oscillations of high order RBC schemes. In a second part of the paper, exact solutions are also derived for equivalent differential equations (EDE) approximating RBC2p-1 schemes (subscript denotes the accuracy order) at orders 2p and 2p+1. Although EDE involves Taylor expansions around steep structures, agreement between the exact EDE shock-profiles and the discrete shocks is remarkably good for RBC5 and RBC7 schemes. In addition, a strong similarity is demonstrated between the analytical expressions of discrete shocks and EDE shock profiles.

Lerat, Alain

2014-09-01

295

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

296

Summation by parts operators for finite difference approximations of second derivatives

International Nuclear Information System (INIS)

Finite difference operators approximating second derivatives and satisfying a summation by parts rule have been derived for the fourth, sixth and eighth order case by using the symbolic mathematics software Maple. The operators are based on the same norms as the corresponding approximations of the first derivative, which makes the construction of stable approximations to general parabolic problems straightforward. The error analysis shows that the second derivative approximation can be closed at the boundaries with an approximation two orders less accurate than the internal scheme, and still preserve the internal accuracy

297

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.

Tadghighi, Hormoz; Hassan, Ahmed A.; Charles, Bruce

1990-01-01

298

Numerical simulation of shock wake propagation using the finite difference lattice Boltzmann method

International Nuclear Information System (INIS)

The shock wave process represents an abrupt change in fluid properties, in which finite variations in pressure, temperature, and density occur over the shock thickness which is comparable to the mean free path of the gas molecules involved. This shock wave fluid phenomenon is simulated by using the Finite Difference Lattice Boltzmann Method (FDLBM). In this paper, a new model is proposed using the lattice BGK compressible fluid model in FDLBM for the purpose of speeding up the calculation as well as stabilizing the numerical scheme. The numerical results of the proposed model show good agreement with the theoretical predictions

299

Directory of Open Access Journals (Sweden)

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.

Hai Fang

2013-11-01

300

Replica-symmetry breaking: discrete and continuous schemes in the Sherrington-Kirkpatrick model

International Nuclear Information System (INIS)

We study hierarchies of replica-symmetry-breaking solutions of the Sherrington-Kirkpatrick model. Stationarity equations for order parameters of solutions with an arbitrary number of hierarchies are set and the limit to infinite number of hierarchical levels is discussed. In particular, we demonstrate how the continuous replica-symmetry breaking scheme of Parisi emerges and how the limit to infinite-many hierarchies leads to equations for the order-parameter function of the continuous solution. The general analysis is accompanied by an explicit asymptotic solution near the de Almeida-Thouless instability line in the nonzero magnetic field

301

Discrete Wavelet Transform Method: A New Optimized Robust Digital Image Watermarking Scheme

In this paper, a wavelet-based logo watermarking scheme is presented. The logo watermark is embedded into all sub-blocks of the LLn sub-band of the transformed host image, using quantization technique. Extracted logos from all sub-blocks are mixed to make the extracted watermark from distorted watermarked image. Knowing the quantization step-size, dimensions of logo and the level of wavelet transform, the watermark is extracted, without any need to have access to the original image. Robustnes...

Hassan Talebi; Behzad Poursoleyman

2012-01-01

302

Higher order rotated iterative scheme for the 2D Helmholtz equation

Improved techniques derived from the rotated finite difference operators have been developed over the last few years in solving the linear systems that arise from the discretization of various partial differential equations (PDEs). Furthermore, a higher order system can be generated from discretization of the finite difference scheme using the fourth order compact scheme generated from the second order central difference. By using compact finite differences, a new rotated point scheme with fourth-order accuracy for the two-dimensional (2D) Helmholtz equation is formed. On the other hand, the multiscale multigrid method combined with Richardson's extrapolation is first introduced by Zhang to solve the 2D Poisson equation. By combining the fourth-order rotated scheme and multiscale multigrid method with Richardson's extrapolation in the solution of the 2D Helmholtz equation, the order of accuracy of the approximation can be improved up to sixth order, and with larger mesh size, the convergence rate of these iterative methods is faster as well. Numerical experiments are conducted on the rotated scheme combined with multiscale multigrid method and Richardson's extrapolation, and the result is compared with existing point methods and multigrid method. The results show the improvements in the convergence rate and the efficiency of the newly formulated iterative scheme.

Ping, Teng Wai; Ali, Norhashidah Hj. Mohd.

2014-07-01

303

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.

Gladden, Herbert J.; Ko, Ching L.; Boddy, Douglas E.

1995-01-01

304

Discrete Wavelet Transform Method: A New Optimized Robust Digital Image Watermarking Scheme

Directory of Open Access Journals (Sweden)

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.

Hassan Talebi

2012-11-01

305

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.

Franzè, Giuseppe; Lucia, Walter; Tedesco, Francesco

2014-12-01

306

A three-dimensional fourth-order finite-difference time-domain (FDTD) program with a symplectic integrator scheme has been developed to solve the problem of light scattering by small particles. The symplectic scheme is nondissipative and requires no more storage than the conventional second-order FDTD scheme. The total-field and scattered-field technique is generalized to provide the incident wave source conditions in the symplectic FDTD (SFDTD) scheme. The perfectly matched layer absorbing boundary condition is employed to truncate the computational domain. Numerical examples demonstrate that the fourth-order SFDTD scheme substantially improves the precision of the near-field calculation. The major shortcoming of the fourth-order SFDTD scheme is that it requires more computer CPU time than a conventional second-order FDTD scheme if the same grid size is used. Thus, to make the SFDTD method efficient for practical applications, one needs to parallelize the corresponding computational code. PMID:15813268

Zhai, Peng-Wang; Kattawar, George W; Yang, Ping; Li, Changhui

2005-03-20

307

Asymptotic analysis of numerical wave propagation in finite difference equations

An asymptotic technique is developed for analyzing the propagation and dissipation of wave-like solutions to finite difference equations. It is shown that for each fixed complex frequency there are usually several wave solutions with different wavenumbers and the slowly varying amplitude of each satisfies an asymptotic amplitude equation which includes the effects of smoothly varying coefficients in the finite difference equations. The local group velocity appears in this equation as the velocity of convection of the amplitude. Asymptotic boundary conditions coupling the amplitudes of the different wave solutions are also derived. A wavepacket theory is developed which predicts the motion, and interaction at boundaries, of wavepackets, wave-like disturbances of finite length. Comparison with numerical experiments demonstrates the success and limitations of the theory. Finally an asymptotic global stability analysis is developed.

Giles, M.; Thompkins, W. T., Jr.

1983-01-01

308

Variational integrators on fractional Lagrangian systems in the framework of discrete embedddings

In this paper, we introduce the notion of discrete embedding which is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. We first define the Gauss finite differences embedding. In this setting, we study variational integrator on classical Lagrangian systems. Finally, we extend these constructions to the fractional case. In particular, we define the Gauss Gr\\"unwald-Letnikov embedding and the corresponding variational integrator on fractional Lagrangian systems.

Bourdin, Loïc; Greff, Isabelle; Inizan, Pierre

2011-01-01

309

Finite-difference modeling experiments for seismic interferometry :

In passive seismic interferometry, new reflection data can be retrieved by crosscorrelating recorded noise data. The quality of the retrieved reflection data is, among others, dependent on the duration and number of passive sources present during the recording time, the source distribution, and the source strength. To investigate these relations we set up several numerical modeling studies. To carry out the modeling in a feasible time, we design a finite-difference algorithm for the simulatio...

Thorbecke, J. W.; Draganov, D.

2011-01-01

310

Non-finite-difference algorithm for integrating Newton's motion equations

We have presented some practical consequences on the molecular-dynamics simulations arising from the numerical algorithm published recently in paper Int. J. Mod. Phys. C 16, 413 (2005). The algorithm is not a finite-difference method and therefore it could be complementary to the traditional numerical integrating of the motion equations. It consists of two steps. First, an analytic form of polynomials in some formal parameter $\\lambda$ (we put $\\lambda=1$ after all) is deriv...

Brzostowski, B.; Dudek, M. R.; Grabiec, B.; Nadzieja, T.

2007-01-01

311

Finite difference time domain calculations of antenna mutual coupling

The Finite Difference Time Domain (FDTD) technique has been applied to a wide variety of electromagnetic analysis problems, including shielding and scattering. However, the method has not been extensively applied to antennas. In this short paper calculations of self and mutual admittances between wire antennas are made using FDTD and compared with results obtained using the Method of Moments. The agreement is quite good, indicating the possibilities for FDTD application to antenna impedance and coupling.

Luebbers, Raymond J.; Kunz, Karl S.

1991-01-01

312

Calculating rotordynamic coefficients of seals by finite-difference techniques

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.

Dietzen, F. J.; Nordmann, R.

1987-01-01

313

Finite Difference Time Domain Method For Grating Structures

The aim of this chapter is to present the principle of the FDTD method when applied to the resolution of Maxwell equations. Centered finite differences are used to approximate the value of both time and space derivatives that appear in these equations. The convergence criteria in addition to the boundary conditions (periodic or absorbing ones) are given. The special case of bi-periodic structures illuminated at oblique incidence is solved with the SFM (split field method) technique. In all ca...

Baida, F. I.; Belkhir, A.

2012-01-01

314

Convergence analysis of the coarse mesh finite difference method

The convergence rates of the nonlinear coarse mesh finite difference (CMFD) and the coarse mesh rebalance (CMR) methods are derived analytically for one- and two-dimensional geometries and one- and two-energy group solutions of the fixed source diffusion problem in a non-multiplying infinite homogeneous medium. The derivation is performed by linearizing the nonlinear algorithm and by applying Fourier error analysis to the linearized algorithm. The mesh size measured in units of the diffusion ...

Lee, Deokjung

2003-01-01

315

Real space finite difference method for conductance calculations

We present a general method for calculating coherent electronic transport in quantum wires and tunnel junctions. It is based upon a real space high order finite difference representation of the single particle Hamiltonian and wave functions. Landauer's formula is used to express the conductance as a scattering problem. Dividing space into a scattering region and left and right ideal electrode regions, this problem is solved by wave function matching (WFM) in the boundary zon...

Khomyakov, Petr A.; Brocks, Geert

2004-01-01

316

Using finite difference method to simulate casting thermal stress

Thermal stress simulation can provide a scientific reference to eliminate defects such as crack, residual stress centralization and deformation etc., caused by thermal stress during casting solidification. To study the thermal stress distribution during casting process, a unilateral thermal-stress coupling model was employed to simulate 3D casting stress using Finite Difference Method (FDM), namely all the traditional thermal-elastic-plastic equations are numerically and differentially discre...

Liao Dunming; Zhang Bin; Zhou Jianxin

2011-01-01

317

Here we describe a new staggered grid formulation for discretizing incompressible Stokes flow which has been specifically designed for use on adaptive quadtree-type meshes. The key to our new adaptive staggered grid (ASG) stencil is in the form of the stress-conservative finite difference constraints which are enforced at the "hanging" velocity nodes between resolution transitions within the mesh. The new ASG discretization maintains a compact stencil, thus preserving the sparsity within the matrix which both minimizes the computational cost and enables the discrete system to be efficiently solved via sparse direct factorizations or iterative methods. We demonstrate numerically that the ASG stencil (1) is stable and does not produce spurious pressure oscillations across regions of grid refinement, which intersect discontinuous viscosity structures, and (2) possesses the same order of accuracy as the classical nonadaptive staggered grid discretization. Several pragmatic error indicators that are used to drive adaptivity are introduced in order to demonstrate the superior performance of the ASG stencil over traditional nonadaptive grid approaches. Furthermore, to demonstrate the potential of this new methodology, we present geodynamic examples of both lithospheric and planetary scales models.

Gerya, T. V.; May, D. A.; Duretz, T.

2013-04-01

318

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 (?-specified) or Neumann (?? 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)

319

A fully implicit finite difference scheme has been developed to solve the hydrodynamic equations coupled with radiation transport. Solution of the time dependent radiation transport equation is obtained using the discrete ordinates method and the energy flow into the Lagrangian meshes as a result of radiation interaction is fully accounted for. A tridiagonal matrix system is solved at each time step to determine the hydrodynamic variables implicitly. The results obtained fro...

Ghosh, Karabi; Menon, S. V. G.

2008-01-01

320

Higher-order accurate Osher schemes with application to compressible boundary layer stability

Two fourth order accurate Osher schemes are presented which maintain higher order accuracy on nonuniform grids. They use either a conservative finite difference or finite volume discretization. Both methods are successfully used for direct numerical simulations of flat plate boundary layer instability at different Mach numbers. Results of growth rates of Tollmien-Schlichting waves compare well with direct simulations of incompressible flow and for compressible flow with results obtained by solving the parabolic stability equations.

Van Der Vegt, J. J. W.

1993-01-01

321

Two discretizations of a 9-velocity Boltzmann equation with a BGK collision operator are studied. A Chapman-Enskog expansion of the PDE system predicts that the macroscopic behavior corresponds to the incompressible Navier-Stokes equations with additional terms of order Mach number squared. We introduce a fourth-order scheme and compare results with those of the commonly used lattice Boltzmann discretization and with finite-difference schemes applied to the incompressible Navier-Stokes equations in primitive-variable form. We numerically demonstrate convergence of the BGK schemes to the incompressible Navier-Stokes equations and quantify the errors associated with compressibility and discretization effects. When compressibility error is smaller than discretization error, convergence in both grid spacing and time step is shown to be second-order for the LB method and is confirmed to be fourth-order for the fourth-order BGK solver. However, when the compressibility error is simultaneously reduced as the grid is...

Reider, M B; Reider, Marc B.; Sterling, James D.

1993-01-01

322

Seismic imaging using finite-differences and parallel computers

Energy Technology Data Exchange (ETDEWEB)

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.

Ober, C.C. [Sandia National Labs., Albuquerque, NM (United States)

1997-12-31

323

An improvement to the numerical approach and the underlying physical model made by Hiller and Ruiz in their attempt to generate musical sounds by solving the equations of vibrating strings by means of Finite Difference Methods (FDM) in order to simulate the notion of the piano string with a high degree of realism, is shown. Starting from the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, a numerical finite difference scheme is derived, from which the time histories of string displacement and velocity for each point of the string are computed in the time domain. The interacting force between hammer and string, and the force acting on the bridge, are given by the same scheme. The performance of the model is illustrated by examples of simulated string waveforms. Aspects of numerical stability and dispersion are discussed with reference to the proper choice of sampling parameters.

Chaigne, Antoine; Askenfelt, Anders

1993-04-01

324

Directory of Open Access Journals (Sweden)

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.

Md. Kamal Hossain

2010-10-01

325

Finite difference evolution equations and quantum dynamical semigroups

International Nuclear Information System (INIS)

We consider the recently proposed [Bonifacio, Lett. Nuovo Cimento, 37, 481 (1983)] coarse grained description of time evolution for the density operator rho(t) through a finite difference equation with steps tau, and we prove that there exists a generator of the quantum dynamical semigroup type yielding an equation giving a continuous evolution coinciding at all time steps with the one induced by the coarse grained description. The map rho(0)?rho(t) derived in this way takes the standard form originally proposed by Lindblad [Comm. Math. Phys., 48, 119 (1976)], even when the map itself (and, therefore, the corresponding generator) is not bounded. (author)

326

Thermal stress in a composite cylinder by finite difference technique

Temperature and stress time series have been generated in the tubular heat exchanger of a concentrator type solar collector. Hourly measurements of ambient temperature, solar radiation and wind speed were used as random, input time series to a finite difference solution of the heat transfer problem. The results indicate that significant alternating thermal stresses are generated. Tangential stress in the copper tube has been found to be the greatest stress component. Given the fact that the collector is subjected to alternating temperature changes daily and seasonally. As a result, over a long period of time, the induced stress may lead to life limiting fatigue.

Con, V. N.; Heller, R. A.; Singh, M. P.; Tuyen, L. D.

1980-07-01

327

Application of a finite difference technique to thermal wave propagation

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.

Baumeister, K. J.

1975-01-01

328

Finite difference program for calculating hydride bed wall temperature profiles

International Nuclear Information System (INIS)

A QuickBASIC finite difference program was written for calculating one dimensional temperature profiles in up to two media with flat, cylindrical, or spherical geometries. The development of the program was motivated by the need to calculate maximum temperature differences across the walls of the Tritium metal hydrides beds for thermal fatigue analysis. The purpose of this report is to document the equations and the computer program used to calculate transient wall temperatures in stainless steel hydride vessels. The development of the computer code was motivated by the need to calculate maximum temperature differences across the walls of the hydrides beds in the Tritium Facility for thermal fatigue analysis

329

There is a large and growing need for accurate full-wave optical simulations of complex systems such as photovoltaic (PV) cells, particularly at the nanoscale. A finite-difference time-domain tool known as MEEP offers this capability in principle, through C++ libraries and the Scheme programming language. For expert users, this approach has been quite successful, but there is also great interest from new and less frequent users in starting to use MEEP. In order to facilitate this process, we ...

Tee, Xin Tze; Bermel, Peter

2013-01-01

330

A RBF Based Local Gridfree Scheme for Unsteady Convection-Diffusion Problems

Directory of Open Access Journals (Sweden)

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.

Sanyasiraju VSS Yedida

2009-12-01

331

A Variational Finite Difference Method for Time-Dependent Stokes Flow on Irregular Domains

We restate time-dependent Stokes flow for incompressible Newtonian fluids as a variational problem relating velocity, pressure, and deviatoric stress variables, which leads to a simple weighted finite difference discretization on staggered Cartesian grids. The method easily handles irregular domains involving both free surfaces and moving solid boundaries by exploiting natural boundary conditions, while supporting spatially varying viscosity and density. Due to its basis in extremizing a well-posed quadratic functional, the resulting linear system is sparse and symmetric indefinite. It can also be converted to an equivalent sparse, symmetric positive-definite system by applying a simple and inexpensive algebraic manipulation, allowing the use of a wide range of efficient linear solvers. We demonstrate that the method achieves first order convergence in velocity on a range of test cases. In addition, we apply our method as part of a simple Navier-Stokes solver to illustrate that it can reproduce the characteri...

Batty, Christopher

2010-01-01

332

Finite difference method to find period-one gait cycles of simple passive walkers

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.

Dardel, Morteza; Safartoobi, Masoumeh; Pashaei, Mohammad Hadi; Ghasemi, Mohammad Hassan; Navaei, Mostafa Kazemi

2015-01-01

333

Dynamic rupture simulation of non-planar faults with a finite-difference approach

Two-dimensional (2-D) modelling of dynamic seismic rupture is performed using a recent staggered-grid finite-difference formulation. Rupture boundary conditions are applied only inside the crack, without assuming any symmetry with respect to the rupture surface. By a simple rotation of the stress tensor, the local orientation of the crack is taken into consideration at each stress point. The grid size is controlled by the source discretization. The greater the number of grid nodes discretizing the finite source, the lower the grid size could be. Below the lower bound value associated with a given discretization, numerical artefacts are not negligible with respect to the spatial frequency content of the dynamic solution. Solutions converge for both point and finite sources by densifying the number of stress points in the source. Numerical scaling of boundary conditions is an important element of this convergence and allows the removal of high-frequency spurious effects of dynamic rupture conditions. For the self-similar crack, a comparison with Kostrov's analytical solution shows that accurate stress singularities are obtained for various crack orientations with respect to the numerical grid. For spontaneous rupture modelling assuming a slip-weakening constitutive law, similar solutions are found for both rupture kinematics and excited wavefield in planar faults with any orientation. Finally, based on these results, rupture propagation over an arbitrary non-planar fault is justified and then performed in the presence of heterogeneous medium.

Cruz-Atienza, V. M.; Virieux, J.

2004-09-01

334

a Mapped Finite Difference Study of Noise Transmission in Nonuniform Ducts

The primary objective of this work was to study a class of problems involving noise propagation in acoustically lined variable area ducts with or without mean fluid flow. The method of study was numerical in nature and included body -fitted grid mapping procedures in conjunction with implicit finite difference techniques. The work resulted in several general FORTRAN programs that were tested for cases with or without mean fluid flow, including soft wall or hard wall acoustic liner conditions, and plane wave or far field exit conditions. The results were compared to available theoretical and experimental data. The automated, body-fitted grid mapping procedure was found to be robust, simple to use, and capable of mapping very complicated geometries simply by defining the grid distribution on the boundaries. In general, the solution of the wave equation was found to be successful when using a plane wave exit condition, whereas a problem was encountered with reflections from the particular far field exit condition being applied. The problem was determined to be the result of the proximity of the far field boundary to the noise source as well as its applicability to exactly cylindrical wave expansions only. The fully-coupled solution of the linearized gas dynamic equations was successful for both positive and negative Mach numbers as well as for hard and soft wall conditions. The mean fluid flow considered was two-dimensional, inviscid, irrotational, incompressible, and nonheat conducting. The factored-implicit finite difference technique used did give rise to short wavelength perturbations, but these were dampened by the introduction of higher order artificial dissipation terms into the scheme. In the different problems that this study considered, the finite difference theory was found to be well-suited for the simulation of noise transmission in nonuniform ducts.

Raad, Peter Emile

335

Finite-Difference Frequency-Domain Method in Nanophotonics

DEFF Research Database (Denmark)

Optics and photonics are exciting, rapidly developing fields building their success largely on use of more and more elaborate artificially made, nanostructured materials. To further advance our understanding of light-matter interactions in these complicated artificial media, numerical modeling is often indispensable. This thesis presents the development of rigorous finite-difference method, a very general tool to solve Maxwell’s equations in arbitrary geometries in three dimensions, with an emphasis on the frequency-domain formulation. Enhanced performance of the perfectly matched layers is obtained through free space squeezing technique, and nonuniform orthogonal grids are built to greatly improve the accuracy of simulations of highly heterogeneous nanostructures. Examples of the use of the finite-difference frequency-domain method in this thesis range from simulating localized modes in a three-dimensional photonic-crystal membrane-based cavity, a quasi-one-dimensional nanobeam cavity and arrays of side-coupled nanobeam cavities, to modeling light propagation through metal films with single or periodically arranged multiple subwavelength slits.

Ivinskaya, Aliaksandra

2011-01-01

336

OBTAINING POTENTIAL FIELD SOLUTIONS WITH SPHERICAL HARMONICS AND FINITE DIFFERENCES

International Nuclear Information System (INIS)

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: (1) 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; (2) using an iterative finite difference algorithm to solve for the potential field. The naive and the improved numerical solutions are compared for actual magnetograms and the differences are found to be rather dramatic. We made our new Finite Difference Iterative Potential-field Solver (FDIPS) a publicly available code so that other researchers can also use it as an alternative to the spherical harmonics approach.

337

Finite difference analysis of curved deep beams on Winkler foundation

Directory of Open Access Journals (Sweden)

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.

Adel A. Al-Azzawi

2011-03-01

338

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.

Farhat, Charbel; Geuzaine, Philippe; Grandmont, Céline

2001-12-01

339

In this paper, a novel, unified model-based fault-detection and prediction (FDP) scheme is developed for nonlinear multiple-input-multiple-output (MIMO) discrete-time systems. The proposed scheme addresses both state and output faults by considering separate time profiles. The faults, which could be incipient or abrupt, are modeled using input and output signals of the system. The fault-detection (FD) scheme comprises online approximator in discrete time (OLAD) with a robust adaptive term. An output residual is generated by comparing the FD estimator output with that of the measured system output. A fault is detected when this output residual exceeds a predefined threshold. Upon detecting the fault, the robust adaptive terms and the OLADs are initiated wherein the OLAD approximates the unknown fault dynamics online while the robust adaptive terms help in ensuring asymptotic stability of the FD design. Using the OLAD outputs, a fault diagnosis scheme is introduced. A stable parameter update law is developed not only to tune the OLAD parameters but also to estimate the time to failure (TTF), which is considered as a first step for prognostics. The asymptotic stability of the FDP scheme enhances the detection and TTF accuracy. The effectiveness of the proposed approach is demonstrated using a fourth-order MIMO satellite system. PMID:20106734

Thumati, Balaje T; Jagannathan, S

2010-03-01

340

In this work we construct a high-order, single-stage, single-step positivity-preserving method for the compressible Euler equations. Space is discretized with the finite difference weighted essentially non-oscillatory (WENO) method. Time is discretized through a Lax-Wendroff procedure that is constructed from the Picard integral formulation (PIF) of the partial differential equation. The method can be viewed as a modified flux approach, where a linear combination of a low- a...

Seal, David C.; Tang, Qi; Xu, Zhengfu; Christlieb, Andrew J.

2014-01-01

341

Conservative high-order-accurate finite-difference methods for curvilinear grids

Two fourth-order-accurate finite-difference methods for numerically solving hyperbolic systems of conservation equations on smooth curvilinear grids are presented. The first method uses the differential form of the conservation equations; the second method uses the integral form of the conservation equations. Modifications to these schemes, which are required near boundaries to maintain overall high-order accuracy, are discussed. An analysis that demonstrates the stability of the modified schemes is also provided. Modifications to one of the schemes to make it total variation diminishing (TVD) are also discussed. Results that demonstrate the high-order accuracy of both schemes are included in the paper. In particular, a Ringleb-flow computation demonstrates the high-order accuracy and the stability of the boundary and near-boundary procedures. A second computation of supersonic flow over a cylinder demonstrates the shock-capturing capability of the TVD methodology. An important contribution of this paper is the dear demonstration that higher order accuracy leads to increased computational efficiency.

Rai, Man M.; Chakrvarthy, Sukumar

1993-01-01

342

A finite difference solution for the propagation of sound in near sonic flows

An explicit time/space finite difference procedure is used to model the propagation of sound in a quasi one-dimensional duct containing high Mach number subsonic flow. Nonlinear acoustic equations are derived by perturbing the time-dependent Euler equations about a steady, compressible mean flow. The governing difference relations are based on a fourth-order, two-step (predictor-corrector) MacCormack scheme. The solution algorithm functions by switching on a time harmonic source and allowing the difference equations to iterate to a steady state. The principal effect of the non-linearities was to shift aocustical energy to higher harmonics. With increased source strength, wave steepening was observed. This phenomenon suggests that the acoustical response may approach a shock behavior at higher sound pressure level as the throat Mach number approaches unity. On a peak level basis, good agreement between the nonlinear finite difference and linear finite element solutions was observed, even through a peak sound pressure level of about 150 dB occurred in the throat region. Nonlinear steady state waveform solutions are shown to be in excellent agreement with a nonlinear asymptotic theory. Previously announced in STAR as N83-30167

Hariharan, S. I.; Lester, H. C.

1984-01-01

343

Finite-difference time-domain simulation of fusion plasmas at radiofrequency time scales

International Nuclear Information System (INIS)

Simulation of dense plasmas in the radiofrequency range are typically performed in the frequency domain, i.e., by solving Laplace-transformed Maxwell's equations. This technique is well-suited for the study of linear heating and quasilinear evolution, but does not generalize well to the study of nonlinear phenomena. Conversely, time-domain simulation in this range is difficult because the time scale is long compared to the electron plasma wave period, and in addition, the various cutoff and resonance behaviors within the plasma insure that any explicit finite-difference scheme would be numerically unstable. To resolve this dilemma, explicit finite-difference Maxwell terms are maintained, but a carefully time-centered locally implicit method is introduced to treat the plasma current, such that all linear plasma dispersion behavior is faithfully reproduced at the available temporal and spatial resolution, despite the fact that the simulation time step may exceed the electron gyro and electron plasma time scales by orders of magnitude. Demonstrations are presented of the method for several classical benchmarks, including mode conversion to ion cyclotron wave, cyclotron resonance, propagation into a plasma-wave cutoff, and tunneling through low-density edge plasma

344

A coarse-mesh nodal method-diffusive-mesh finite difference method

International Nuclear Information System (INIS)

Modern nodal methods have been successfully used for conventional light water reactor core analyses where the homogenized, node average cross sections (XSs) and the flux discontinuity factors (DFs) based on equivalence theory can reliably predict core behavior. For other types of cores and other geometries characterized by tightly-coupled, heterogeneous core configurations, the intranodal flux shapes obtained from a homogenized nodal problem may not accurately portray steep flux gradients near fuel assembly interfaces or various reactivity control elements. This may require extreme values of DFs (either very large, very small, or even negative) to achieve a desired solution accuracy. Extreme values of DFs, however, can disrupt the convergence of the iterative methods used to solve for the node average fluxes, and can lead to a difficulty in interpolating adjacent DF values. Several attempts to remedy the problem have been made, but nothing has been satisfactory. A new coarse-mesh nodal scheme called the Diffusive-Mesh Finite Difference (DMFD) technique, as contrasted with the coarse-mesh finite difference (CMFD) technique, has been developed to resolve this problem. This new technique and the development of a few-group, multidimensional kinetics computer program are described in this paper

345

International Nuclear Information System (INIS)

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)

346

Directory of Open Access Journals (Sweden)

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.

Vineet K. Srivastava

2014-03-01

347

Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows

A higher order accurate numerical procedure has been developed for solving incompressible Navier-Stokes equations for 2D or 3D fluid flow problems. It is based on low-storage Runge-Kutta schemes for temporal discretization and fourth and sixth order compact finite-difference schemes for spatial discretization. The particular difficulty of satisfying the divergence-free velocity field required in incompressible fluid flow is resolved by solving a Poisson equation for pressure. It is demonstrated that for consistent global accuracy, it is necessary to employ the same order of accuracy in the discretization of the Poisson equation. Special care is also required to achieve the formal temporal accuracy of the Runge-Kutta schemes. The accuracy of the present procedure is demonstrated by application to several pertinent benchmark problems.

Wilson, Robert V.; Demuren, Ayodeji O.; Carpenter, Mark

1998-01-01

348

Computational electrodynamics the finite-difference time-domain method

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.

Taflove, Allen

2005-01-01

349

Parallel finite-difference time-domain method

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.

Yu, Wenhua

2006-01-01

350

Visualization of elastic wavefields computed with a finite difference code

Energy Technology Data Exchange (ETDEWEB)

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.

Larsen, S. [Lawrence Livermore National Lab., CA (United States); Harris, D.

1994-11-15

351

Finite-difference modeling of commercial aircraft using TSAR

Energy Technology Data Exchange (ETDEWEB)

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.

Pennock, S.T.; Poggio, A.J.

1994-11-15

352

Obtaining Potential Field Solution with Spherical Harmonics and Finite Differences

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...

Toth, Gabor; Huang, Zhenguang; 10.1088/0004-637X/732/2/102

2011-01-01

353

Effects of sources on time-domain finite difference models.

Recent work on excitation mechanisms in acoustic finite difference models focuses primarily on physical interpretations of observed phenomena. This paper offers an alternative view by examining the properties of models from the perspectives of linear algebra and signal processing. Interpretation of a simulation as matrix exponentiation clarifies the separate roles of sources as boundaries and signals. Boundary conditions modify the matrix and thus its modal structure, and initial conditions or source signals shape the solution, but not the modal structure. Low-frequency artifacts are shown to follow from eigenvalues and eigenvectors of the matrix, and previously reported artifacts are predicted from eigenvalue estimates. The role of source signals is also briefly discussed. PMID:24993210

Botts, Jonathan; Savioja, Lauri

2014-07-01

354

Stability analysis for acoustic wave propagation in tilted TI media by finite differences

Several papers in recent years have reported instabilities in P-wave modelling, based on an acoustic approximation, for inhomogeneous transversely isotropic media with tilted symmetry axis (TTI media). In particular, instabilities tend to occur if the axis of symmetry varies rapidly in combination with strong contrasts of medium parameters, which is typically the case at the foot of a steeply dipping salt flank. In a recent paper, we have proposed and demonstrated a P-wave modelling approach for TTI media, based on rotated stress and strain tensors, in which the wave equations reduce to a coupled set of two second-order partial differential equations for two scalar stress components: a normal component along the variable axis of symmetry and a lateral component of stress in the plane perpendicular to that axis. Spatially constant density is assumed in this approach. A numerical discretization scheme was proposed which uses discrete second-derivative operators for the non-mixed second-order derivatives in the wave equations, and combined first-derivative operators for the mixed second-order derivatives. This paper provides a complete and rigorous stability analysis, assuming a uniformly sampled grid. Although the spatial discretization operator for the TTI acoustic wave equation is not self-adjoint, this operator still defines a complete basis of eigenfunctions of the solution space, provided that the solution space is somewhat restricted at locations where the medium is elliptically anisotropic. First, a stability analysis is given for a discretization scheme, which is purely based on first-derivative operators. It is shown that the coefficients of the central difference operators should satisfy certain conditions. In view of numerical artefacts, such a discretization scheme is not attractive, and the non-mixed second-order derivatives of the wave equation are discretized directly by second-derivative operators. It is shown that this modification preserves stability, provided that the central difference operators of the second-order derivatives dominate over the twice applied operators of the first-order derivatives. In practice, it turns out that this is almost the case. Stability of the desired discretization scheme is enforced by slightly weighting down the mixed second-order derivatives in the wave equation. This has a minor, practically negligible, effect on the kinematics of wave propagation. Finally, it is shown that non-reflecting boundary conditions, enforced by applying a taper at the boundaries of the grid, do not harm the stability of the discretization scheme.

Bakker, Peter M.; Duveneck, Eric

2011-05-01

355

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 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

356

High-order finite difference solution for 3D nonlinear wave-structure interaction

DEFF Research Database (Denmark)

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 cylinder are presented and compared to the fully nonlinear potential code XWAVE as well as to experiments.

Ducrozet, Guillaume; Bingham, Harry B.

2010-01-01

357

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...

Chernyshenko, Dmitri

2014-01-01

358

Preliminary results are presented of a finite element/finite difference method (semidiscrete Galerkin method) used to calculate compressible boundary layer flow about airfoils, in which the group finite element scheme is applied to the Dorodnitsyn formulation of the boundary layer equations. The semidiscrete Galerkin (SDG) method promises to be fast, accurate and computationally efficient. The SDG method can also be applied to any smoothly connected airfoil shape without modification and possesses the potential capability of calculating boundary layer solutions beyond flow separation. Results are presented for low speed laminar flow past a circular cylinder and past a NACA 0012 airfoil at zero angle of attack at a Mach number of 0.5. Also shown are results for compressible flow past a flat plate for a Mach number range of 0 to 10 and results for incompressible turbulent flow past a flat plate. All numerical solutions assume an attached boundary layer.

Strong, Stuart L.; Meade, Andrew J., Jr.

1992-01-01

359

GPU Accelerated 2-D Staggered-grid Finite Difference Seismic Modelling

Directory of Open Access Journals (Sweden)

Full Text Available The staggered-grid finite difference (FD method demands significantly computational capability and is inefficient for seismic wave modelling in 2-D viscoelastic media on a single PC. To improve computation speedup, a graphic processing units (GPUs accelerated method was proposed, for modern GPUs have now become ubiquitous in desktop computers and offer an excellent cost-to-performance-ratio parallelism. The geophysical model is decomposed into subdomains for PML absorbing conditions. The vertex and fragment processing are fully used to solve FD schemes in parallel and the latest updated frames are swapped in Framebuffer Object (FBO attachments as inputs for the next simulation step. The seismic simulation program running on modern PCs provides significant speedup over a CPU implementation, which makes it possible to simulate realtime complex seismic propagation in high resolution of 2048*2048 gridsizes on low-cost PCs.

Zhangang Wang

2011-08-01

360

Black-Scholes finite difference modeling in forecasting of call warrant prices in Bursa Malaysia

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.

Mansor, Nur Jariah; Jaffar, Maheran Mohd

2014-07-01

361

We propose numerical simulations of viscoelastic fluids based on a hybrid algorithm combining Lattice-Boltzmann models (LBM) and Finite Differences (FD) schemes, the former used to model the macroscopic hydrodynamic equations, and the latter used to model the polymer dynamics. The kinetics of the polymers is introduced using constitutive equations for viscoelastic fluids with finitely extensible non-linear elastic dumbbells with Peterlin's closure (FENE-P). The numerical model is first benchmarked by characterizing the rheological behaviour of dilute homogeneous solutions in various configurations, including steady shear, elongational flows, transient shear and oscillatory flows. As an upgrade of complexity, we study the model in presence of non-ideal multicomponent interfaces, where immiscibility is introduced in the LBM description using the "Shan-Chen" model. The problem of a confined viscoelastic (Newtonian) droplet in a Newtonian (viscoelastic) matrix under simple shear is investigated and numerical resu...

Gupta, A; Scagliarini, A

2014-01-01

362

A New Time-Dependent Finite Difference Method for Relativistic Shock Acceleration

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.

Delaney, Sean; Duffy, Peter; Downes, Turlough P

2011-01-01

363

Finite difference preserving the energy properties of a coupled system of diffusion equations

Scientific Electronic Library Online (English)

Full Text Available SciELO Brazil | Language: English Abstract in portuguese Neste trabalho, nós provamos a propriedade de decaimento exponencial da energia numérica associada a um particular esquema numérico em diferenças finitas aplicado a um sistema acoplado de equações de difusão. Ao nível da dinâmica do contínuo, é bem conhecido que a energia do sistema é decrescente e [...] exponencialmente estável. Aqui nós apresentamos em detalhes a análise numérica de decaimento exponencial da energia numérica desde que obedecido o critério de estabilidade. Abstract in english In this paper we proved the exponential decay of the energy of a numerical scheme in finite difference applied to a coupled system of diffusion equations. At the continuous level, it is well-known that the energy is decreasing and stable in the exponential sense. We present in detail the numerical a [...] nalysis of exponential decay to numerical energy since holds the stability criterion.

A.J.A., Ramos; D.S., Almeida Jr..

2013-08-01

364

This paper presents a new approximation of elastodynamic frictionless contact problems based both on the finite element method and on an adaptation of Nitsche's method which was initially designed for Dirichlet's condition. A main interesting characteristic is that this approximation produces well-posed space semi-discretizations contrary to standard finite element discretizations. This paper is then mainly devoted to present an analysis of the semi-discrete problem in terms of consistency, w...

Chouly, Franz; Hild, Patrick; Renard, Yves

2014-01-01

365

Energy Technology Data Exchange (ETDEWEB)

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.

Ewing, R.E.; Saevareid, O.; Shen, J. [Texas A& M Univ., College Station, TX (United States)

1994-12-31

366

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), we establish the theory of a straightforward finite difference scheme for the 1D, monoenergetic neutron diffusion equation in plane media. We also demonstrate an analytically enhanced version that leads directly to the analytical solution. The second part of our presentation (Part II, in these proceedings) is concerned with numerical implementation and application of the finite difference solutions. There, 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)

367

Directory of Open Access Journals (Sweden)

Full Text Available Results of a sensitivity study are presented from various configurations of the NEMO ocean model in the Black Sea. The standard choices of vertical discretization, viz. z levels, s coordinates and enveloped s coordinates, all show their limitations in the areas of complex topography. Two new hybrid vertical coordinate schemes are presented: the "s-on-top-of-z" and its enveloped version. The hybrid grids use s coordinates or enveloped s coordinates in the upper layer, from the sea surface to the depth of the shelf break, and z-coordinates are set below this level. The study is carried out for a number of idealised and real world settings. The hybrid schemes help reduce errors generated by the standard schemes in the areas of steep topography. Results of sensitivity tests with various horizontal diffusion formulations are used to identify the optimum value of Smagorinsky diffusivity coefficient to best represent the mesoscale activity.

G. Shapiro

2013-03-01

368

Directory of Open Access Journals (Sweden)

Full Text Available Este trabalho foi desenvolvido com o objetivo de se apresentar a aplicação de um esquema de discretização mais eficiente para volumes finitos, denominado Flux-Spline utilizando-se, para tal, de dois casos de transporte difusivo de umidade e calor, através de um meio poroso capilar. Os resultados da solução numérica do sistema de equações formado pelas equações de Luikov mostram desempenho adequado do esquema para este tipo de problema, quando comparado ao tradicional esquema de diferença central e ao método da transformada integral.This study was conducted with the objective to present a more efficient discretization scheme to finite volumes method called Flux-Spline, utilising for the purpose two cases of pure diffusion in capillary porous media. The results of numerical simulation of the equations system formed by Luikov equations showed a good performance of the scheme in comparison to the Central Difference Scheme and Generalised Integral Transform Technique method.

Paulo C. Oliveira

2003-04-01

369

Results of a sensitivity study are presented from various configurations of the NEMO ocean model in the Black Sea. The standard choices of vertical discretization, viz. z levels, s coordinates and enveloped s coordinates, all show their limitations in the areas of complex topography. Two new hybrid vertical coordinate schemes are presented: the "s-on-top-of-z" and its enveloped version. The hybrid grids use s coordinates or enveloped s coordinates in the upper layer, from the sea surface to the depth of the shelf break, and z-coordinates are set below this level. The study is carried out for a number of idealised and real world settings. The hybrid schemes help reduce errors generated by the standard schemes in the areas of steep topography. Results of sensitivity tests with various horizontal diffusion formulations are used to identify the optimum value of Smagorinsky diffusivity coefficient to best represent the mesoscale activity.

Shapiro, G.; Luneva, M.; Pickering, J.; Storkey, D.

2013-03-01

370

Energy Technology Data Exchange (ETDEWEB)

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)

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

371

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

372

3D anisotropic modeling for airborne EM systems using finite-difference method

Most current airborne EM data interpretations assume an isotropic model, which is sometimes inappropriate, especially in regions with distinct dipping anisotropy due to strong layering and stratifications. In this paper, we investigate airborne EM modeling and interpretation for a 3D earth with arbitrarily electrical anisotropy. We implement the staggered finite-difference algorithm to solve the coupled partial differential equations for the scattered electrical fields. Whereas the current density that is connected to the diagonal elements of the anisotropic conductivity tensor is discretized by using the volume weighted average, the current density that is connected to the non-diagonal elements is discretized by using the volume current density average. Further, we apply a divergence correction technique designed specifically for 3D anisotropic models to speed up the modeling process. For numerical experiments, we take both VMD and HMD transmitting dipoles for two typical anisotropic cases: 1) anisotropic anomalous inhomogeneities embedded in an isotropic half-space; and 2) isotropic anomalous inhomogeneities embedded in an anisotropic host rock. Model experiments show that our algorithm has high calculation accuracy, the divergence correction technique used in the modeling can greatly improve the convergence of the solutions, accelerating the calculation speed up to 2 times for the model presented in the paper. The characteristics inside the anisotropic earth, like the location of the anomalous body and the principal axis orientations, can also be clearly identified from AEM area surveys.

Liu, Yunhe; Yin, Changchun

2014-10-01

373

We propose an efficient split-step compact finite difference method for the cubic-quintic complex Ginzburg-Landau (CQ CGL) equations both in one dimension and in multi-dimensions. The key point of this method is to separate the original CQ CGL equations into two nonlinear subproblems and one or several linear ones. The linear subproblems are solved by the compact finite difference schemes. As the nonlinear subproblems cannot be solved exactly, the Runge-Kutta method is applied and the total accuracy order is not reduced. The proposed method is convergent of second-order in time and fourth-order in space, which is confirmed numerically. Extensive numerical experiments are carried out to examine the performance of this method for the nonlinear Schrödinger equations, the cubic complex Ginzburg-Landau equation, and the CQ CGL equations. It is shown from all the numerical tests that the present method is efficient and reliable.

Wang, Shanshan; Zhang, Luming

2013-06-01

374

Seismic wave modeling is a cornerstone of geophysical data acquisition, processing, and interpretation, for which finite-difference methods are often applied. In this paper, we extend the velocity-pressure formulation of the acoustic wave equation to marine seismic modeling using the staggered-grid finite-difference method. The scheme is developed using a fourth-order spatial and a second-order temporal operator. Then, we define a stability coefficient (SC) and calculate its maximum value under the stability condition. Based on the dispersion relationship, we conduct a detailed dispersion analysis for submarine sediments in terms of the phase and group velocity over a range of angles, stability coefficients, and orders. We also compare the numerical solution with the exact solution for a P-wave line source in a homogeneous submarine model. Additionally, the numerical results determined by a Marmousi2 model with a rugged seafloor indicate that this method is sufficient for modeling complex submarine structures.

Qian, Jin; Wu, Shiguo; Cui, Ruofei

2013-01-01

375

A standard parabolic equation is used to approximate the Helmholtz equation for electromagnetic propagation in an inhomogeneneous atmosphere. An implicit finite difference (IFD) scheme to solve the SPE is applied between the irregularly shaped ground and an altitude z = z(h) below which all inhomogeneities of the medium are assumed localized. The boundary condition at z = z(h) is obtained by matching the IFD solution to a surface Green's function solution within the uniform region above z = z(h). For ground slopes above about 1 percent the IFD implementation of the impedance boundary condition at the ground is shown to maintain the validity of the approximation only for vertically polarized waves. Predictions using this hybrid finite difference-surface Green's function method agree well with results obtained using other computational methods.

Marcus, Sherman W.

1992-12-01

376

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.

Coelho, Pedro J.

2014-08-01

377

A 3D Mimetic Finite Difference Method for Rupture Dynamics

We are developing a method for solving earthquake rupture dynamics problems on structured curvilinear meshes. The advantage of a curvilinear mesh over a rectangular mesh is that it can accommodate free-surface topography as well as non-planar fault geometry. The advantages of using a structured mesh over an unstructured mesh (as used in many finite element methods) is simplicity and computational efficiency. Structured meshes also make a number of computational tasks easier, such as parallelization, or coupling with other codes that use similar structured meshes. To build the discretized equations of motion on a structured, yet non Cartesian mesh, we use a mimetic method, so named because it takes special care to mimic the important conservation properties of the original equations of motion. We begin by writing the equations of motion in terms of gradient and divergence operators. We then derive a discrete grad (or div) operator by differentiating an interpolation function of the discrete variable. Next, that grad (or div) operator is plugged into a discrete analog of Gauss' Identity and manipulated to find the adjoint div (or grad) operator. We use a computer algebra system to handle the manipulations, which is practically essential for the 3D case because of the extreme lengths of the expressions to be coded. The code is currently implemented as a "rapid prototype" in MATLAB and undergoing validation prior to conversion to a high performance language. We compare results for simple types of rupture that have analytical solutions.

Ely, G.; Minster, J.; Day, S.

2004-12-01

378

Non-finite-difference algorithm for integrating Newton's motion equations

We have presented some practical consequences on the molecular-dynamics simulations arising from the numerical algorithm published recently in paper Int. J. Mod. Phys. C 16, 413 (2005). The algorithm is not a finite-difference method and therefore it could be complementary to the traditional numerical integrating of the motion equations. It consists of two steps. First, an analytic form of polynomials in some formal parameter $\\lambda$ (we put $\\lambda=1$ after all) is derived, which approximate the solution of the system of differential equations under consideration. Next, the numerical values of the derived polynomials in the interval, in which the difference between them and their truncated part of smaller degree does not exceed a given accuracy $\\epsilon$, become the numerical solution. The particular examples, which we have considered, represent the forced linear and nonlinear oscillator and the 2D Lennard-Jones fluid. In the latter case we have restricted to the polynomials of the first degree in formal parameter $\\lambda$. The computer simulations play very important role in modeling materials with unusual properties being contradictictory to our intuition. The particular example could be the auxetic materials. In this case, the accuracy of the applied numerical algorithms as well as various side-effects, which might change the physical reality, could become important for the properties of the simulated material.

Brzostowski, B.; Dudek, M. R.; Grabiec, B.; Nadzieja, T.

2007-03-01

379

Analysis of numerical seismic source functions by finite difference method

International Nuclear Information System (INIS)

The finite difference synthetic seismograms are tested for a number of seismic sources to understand their stability characteristics. Processing techniques such as frequency filtering and gain application are applied to improve the model resonance. The grid dispersion due to high frequencies contained in source appears to be controllable by high cut filtering the model output. A single velocity distribution model is used to prepare synthetic seismograms with different source functions. The results seem to be in agreement with the previous work. The output of modeling algorithm gives correct arrival times but when the model becomes unstable, the relative amplitude information of different arrivals seems to be lost. The use of different seismic source wavelets with same central frequency indicates that model stability and numerical anisotropy also depend on the pulse shape or phase characteristics of the source. From the present study it is concluded that in addition to previous work, which showed the numerical stability a frequency dependent phenomenon, it also depends on the phase spectrum of the input source wavelet. (author)

380

Finite-Difference Simulations of the 1927 Jericho Earthquake

Four possible scenarios of the 1927 Jericho earthquake are tested by simulating 75 seconds of 1.5 Hz-wave propagation in a 3D model of the Dead Sea Basin (DSB) substructure. The scenarios examine the effects of various source and rupture parameters, since the original parameters could not be constrained by the sparse data gathered in 1927. The simulations are carried out using a fourth-order staggered-grid finite-difference (FD) method. Peak ground velocities and spectral accelerations (at 0.5 Hz, 1 Hz, and 1.5 Hz) are determined from the time-histories. Finally those are compared to an intensity map, which shows a considerable heterogeneous distribution of large intensities. The purpose of this study is (a) to find possible explanations for the heterogeneous intensity distribution and (b) to determine the best fitting source and rupture parameters for this event. We find the best overall agreement to correspond to scenario 2, a unilateral rupture on a vertical strike-slip fault with a fault plane of 12 12 km2. The rupture starts at a depth of 7 km at the southern end of the fault and propagates with a velocity that amounts to 90% of the local shear wave speed.

Gottschämmer, Ellen; Wenzel, Friedemann; Wust-Bloch, Hillel; Ben-Avraham, Zvi

381

A finite-difference contrast source inversion method

International Nuclear Information System (INIS)

We present a contrast source inversion (CSI) algorithm using a finite-difference (FD) approach as its backbone for reconstructing the unknown material properties of inhomogeneous objects embedded in a known inhomogeneous background medium. Unlike the CSI method using the integral equation (IE) approach, the FD-CSI method can readily employ an arbitrary inhomogeneous medium as its background. The ability to use an inhomogeneous background medium has made this algorithm very suitable to be used in through-wall imaging and time-lapse inversion applications. Similar to the IE-CSI algorithm the unknown contrast sources and contrast function are updated alternately to reconstruct the unknown objects without requiring the solution of the full forward problem at each iteration step in the optimization process. The FD solver is formulated in the frequency domain and it is equipped with a perfectly matched layer (PML) absorbing boundary condition. The FD operator used in the FD-CSI method is only dependent on the background medium and the frequency of operation, thus it does not change throughout the inversion process. Therefore, at least for the two-dimensional (2D) configurations, where the size of the stiffness matrix is manageable, the FD stiffness matrix can be inverted using a non-iterative inversion matrix approach such as a Gauss elimination method for the sparse matrix. In this case, an LU decomposition needs to be done only once and can then be reused for multiple souce and can then be reused for multiple source positions and in successive iterations of the inversion. Numerical experiments show that this FD-CSI algorithm has an excellent performance for inverting inhomogeneous objects embedded in an inhomogeneous background medium

382

Coarse mesh finite difference formulation for accelerated Monte Carlo eigenvalue calculation

International Nuclear Information System (INIS)

Highlights: • Coarse Mesh Finite Difference (CMFD) formulation is applied to Monte Carlo (MC) calculations. • CMFD leads very rapid convergence of the MC fission source distribution. • The variance bias problem is less significant in three dimensional problems for local tallies. • CMFD-MC enables using substantially many particles without causing waste in inactive cycles. • CMFD-MC is suitable for power reactor calculations requiring many particles per cycle. - Abstract: An efficient Monte Carlo (MC) eigenvalue calculation method for source convergence acceleration and stabilization is developed by employing the Coarse Mesh Finite Difference (CMFD) formulation. The detailed methods for constructing the CMFD system using proper MC tallies are devised such that the coarse mesh homogenization parameters are dynamically produced. These involve the schemes for tally accumulation and periodic reset of the CMFD system. The method for feedback which is to adjust the MC fission source distribution (FSD) using the CMFD global solution is then introduced through a weight adjustment scheme. The CMFD accelerated MC (CMFD-MC) calculation is examined first for a simple one-dimensional multigroup problem to investigate the effectiveness of the accelerated fission source convergence process and also to analyze the sensitivity of the CMFD-MC solutions on the size of coarse meshes and on the number of CMFD energy groups. The performance of CMFD acceleration is then assessed for a set of two-dimensional and three-dimensional multigroup (3D) pressurized water reactor core problems. It is demonstrated that very rapid convergence of the MC FSD is possible with the CMFD formulation in that a sufficiently converged MC FSD can be obtained within 20 cycles even for large three-dimensional problems which would require more than 600 inactive cycles with the standard MC fission source iteration scheme. It is also shown that the optional application of the CMFD formulation in the active cycles can stabilize FSDs such that the real-to-apparent variance ratio of the local tallies can be reduced. However, due to the reduced importance of the variance bias in fine local tallies of 3D MC eigenvalue problems, the effectiveness of CMFD in tally stabilization turns out to be not so great

383

In this paper, a fully discrete local discontinuous Galerkin (LDG) finite element method is considered for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation. The scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. We prove that our scheme is unconditional stable and $L^2$ error estimate for the linear case with the convergence rate $O(h^{k+1}+(\\Delta t)^2+(\\Delta t)^\\frac{\\alpha}{2}h^{k+1/2})$. Numerical examples are presented to show the efficiency and accuracy of our scheme.

Wei, Leilei

2012-01-01

384

3D frequency-domain finite-difference modeling of acoustic wave propagation

We present a 3D frequency-domain finite-difference method for acoustic wave propagation modeling. This method is developed as a tool to perform 3D frequency-domain full-waveform inversion of wide-angle seismic data. For wide-angle data, frequency-domain full-waveform inversion can be applied only to few discrete frequencies to develop reliable velocity model. Frequency-domain finite-difference (FD) modeling of wave propagation requires resolution of a huge sparse system of linear equations. If this system can be solved with a direct method, solutions for multiple sources can be computed efficiently once the underlying matrix has been factorized. The drawback of the direct method is the memory requirement resulting from the fill-in of the matrix during factorization. We assess in this study whether representative problems can be addressed in 3D geometry with such approach. We start from the velocity-stress formulation of the 3D acoustic wave equation. The spatial derivatives are discretized with second-order accurate staggered-grid stencil on different coordinate systems such that the axis span over as many directions as possible. Once the discrete equations were developed on each coordinate system, the particle velocity fields are eliminated from the first-order hyperbolic system (following the so-called parsimonious staggered-grid method) leading to second-order elliptic wave equations in pressure. The second-order wave equations discretized on each coordinate system are combined linearly to mitigate the numerical anisotropy. Secondly, grid dispersion is minimized by replacing the mass term at the collocation point by its weighted averaging over all the grid points of the stencil. Use of second-order accurate staggered- grid stencil allows to reduce the bandwidth of the matrix to be factorized. The final stencil incorporates 27 points. Absorbing conditions are PML. The system is solved using the parallel direct solver MUMPS developed for distributed-memory computers. The MUMPS solver is based on a multifrontal method for LU factorization. We used the METIS algorithm to perform re-ordering of the matrix coefficients before factorization. Four grid points per minimum wavelength is used for discretization. We applied our algorithm to the 3D SEG/EAGE synthetic onshore OVERTHRUST model of dimensions 20 x 20 x 4.65 km. The velocities range between 2 and 6 km/s. We performed the simulations using 192 processors with 2 Gbytes of RAM memory per processor. We performed simulations for the 5 Hz, 7 Hz and 10 Hz frequencies in some fractions of the OVERTHRUST model. The grid interval was 100 m, 75 m and 50 m respectively. The grid dimensions were 207x207x53, 275x218x71 and 409x109x102 respectively corresponding to 100, 80 and 25 percents of the model respectively. The time for factorization is 20 mn, 108 mn and 163 mn respectively. The time for resolution was 3.8, 9.3 and 10.3 s per source. The total memory used during factorization is 143, 384 and 449 Gbytes respectively. One can note the huge memory requirement for factorization and the efficiency of the direct method to compute solutions for a large number of sources. This highlights the respective drawback and merit of the frequency-domain approach with respect to the time- domain counterpart. These results show that 3D acoustic frequency-domain wave propagation modeling can be performed at low frequencies using direct solver on large clusters of Pcs. This forward modeling algorithm may be used in the future as a tool to image the first kilometers of the crust by frequency-domain full-waveform inversion. For larger problems, we will use the out-of-core memory during factorization that has been implemented by the authors of MUMPS.

Operto, S.; Virieux, J.

2006-12-01

385

Unconditionally stable finite-difference time-domain methods for modeling the Sagnac effect

We present two unconditionally stable finite-difference time-domain (FDTD) methods for modeling the Sagnac effect in rotating optical microsensors. The methods are based on the implicit Crank-Nicolson scheme, adapted to hold in the rotating system reference frame—the rotating Crank-Nicolson (RCN) methods. The first method (RCN-2) is second order accurate in space whereas the second method (RCN-4) is fourth order accurate. Both methods are second order accurate in time. We show that the RCN-4 scheme is more accurate and has better dispersion isotropy. The numerical results show good correspondence with the expression for the classical Sagnac resonant frequency splitting when using group refractive indices of the resonant modes of a microresonator. Also we show that the numerical results are consistent with the perturbation theory for the rotating degenerate microcavities. We apply our method to simulate the effect of rotation on an entire Coupled Resonator Optical Waveguide (CROW) consisting of a set of coupled microresonators. Preliminary results validate the formation of a rotation-induced gap at the center of a transfer function of a CROW.

Novitski, Roman; Scheuer, Jacob; Steinberg, Ben Z.

2013-02-01

386

Unconditionally stable finite-difference time-domain methods for modeling the Sagnac effect.

We present two unconditionally stable finite-difference time-domain (FDTD) methods for modeling the Sagnac effect in rotating optical microsensors. The methods are based on the implicit Crank-Nicolson scheme, adapted to hold in the rotating system reference frame-the rotating Crank-Nicolson (RCN) methods. The first method (RCN-2) is second order accurate in space whereas the second method (RCN-4) is fourth order accurate. Both methods are second order accurate in time. We show that the RCN-4 scheme is more accurate and has better dispersion isotropy. The numerical results show good correspondence with the expression for the classical Sagnac resonant frequency splitting when using group refractive indices of the resonant modes of a microresonator. Also we show that the numerical results are consistent with the perturbation theory for the rotating degenerate microcavities. We apply our method to simulate the effect of rotation on an entire Coupled Resonator Optical Waveguide (CROW) consisting of a set of coupled microresonators. Preliminary results validate the formation of a rotation-induced gap at the center of a transfer function of a CROW. PMID:23496635

Novitski, Roman; Scheuer, Jacob; Steinberg, Ben Z

2013-02-01

387

It is widely accepted that they are oversampled in spatial grid spacing and temporal time step in the high speed medium if uniform grids are used for the numerical simulation. This oversampled grid spacing and time step will lower the efficiency of the calculation, especially high velocity contrast exists. Based on the collocated-grid finite-difference method (FDM), we present an algorithm of spatial discontinuous grid, with localized grid blocks and locally varying time steps, which will increase the efficiency of simulation of seismic wave propagation and earthquake strong ground motion. According to the velocity structure, we discretize the model into discontinuous grid blocks, and the time step of each block is determined according to the local stability. The key problem of the discontinuous grid method is the connection between grid blocks with different grid spacing. We use a transitional area overlapped by both of the finer and the coarser grids to deal with the problem. In the transitional area, the values of finer ghost points are obtained by interpolation from the coarser grid in space and time domain, while the values of coarser ghost points are obtained by downsampling from the finer grid. How to deal with coarser ghost points can influent the stability of long time simulation. After testing different downsampling methods and finally we choose the Gaussian filtering. Basically, 4th order Rung-Kutta scheme will be used for the time integral for our numerical method. For our discontinuous grid FDM, discontinuous time steps for the coarser and the finer grids will be used to increase the simulation efficiency. Numerical tests indicate that our method can provide a stable solution even for the long time simulation without any additional filtration for grid spacing ratio n=2. And for larger grid spacing ratio, Gaussian filtration could be used to preserve the stability. With the collocated-grid FDM, which is flexible and accurate in implementation of free surface condition with topography, our method has high advantage in the simulation of strong ground motion of real earthquake.

Li, H.; Zhang, Z.; Chen, X.

2012-12-01

388

Results of a sensitivity study are presented from various configurations of the NEMO ocean model in the Black Sea. The standard choices of vertical discretization, viz. z-levels, s-coordinates and enveloped s-coordinates, all show their limitations in the areas of complex topography. Two new hydrid vertical coordinate schemes are presented: the "s-on-top-of-z" and its enveloped version. The hybrid grids use s-coordinates or enveloped s-coordinates in the upper layer, from the sea surface ...

Shapiro, G.; Luneva, M.; Pickering, J.; Storkey, D.

2012-01-01

389

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

390

Stability analysis and dynamics preserving nonstandard finite difference schemes for a malaria model

When both human and mosquito populations vary, forward bifurcation occurs if the basic reproduction number R0 is less than one in the absence of disease-induced death. When the disease-induced death rate is large enough R0 = 1 is a subcritical backward bifurcation point. The domain for the study of the dynamics is reduced to a compact and feasible region, where the system admits a speci c algebraic decomposition into infective and non-infected humans and mosquitoes. Stability ...

Anguelov, Roumen; Dumont, Yves; Lubuma, Jean M. -s; Mureithi, Eunice W.

2013-01-01

391

A symbolic approach to finite difference schemes of partial differential equations

Zusammenfassung. In dieser Arbeit werden Probleme der numerischen Lösung finiter Differenzenverfahren partieller Differentialgleichungen in einem algebraischen Ansatz behandelt. Es werden sowohl theoretische Ergebnisse präsentiert als auch die praktische Implementierung mithilfe der Systeme SINGULAR und QEPCAD vorgeführt. Dabei beziehen sich die algebraischen Methoden auf zwei unterschiedliche Aspekte bei finiten Differenzenverfahren: die Erzeugung von Schemata mithilfe von Gröbnerbasen u...

Dingler, Christian

2010-01-01

392

International Nuclear Information System (INIS)

Seismic waves radiated from an earthquake propagate in the Earth and the ground shaking is felt and recorded at (or near) the ground surface. Understanding the wave propagation with respect to the Earth's structure and the earthquake mechanisms is one of the main objectives of seismology, and predicting the strong ground shaking for moderate and large earthquakes is essential for quantitative seismic hazard assessment. The finite difference scheme for solving the wave propagation problem in elastic (sometimes anelastic) media has been more widely used since the 1970s than any other numerical methods, because of its simple formulation and implementation, and its easy scalability to large computations. This paper briefly overviews the advances in finite difference simulations, focusing particularly on earthquake mechanics and the resultant wave radiation in the near field. As the finite difference formulation is simple (interpolation is smooth), an easy coupling with other approaches is one of its advantages. A coupling with a boundary integral equation method (BIEM) allows us to simulate complex earthquake source processes

393

A convergent finite difference method for a nonlinear variational wave equation

We establish rigorously convergence of a semi-discrete upwind scheme for the nonlinear variational wave equation $u_{tt} - c(u)(c(u) u_x)_x = 0$ with $u|_{t=0}=u_0$ and $u_t|_{t=0}=v_0$. Introducing Riemann invariants $R=u_t+c u_x$ and $S=u_t-c u_x$, the variational wave equation is equivalent to $R_t-c R_x=\\tilde c (R^2-S^2)$ and $S_t+c S_x=-\\tilde c (R^2-S^2)$ with $\\tilde c=c'/(4c)$. An upwind scheme is defined for this system. We assume that the the speed $c$ is positive, increasing and both $c$ and its derivative are bounded away from zero and that $R|_{t=0}, S|_{t=0}\\in L^1\\cap L^3$ are nonpositive. The numerical scheme is illustrated on several examples.

Holden, H; Risebro, N H

2007-01-01

394

Finite-Difference Numerical Simulation of Seismic Gradiometry

We use the phrase seismic gradiometry to refer to the developing research area involving measurement, modeling, analysis, and interpretation of spatial derivatives (or differences) of a seismic wavefield. In analogy with gradiometric methods used in gravity and magnetic exploration, seismic gradiometry offers the potential for enhancing resolution, and revealing new (or hitherto obscure) information about the subsurface. For example, measurement of pressure and rotation enables the decomposition of recorded seismic data into compressional (P) and shear (S) components. Additionally, a complete observation of the total seismic wavefield at a single receiver (including both rectilinear and rotational motions) offers the possibility of inferring the type, speed, and direction of an incident seismic wave. Spatially extended receiver arrays, conventionally used for such directional and phase speed determinations, may be dispensed with. Seismic wave propagation algorithms based on the explicit, time-domain, finite-difference (FD) numerical method are well-suited for investigating gradiometric effects. We have implemented in our acoustic, elastic, and poroelastic algorithms a point receiver that records the 9 components of the particle velocity gradient tensor. Pressure and particle rotation are obtained by forming particular linear combinations of these tensor components, and integrating with respect to time. All algorithms entail 3D O(2,4) FD solutions of coupled, first- order systems of partial differential equations on uniformly-spaced staggered spatial and temporal grids. Numerical tests with a 1D model composed of homogeneous and isotropic elastic layers show isolation of P, SV, and SH phases recorded in a multiple borehole configuration, even in the case of interfering events. Synthetic traces recorded by geophones and rotation receivers in a shallow crosswell geometry with randomly heterogeneous poroelastic models also illustrate clear P (fast and slow) and S separation. Finally, numerical tests of the "point seismic array" concept are oriented toward understanding its potential and limitations. Sandia National Laboratories is a multiprogram science and engineering facility operated by Sandia Corporation, a Lockheed-Martin company, for the United States Department of Energy under contract DE- AC04-94AL85000.

Aldridge, D. F.; Symons, N. P.; Haney, M. M.

2006-12-01

395

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.

Ouyang, Chaojun; He, Siming; Xu, Qiang; Luo, Yu; Zhang, Wencheng

2013-03-01

396

Fast finite difference Poisson solvers on heterogeneous architectures

In this paper we propose and evaluate a set of new strategies for the solution of three dimensional separable elliptic problems on CPU-GPU platforms. The numerical solution of the system of linear equations arising when discretizing those operators often represents the most time consuming part of larger simulation codes tackling a variety of physical situations. Incompressible fluid flows, electromagnetic problems, heat transfer and solid mechanic simulations are just a few examples of application areas that require efficient solution strategies for this class of problems. GPU computing has emerged as an attractive alternative to conventional CPUs for many scientific applications. High speedups over CPU implementations have been reported and this trend is expected to continue in the future with improved programming support and tighter CPU-GPU integration. These speedups by no means imply that CPU performance is no longer critical. The conventional CPU-control-GPU-compute pattern used in many applications wastes much of CPU's computational power. Our proposed parallel implementation of a classical cyclic reduction algorithm to tackle the large linear systems arising from the discretized form of the elliptic problem at hand, schedules computing on both the GPU and the CPUs in a cooperative way. The experimental result demonstrates the effectiveness of this approach.

Valero-Lara, Pedro; Pinelli, Alfredo; Prieto-Matias, Manuel

2014-04-01

397

The TSN (Traction-at-Split-Nodes) method has been developed independently by Andrews (1973, 1976, 1999) and Day (1977, 1982). Andrews implemented his TSN formulation in the finite-difference scheme in which spatial differentiation is equivalent to the 2nd-order finite-element method. Day implemented his slightly different formulation of the TSN method in the 2nd-order partly-staggered finite-difference scheme. Dalguer and Day (2006) adapted the TSN method to the velocity-stress staggered-grid finite-difference scheme. Whereas the 4th-order spatial differencing is applied outside the fault, the 2nd-order differencing is applied along the fault plane. We present two implementations of the Day's TSN formulation in the velocity-stress staggered-grid finite-difference scheme for a 3D viscoelastic medium. In the first one we apply the 2nd-order spatial differencing everywhere in the grid including derivatives at the fault in the direction perpendicular to the fault plane. In the second implementation we similarly apply the 4th-order spatial differencing. In both cases we use the adjusted finite-difference approximations (AFDA, Kristek et al. 2002, Moczo et al. 2004) to derivatives in the direction perpendicular to the fault plane in order to have the same order of approximation everywhere. We numerically investigate convergence rates of both implementations with respect to rupture-time, final-slip, and peak-slip-rate metrics. Moreover, we compare the numerical solutions to those obtained by the finite-element implementation of the TSN method.

Kristek, J.; Moczo, P.; Galis, M.

2006-12-01

398

Dynamic Rupture Simulation of Bent Faults with a New Finite Difference Approach

Many questions about physical parameters governing the rupture propagation of earthquakes find their answers within realistic dynamic considerations. For instance, sophisticated constitutive relations based on laboratory experiments have led to a better understanding of rupture evolution from its very beginning to its arrest. In fact, large amount of field observations as well as recent simulations have shown the importance of considering more reasonable geological settings (e.g., bent and step-over fault geometries; heterogeneous surrounding media). However, despite the development of powerful numerical tools, one important question remains unanswered. How important is in rupture process the feedback coming from a heterogeneous structure if the fault geometry is complex? To start answering this question, we propose a numerical approach based in a new staggered-grid finite-difference technique. In this work, we use a recently proposed four-order staggered-grid finite-difference scheme to dynamically model in-plane (mode II) shear fracturing propagation in faults with any pre-established geometry. In contrast with the classical 2-D staggered grid elementary cell in which all the elastic fields are defined in different positions (except the normal stresses), the stencil used here considers the velocity and stress fields separately in only two staggered grids. Such an elementary structure is a straightforward consequence of a new definition of the four-order spatial differential operators: they are decoupled into two 45-degree rotated operators. This approach permits efficient treatment of boundary conditions to impose the shear stress drop in the nodes where the entire stress tensor is located. Furthermore, this procedure reduces numerical anisotropy along preferred directions and provides stable solutions for any fault orientation. The fault is defined as a set of point sources placed in the middle of the grid without using any ad hoc numerical ghost plane often used in finite-difference approaches. Thus, the fault is a sum of point sources that are taken as close as possible from the given analytical fault line. One point source is defined as a cluster of stress-grid points. Numerically, there exist different point-source configurations that are equivalent. However, configurations with more stress-grid points allow better rupture simulations for any fault geometry. Simulations with the standard slip-weakening constitutive law are presented. To quantify the slip upon every point source having any fault orientation, we define a slip function which depends on the displacement field around every point source but not in the fault plane orientation. We also present spontaneous and velocity fixed self-similar simulations. As we are dealing with a finite-difference approach, the proposed methodology makes possible to analyze the effect of arbitrary heterogeneous media surrounding the fault region in the dynamics of seismic sources evolution.

Cruz-Atienza, V. M.; Virieux, J.; Operto, S.

2003-04-01

399

Finite difference time domain modeling of light matter interaction in light-propelled microtools

DEFF Research Database (Denmark)

Direct laser writing and other recent fabrication techniques offer a wide variety in the design of microdevices. Hence, modeling such devices requires analysis methods capable of handling arbitrary geometries. Recently, we have demonstrated the potential of microtools, optically actuated microstructures with functionalities geared towards biophotonics applications. Compared to dynamic beam shaping alone, microtools allow more complex interactions between the shaped light and the biological samples at the receiving end. For example, strongly focused light coming from a tapered tip of a microtool may trigger highly localized non linear processes in the surface of a cell. Since these functionalities are strongly dependent on design, it is important to use models that can handle complexities and take in little simplifying assumptions about the system. Hence, we use the finite difference time domain (FDTD) method which is a direct discretization of the fundamental Maxwell's equations applicable to many optical systems. Using the FDTD, we investigate light guiding through microstructures as well as the field enhancement as light comes out of our tapered wave guide designs. Such calculations save time as it helps optimize the structures prior to fabrication and experiments. In addition to field distributions, optical forces can also be obtained using the Maxwell stress tensor formulation. By calculating the forces on bent waveguides subjected to tailored static light distributions, we demonstrate novel methods of optical micromanipulation which primarily result from the particle's geometry as opposed to the directly moving the light distributions as in conventional trapping.

Bañas, Andrew Rafael; Palima, Darwin

2013-01-01

400

Mimetic finite difference method for the stokes problem on polygonal meshes

Energy Technology Data Exchange (ETDEWEB)

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.

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

401

Finite difference modelling of bulk high temperature superconducting cylindrical hysteresis machines

International Nuclear Information System (INIS)

A mathematical model of the critical state based on averaged fluxon motion has been implemented to solve for the current and field distributions inside a high temperature superconducting hysteresis machine. The machine consists of a rotor made from a solid cylindrical single domain HTS placed in a perpendicular rotating field. The solution technique uses the finite difference approximation for a two-dimensional domain, discretized in cylindrical polar co-ordinates. The torque generated or equivalently the hysteresis loss in such a machine has been investigated using the model. It was found that to maximize the efficiency, the field needs to penetrate the rotor such that B0/?0JcR=0.56, where B0 is the applied field amplitude, Jc is the critical current density and R is the rotor radius. This corresponds to a penetration that is 27% greater than that which reaches the centre of the rotor. An examination of the torque density distributions across the rotor reveal that for situations where the field is less than optimal, a significant increase in the performance can be achieved by removing an inner cylinder from the rotor. (author)

402

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

403

Directory of Open Access Journals (Sweden)

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.

Tsugio Fukuchi

2014-06-01

404

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.

Fukuchi, Tsugio

2014-06-01

405

Analysis of an Il’in Scheme for a System of Singularly Perturbed Convection-Diffusion Equations

Directory of Open Access Journals (Sweden)

Full Text Available In this paper, a numerical solution for a system of singularly perturbed convection-diffusion equations is studied. The system is discretized by the Il’in scheme on a uniform mesh. It is proved that the numerical scheme has first order accuracy, which is uniform with respect to the perturbation parameters. We show that the condition number of the discrete linear system obtained from applying the Il’in scheme for a system of singularly perturbed convection-diffusion equations is O(N and the relevant coefficient matrix is well conditioned in comparison with the matrices obtained from applying upwind finite difference schemes on this problem. Numerical results confirm the theory of the method.

Mohammad Ghorbanzadeh

2011-07-01

406

Energy Technology Data Exchange (ETDEWEB)

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.

Appelo, D; Petersson, N A

2007-12-17

407

In this paper, a new family of implicit compact finite difference schemes for computation of unsteady convection-diffusion equation with variable convection coefficient is proposed. The schemes are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter. The proposed schemes, where transport variable and its first derivatives are carried as the unknowns, combine virtues of compact discretization and Pad\\'{e} scheme for spatial derivative. These schemes which are based on five point stencil with constant coefficients, named as \\emph{(5,5) Constant Coefficient 4th Order Compact} [(5,5)CC-4OC], give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods. These schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable. They efficiently capture both transient and steady solutions of linear and ...

Sen, Shuvam

2012-01-01

408

Vertical Discretization of Hydrostatic Primitive Equations with Finite Element Method

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.

Yi, Tae-Hyeong; Park, Ja-Rin

2014-05-01

409

Digital watermarking is the process to hide digital pattern directly into a digital content. Digital watermarking techniques are used to address digital rights management, protect information and conceal secrets. An invisible non-blind watermarking approach for gray scale images is proposed in this paper. The host image is decomposed into 3-levels using Discrete Wavelet Transform. Based on the parent-child relationship between the wavelet coefficients the Set Partitioning in...

Abdur Shahid; Shahriar Badsha; Md. Rethwan Kabeer; Junaid Ahsan; Mufti Mahmud

2012-01-01

410

On an explicit finite difference method for fractional diffusion equations

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.

Yuste, S B

2003-01-01

411

Directory of Open Access Journals (Sweden)

Full Text Available Results of a sensitivity study are presented from various configurations of the NEMO ocean model in the Black Sea. The standard choices of vertical discretization, viz. z-levels, s-coordinates and enveloped s-coordinates, all show their limitations in the areas of complex topography. Two new hydrid vertical coordinate schemes are presented: the "s-on-top-of-z" and its enveloped version. The hybrid grids use s-coordinates or enveloped s-coordinates in the upper layer, from the sea surface to the depth of the shelf break, and z-coordinates are set below this level. The study is carried out for a number of idealised and real world settings. The hybrid schemes help reduce errors generated by the standard schemes in the areas of steep topography. Results of sensitivity tests with various horizontal diffusion formulations show that the mesoscale activity is better captured with a significantly smaller value of Smagorinsky viscosity coefficient than it was previously suggested.

G. Shapiro

2012-11-01

412

Scientific Electronic Library Online (English)

Full Text Available SciELO Brazil | Language: Portuguese Abstract in portuguese Este trabalho foi desenvolvido com o objetivo de se apresentar a aplicação de um esquema de discretização mais eficiente para volumes finitos, denominado Flux-Spline utilizando-se, para tal, de dois casos de transporte difusivo de umidade e calor, através de um meio poroso capilar. Os resultados da [...] solução numérica do sistema de equações formado pelas equações de Luikov mostram desempenho adequado do esquema para este tipo de problema, quando comparado ao tradicional esquema de diferença central e ao método da transformada integral. Abstract in english This study was conducted with the objective to present a more efficient discretization scheme to finite volumes method called Flux-Spline, utilising for the purpose two cases of pure diffusion in capillary porous media. The results of numerical simulation of the equations system formed by Luikov equ [...] ations showed a good performance of the scheme in comparison to the Central Difference Scheme and Generalised Integral Transform Technique method.

Paulo C., Oliveira; José L., Lima.

2003-04-01

413

Results of a sensitivity study are presented from various configurations of the NEMO ocean model in the Black Sea. The standard choices of vertical discretization, viz. z-levels, s-coordinates and enveloped s-coordinates, all show their limitations in the areas of complex topography. Two new hydrid vertical coordinate schemes are presented: the "s-on-top-of-z" and its enveloped version. The hybrid grids use s-coordinates or enveloped s-coordinates in the upper layer, from the sea surface to the depth of the shelf break, and z-coordinates are set below this level. The study is carried out for a number of idealised and real world settings. The hybrid schemes help reduce errors generated by the standard schemes in the areas of steep topography. Results of sensitivity tests with various horizontal diffusion formulations show that the mesoscale activity is better captured with a significantly smaller value of Smagorinsky viscosity coefficient than it was previously suggested.

Shapiro, G.; Luneva, M.; Pickering, J.; Storkey, D.

2012-11-01

414

Energy Technology Data Exchange (ETDEWEB)

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.)

Chen, Y.

2005-04-01

415

The high order finite difference and multigrid methods have been successfully applied to direct numerical simulation (DNS) for flow transition in 3D channels and 3D boundary layers with 2D and 3D isolated and distributed roughness in a curvilinear coordinate system. A fourth-order finite difference technique on stretched and staggered grids, a fully-implicit time marching scheme, a semicoarsening multigrid method associated with line distributive relaxation scheme, and a new treatment of the outflow boundary condition, which needs only a very short buffer domain to damp all wave reflection, are developed. These approaches make the multigrid DNS code very accurate and efficient. This makes us not only able to do spatial DNS for the 3D channel and flat plate at low computational costs, but also able to do spatial DNS for transition in the 3D boundary layer with 3D single and multiple roughness elements. Numerical results show good agreement with the linear stability theory, the secondary instability theory, and a number of laboratory experiments.

Liu, C.; Liu, Z.

1993-01-01

416

Mixed five-point/nine-point finite-difference formulation of multiphase flow in petroleum reservoirs

Energy Technology Data Exchange (ETDEWEB)

Nine-point finite-difference formulation of multiphase flow in the x-y pane is well-known to reduce grid-orientation effects. This paper proposes a new implicit pressure, explicit saturation (IMPES) formulation that uses implicit pressures for the customary five points of the x-y grid system, explicit pressures for the four points on the diagonals, and explicit saturations for all nine points. This proposed split-operator scheme produces virtually identical results as the standard nine-point formulation of Yanosik and McCracken, but can be easily implemented in existing five-point simulators without requiring any changes in the commonly used solution algorithms, such as the D-4 Gauss. The use of this new technique should enhance the accuracy of existing large-scale IMPES simulators with little increase in storage requirement or computing time. The method can be used with any set of nine-point coefficients and can be extended to fully implicit simulators.

Ostebo, B. (Statoil, Stavanger (Norway)); Kazemi, H. (Marathon Oil Co., Petroleum Technology Center, Littleton, CO (United States))

1992-11-01

417

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.

418

Finite difference modeling of coherent wave amplification in the Earth's radiation belts

Modeling of gyroresonant wave-particle interactions in the radiation belts requires solving the Vlasov-Maxwell system of equations in an inhomogenous background geomagnetic field. Previous works have employed particle-in-cell methods or Eulerian solvers (such as the Vlasov Hybrid Simulation code) to provide numerical solutions to the problem. In this report, we provide an alternative numerical approach by utilizing a first order finite difference upwind scheme. When coupled to the narrowband Maxwell's equations, the model reproduces linear as well as nonlinear wave growth of coherent signals. Wave growth is nonlinear growth when the wave amplitude exceeds the minimum value for phase trapping of counterstreaming particles and is linear otherwise. The model also demonstrates free-running frequency variation for a case with a high linear growth rate. In addition, the model confirms the theoretical prediction of a stable "phase-space hole" during the nonlinear growth process. The plasma parameters and L shell used in this study are typical of those associated with the Siple Station wave injection experiment.

Harid, V.; Go?kowski, M.; Bell, T.; Li, J. D.; Inan, U. S.

2014-12-01

419

Accurate 3-D finite difference computation of traveltimes in strongly heterogeneous media

Seismic traveltimes and their spatial derivatives are the basis of many imaging methods such as pre-stack depth migration and tomography. A common approach to compute these quantities is to solve the eikonal equation with a finite-difference scheme. If many recently published algorithms for resolving the eikonal equation do now yield fairly accurate traveltimes for most applications, the spatial derivatives of traveltimes remain very approximate. To address this accuracy issue, we develop a new hybrid eikonal solver that combines a spherical approximation when close to the source and a plane wave approximation when far away. This algorithm reproduces properly the spherical behaviour of wave fronts in the vicinity of the source. We implement a combination of 16 local operators that enables us to handle velocity models with sharp vertical and horizontal velocity contrasts. We associate to these local operators a global fast sweeping method to take into account all possible directions of wave propagation. Our formulation allows us to introduce a variable grid spacing in all three directions of space. We demonstrate the efficiency of this algorithm in terms of computational time and the gain in accuracy of the computed traveltimes and their derivatives on several numerical examples.

Noble, M.; Gesret, A.; Belayouni, N.

2014-12-01

420

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

421

Spectral method and high-order finite differences for the nonlinear cable equation.

We use high-order approximation schemes for the space derivatives in the nonlinear cable equation and investigate the behavior of numerical solution errors by using exact solutions, where available, and grid convergence. The space derivatives are numerically approximated by means of differentiation matrices. Nonlinearity in the equation arises from the Hodgkin-Huxley dynamics of the gating variables for ion channels. We have investigated in particular the effects of synaptic current distribution and compared the accuracy of the spectral solutions with that of finite differencing. A flexible form for the injected current is used that can be adjusted smoothly from a very broad to a narrow peak, which furthermore leads, for the passive cable, to a simple, exact solution. We have used three distinct approaches to assess the numerical solutions: comparison with exact solutions in an unbranched passive cable, the convergence of solutions with progressive refinement of the grid in an active cable, and the simulation of spike initiation in a biophysically realistic single-neuron model. The spectral method provides good numerical solutions for passive cables comparable in accuracy to those from the second-order finite difference method and far greater accuracy in the case of a simulated system driven by inputs that are smoothly distributed in space. It provides faster convergence in active cables and in a realistic neuron model due to better approximation of propagating spikes. PMID:20337534

Omurtag, Ahmet; Lytton, William W

2010-08-01

422

Parallel 3d Finite-Difference Time-Domain Method on Multi-Gpu Systems

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.

Du, Liu-Ge; Li, Kang; Kong, Fan-Min; Hu, Yuan

423

Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs

Energy Technology Data Exchange (ETDEWEB)

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.

Bollig, Evan F., E-mail: bollig@scs.fsu.edu [Department of Scientific Computing, Florida State University, 400 Dirac Science Library, Tallahassee, FL 32306 (United States); Flyer, Natasha, E-mail: flyer@ucar.edu [Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, 1850 Table Mesa Dr., Boulder, CO 80305 (United States); Erlebacher, Gordon, E-mail: gerlebacher@fsu.edu [Department of Scientific Computing, Florida State University, 400 Dirac Science Library, Tallahassee, FL 32306 (United States)

2012-08-30

424

Directory of Open Access Journals (Sweden)

Full Text Available In this article, acoustic signal based traffic flow evaluation method using the calculus of finite differences is considered. The analytical survey for traffic flow evaluations using acoustic signals is performed. The calculus of finite differences application in traffic flow evaluation is investigated. As the result of this work, the considered method of efficiency is evaluated

Posmitnyy Y. V.

2012-12-01

425

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.

Beam, R. M.; Warming, R. F.

1976-01-01

426

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

427

Scientific Electronic Library Online (English)

Full Text Available SciELO Colombia | Language: English Abstract in spanish En este artículo se desarrolla un nuevo esquema de cuatro puntos para la subdivisión interpolante de curvas basado en la primera derivada discreta (DFDS), el cual, reduce la formación de oscilaciones indeseables que pueden surgir en la curva límite cuando los puntos de control no obedecen a una para [...] metrización uniforme. Se empleó un conjunto de 3000 curvas cuyos puntos de control fueron generados aleatoriamente. Curvas suaves fueron obtenidas tras siete pasos de subdivisión empleando los esquemas DFDS, Cuatro-puntos (4P), Nuevo de cuatro-puntos (N4P), Cuatro-puntos ajustado (T4P) y el Esquema interpolante geométricamente controlado (GC4P). Sobre cada curva suave se evaluó la propiedad de tortuosidad. Un análisis de las distribuciones de frecuencia obtenidas para esta propiedad, empleando la prueba de Kruskal-Wallis, revela que el esquema DFDS posee los menores valores de tortuosidad en un rango más estrecho. Abstract in english This paper develops a new scheme of four points for interpolating curve subdivision based on the discrete first derivative (DFDS), which reduces the apparition of undesirable oscillations that can be formed on the limit curve when the control points do not follow a uniform parameterization. We used [...] a set of 3000 curves whose control points were randomly generated. Smooth curves were obtained after seven steps of subdivision using five schemes DFDS, Four-Point (4P), New four-point (N4P), Tight four-point (T4P) and the geometrically controlled scheme (GC4P). The tortuosity property was evaluated on every smooth curve. An analysis for the frequency distributions of this property using the Kruskal-Wallis test reveals that DFDS scheme has the lowest values in a close range.

ALBEIRO, ESPINOSA BEDOYA; GERMÁN, SÁNCHEZ TORRES; JOHN WILLIAN, BRANCH BEDOYA.

2013-08-01