Showing papers on "Finite difference coefficient published in 2016"
TL;DR: In this paper, a linear implicit finite difference scheme for solving the generalized time fractional burgers equation is proposed, which is shown to be globally stable and convergent, and the finite difference method is proved to be unconditional globally stable.
92 citations
TL;DR: Energy stability of the coupled numerical method is proven for the case of curved, nonconforming block-to-block interfaces and a provably energy stable coupling between curvilinear finite difference methods and a curved-triangle discontinuous Galerkin method is demonstrated.
Abstract: A methodology for handling block-to-block coupling of nonconforming, multiblock summation-by-parts finite difference methods is proposed. The coupling is based on the construction of projection operators that move a finite difference grid solution along an interface to a space of piecewise defined functions; we specifically consider discontinuous, piecewise polynomial functions. The constructed projection operators are compatible with the underlying summation-by-parts energy norm. Using the linear wave equation in two dimensions as a model problem, energy stability of the coupled numerical method is proven for the case of curved, nonconforming block-to-block interfaces. To further demonstrate the power of the coupling procedure, we show how it allows for the development of a provably energy stable coupling between curvilinear finite difference methods and a curved-triangle discontinuous Galerkin method. The theoretical results are verified through numerical simulations on curved meshes as well as eigenval...
55 citations
53 citations
01 Jan 2016
TL;DR: In this article, a general procedure for deriving difference approximations to spatial derivatives using Taylor series expansions is presented, and the error incurred in the approximation, the truncation or discretization error is thoroughly analyzed.
Abstract: This chapter provides in-depth coverage of the finite difference method (FDM) in the context of elliptic boundary value problems. The general procedure for deriving difference approximations to spatial derivatives using Taylor series expansions is first presented. The error incurred in the approximation – the truncation or discretization error – is thoroughly analyzed, and the procedure to develop higher-order approximations to derivatives is outlined. The implementation of the three canonical types of boundary conditions, i.e., Dirichlet, Neumann, and Robin, is discussed. Presentation of the matrix form of the discrete equations is finally followed by extension of the FDM to multidimensional geometries, including those described by the cylindrical coordinate system, and generalized curvilinear coordinates (body-fitted mesh).
51 citations
TL;DR: In this paper, a semi-implicit finite difference method is constructed for the time dependent Poisson-Nernst-Planck system and it is proved rigorously the mass conservation and energy decay property of the method and the method can be easily extended to the case of multi-ions.
32 citations
TL;DR: In this paper, an implicit finite difference scheme based on Caputo's definition to the fractional derivatives, and the upper and lower bounds to the spectral radius of the coefficient matrix of the difference scheme are estimated, with which the unconditional stability and convergence are proved.
Abstract: This paper deals with numerical solution to the multi-term time fractional diffusion equation in a finite domain. An implicit finite difference scheme is established based on Caputo's definition to the fractional derivatives, and the upper and lower bounds to the spectral radius of the coefficient matrix of the difference scheme are estimated, with which the unconditional stability and convergence are proved. The numerical results demonstrate the effectiveness of the theoretical analysis, and the method and technique can also be applied to other kinds of time/space fractional diffusion equations.
32 citations
TL;DR: A novel scheme for implementing finite difference by introducing a time-to-space wavelet mapping and demonstrating that this method is superior to standard finite difference methods, while comparable to Zhang’s optimised finite difference scheme.
Abstract: Efficient numerical seismic wavefield modelling is a key component of modern seismic imaging techniques, such as reverse-time migration and full-waveform inversion. Finite difference methods are perhaps the most widely used numerical approach for forward modelling, and here we introduce a novel scheme for implementing finite difference by introducing a time-to-space wavelet mapping. Finite difference coefficients are then computed by minimising the difference between the spatial derivatives of the mapped wavelet and the finite difference operator over all propagation angles. Since the coefficients vary adaptively with different velocities and source wavelet bandwidths, the method is capable to maximise the accuracy of the finite difference operator. Numerical examples demonstrate that this method is superior to standard finite difference methods, while comparable to Zhang’s optimised finite difference scheme.
30 citations
TL;DR: This paper proposes an alternating direction implicit scheme based on L 1 approximation for the one- and two-dimensional time-space fractional sub-diffusion equations and investigates the stability and convergence of the proposed methods.
Abstract: In this paper, we devote to the study of high order finite difference schemes for one- and two-dimensional time-space fractional sub-diffusion equations. A fourth order finite difference scheme is invoked for the spatial fractional derivatives, and the L 1 approximation is applied to the temporal fractional parts. For the two-dimensional case, an alternating direction implicit scheme based on L 1 approximation is proposed. The stability and convergence of the proposed methods are studied. Numerical experiments are performed to verify the effectiveness and accuracy of the proposed difference schemes.
28 citations
TL;DR: An energy estimate for the underlying hyperbolic equation with absorbing boundary conditions is derived and efficiency of the domain decomposition method on the reconstruction of the conductivity function in three dimensions is illustrated.
22 citations
TL;DR: Numerical results in one dimension, two dimensions, three dimensions and four dimensions have shown that the applied compact finite difference scheme has fourth order accuracy and can be efficiently implemented.
Abstract: Fourth-order compact finite difference scheme has been proposed for solving the Poisson equation with Dirichlet boundary conditions for some time. An efficient implementation of such numerical scheme is often desired for practical usage. In this paper, based on fast discrete Sine transform, we design an efficient algorithm to implement this scheme. To do this, Poisson equation is first discretized by fourth-order compact finite difference method. The subsequent discretized system is not solved by the usual method-matrix inversion, instead it is solved with the fast discrete Sine transform. By doing this way, the computational cost of proposed algorithm for such scheme with large grid numbers can be greatly reduced. Detailed numerical algorithm of this fast solver for one-dimensional, two-dimensional and three dimensional Poisson equation has been presented. Numerical results in one dimension, two dimensions, three dimensions and four dimensions have shown that the applied compact finite difference scheme has fourth order accuracy and can be efficiently implemented.
TL;DR: In this article, an unsteady flow of an anomalous Oldroyd-B fluid confined between two infinite parallel plates subject to no-slip condition at boundary is studied.
Abstract: In this article, we study an unsteady flow of an anomalous Oldroyd-B fluid confined between two infinite parallel plates subject to no-slip condition at boundary. The flow is induced by a linear acceleration of the lower plate in its own plane. A standard Galerkin finite element method is adopted to construct an approximate solution blended with a finite difference approximation for Caputo fractional time derivatives. The convergence of the proposed numerical scheme is substantiated, and error estimates are provided in appropriate norms. Some adequate numerical simulations are performed in order to elucidate the dominance of characteristic flow parameters of velocity field in the prescribed configuration.
TL;DR: In this paper, the authors studied the convergence rate of an explicit and an implicit-explicit finite difference scheme for linear stochastic integro-differential equations of parabolic type arising in non-linear filtering of jump-diffusion processes.
TL;DR: Four different ways of obtaining the diffusion propensities are compared theoretically and in numerical experiments, with a finite difference and a finite volume approximation generating the most accurate coefficients.
TL;DR: In this article, the numerical solution of the Burgers-Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme and the positivity, boundedness and local truncation error of the scheme are discussed.
Abstract: In this paper, numerical solution of the Burgers–Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme. At first, two exact finite difference schemes for BH equation obtained. Moreover an NSFD scheme is presented for this equation. The positivity, boundedness and local truncation error of the scheme are discussed. Finally, the numerical results of the proposed method with those of some available methods compared.
TL;DR: This article combines monotone finite difference methods with an adaptive grid refinement technique to produce a PDE discretization and solver which is applied to a broad class of equations, in curved or unbounded domains which include free boundaries.
Abstract: Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic partial differential equations (PDEs). These methods are best suited to regular rectangular grids, which leads to low accuracy near curved boundaries or singularities of solutions. In this article we combine monotone finite difference methods with an adaptive grid refinement technique to produce a PDE discretization and solver which is applied to a broad class of equations, in curved or unbounded domains which include free boundaries. The grid refinement is flexible and adaptive. The discretization is combined with a fast solution method, which incorporates asynchronous time stepping adapted to the spatial scale. The framework is validated on linear problems in curved and unbounded domains. Key applications include the obstacle problem and the one-phase Stefan free boundary problem.
TL;DR: In this article, high order finite difference weighted non-oscillatory (WENO) schemes were designed to maintain the well-balanced property of the blood flow model and verify high order accuracy, maintaining good resolution for smooth and discontinuous solutions.
Abstract: The blood flow model maintains the steady state solutions, in which the flux gradients are non-zero but exactly balanced by the source term. In this paper, we design high order finite difference weighted non-oscillatory (WENO) schemes to this model with such well-balanced property and at the same time keeping genuine high order accuracy. Rigorous theoretical analysis as well as extensive numerical results all indicate that the resulting schemes verify high order accuracy, maintain the well-balanced property, and keep good resolution for smooth and discontinuous solutions.
TL;DR: In this article, a combination of finite difference and numerical integration methods is proposed to determine the characteristics of general full-tensor permeability such as maximum and minimum permeability components, principal direction and anisotropic ratio.
Abstract: Velocity of fluid flow in underground porous media is 6~12 orders of magnitudes lower than that in pipelines. If numerical errors are not carefully controlled in this kind of simulations, high distortion of the final results may occur [1–4]. To fit the high accuracy demands of fluid flow simulations in porous media, traditional finite difference methods and numerical integration methods are discussed and corresponding high-accurate methods are developed. When applied to the direct calculation of full-tensor permeability for underground flow, the high-accurate finite difference method is confirmed to have numerical error as low as 10–5% while the high-accurate numerical integration method has numerical error around 0%. Thus, the approach combining the high-accurate finite difference and numerical integration methods is a reliable way to efficiently determine the characteristics of general full-tensor permeability such as maximum and minimum permeability components, principal direction and anisotropic ratio.
TL;DR: The uniform convergence on compact subsets of the solution of the discrete problems to an approximate problem on the subdomain is proved.
Abstract: Given an orthogonal lattice with mesh length h on a bounded two-dimensional convex domain $${\varOmega }$$Ω, we propose to approximate the Aleksandrov solution of the Monge---Ampere equation by regularizing the data and discretizing the equation in a subdomain using the standard finite difference method. The Dirichlet data is used to approximate the solution in the remaining part of the domain. We prove the uniform convergence on compact subsets of the solution of the discrete problems to an approximate problem on the subdomain. The result explains the behavior of methods based on the standard finite difference method and designed to numerically converge to non-smooth solutions.
TL;DR: A refined error estimate near the boundary and a discrete version of the divergence theorem are introduced, which are applied to prove the second order accuracy of the numerical gradient in arbitrary smooth domains.
Abstract: We consider the standard central finite difference method for solving the Poisson equation with the Dirichlet boundary condition. This scheme is well known to produce second order accurate solutions. From numerous tests, its numerical gradient was reported to be also second order accurate, but the observation has not been proved yet except for few specific domains. In this work, we first introduce a refined error estimate near the boundary and a discrete version of the divergence theorem. Applying the divergence theorem with the estimate, we prove the second order accuracy of the numerical gradient in arbitrary smooth domains.
TL;DR: In this paper, a mixed finite difference method is proposed to solve singularly perturbed differential difference equations with mixed shifts, solutions of which exhibit boundary layer behaviour at the left end of the interval using domain decomposition.
TL;DR: In this paper, the authors proposed two types of multistep finite difference schemes to solve a class of variable coefficient delay parabolic differential equations and proved the solvability, convergence and stability of both schemes.
Abstract: In this paper, we study the high efficient numerical methods to solve a class of variable coefficient delay parabolic differential equations. We propose two types of multistep finite difference schemes while prove the solvability, convergence and stability of both schemes. The convergence orders are O(τ2+h2) and O(τ2+h4), respectively, in the sense of L∞-norm. By some new analytical transformations, we extend our schemes to solve a more general variable coefficient delay convection–diffusion–reaction equations and also apply them to the higher dimensional cases. Furthermore, multistep finite difference method and compact multistep finite difference method for two dimensional variable coefficient diffusion–reaction equations are proposed and the suitable alternate direction implicit (ADI) technique is constructed for the multistep finite difference scheme. At last, several numerical experiments are carried out to illustrate the effectiveness of both schemes.
TL;DR: Numerical investigation shows that the FEM compares favorably with the FDM in terms of accuracy and convergence rate whereas the latter enjoys less computational effort.
Abstract: The paper makes a comparative study of the finite element method (FEM) and the finite difference method (FDM) for two-dimensional fractional advection-dispersion equation (FADE) which has recently been considered a promising tool in modeling non-Fickian solute transport in groundwater. Due to the non-local property of integro-differential operator of the space-fractional derivative, numerical solution of FADE is very challenging and little has been reported in literature, especially for high-dimensional case. In order to effectively apply the FEM and the FDM to the FADE on a rectangular domain, a backward-distance algorithm is presented to extend the triangular elements to generic polygon elements in the finite element analysis, and a variable-step vector Grunwald formula is proposed to improve the solution accuracy of the conventional finite difference scheme. Numerical investigation shows that the FEM compares favorably with the FDM in terms of accuracy and convergence rate whereas the latter enjoys less computational effort.
TL;DR: High-order accurate finite difference schemes are derived for a non-linear soliton model of nerve signal propagation in axons and accuracy and stability properties of the newly derived finite difference approximations are demonstrated for an analytic soliton solution.
19 Apr 2016
TL;DR: The Polya-Schur theory describes the class of hyperbolicity preservers as discussed by the authors, i.e., linear operators acting on univariate polynomials and preserving real-rootedness.
Abstract: The Polya-Schur theory describes the class of hyperbolicity preservers, i.e., the class of linear operators acting on univariate polynomials and preserving real-rootedness. We attempt to develop an ...
TL;DR: In this article, an accurate numerical method for solving fractional Logistic Differential Equation (FLDE) is presented, based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method.
Abstract: This paper presents an accurate numerical method for solving fractional Logistic differential equation (FLDE). The fractional derivative in this problem is in the Caputo sense. The proposed method is so called fractional Chebyshev finite difference method. In this technique, we approximate FLDE with a finite dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The Caputo fractional derivative is replaced by a difference quotient and the integral by a finite sum. The introduced method reduces the proposed problem for solving a system of algebraic equations, and by solving this system, we obtain the solution of FLDE. Special attention is given to study the convergence analysis and estimate an error upper bound of the obtained approximate formula. Illustrative examples are included to demonstrate the validity and the applicability of the proposed technique.
TL;DR: In this paper, the authors developed and tested novel invariant-preserving finite difference schemes for both the Camassa-Holm (CH) equation and one of its 2-component generalizations (2CH).
Abstract: The purpose of this paper is to develop and test novel invariant-preserving finite difference schemes for both the Camassa-Holm (CH) equation and one of its 2-component generalizations (2CH). The considered PDEs are strongly nonlinear, admitting soliton-like peakon solutions which are characterized by a slope discontinuity at the peak in the wave shape, and therefore suitable for modeling both short wave breaking and long wave propagation phenomena. The proposed numerical schemes are shown to preserve two invariants, momentum and energy, hence numerically producing wave solutions with smaller phase error over a long time period than those generated by other conventional methods. We first apply the scheme to the CH equation and showcase the merits of considering such a scheme under a wide class of initial data. We then generalize this scheme to the 2CH equation and test this scheme under several types of initial data.
TL;DR: In this article, a genetic algorithm is used to improve the boundary efficiency and accuracy of a compact finite difference scheme, based on its composite template, for one-dimensional linear wave convection and two-dimensional inviscid vortex convection problems with uniform and curvilinear grids.
TL;DR: In this paper, a finite difference method which is second-order accurate in time and in space is proposed for two-dimensional fractional percolation equations using the Fourier transform, a general approximation for the mixed fractional derivatives is analyzed.
Abstract: A finite difference method which is second-order accurate in time and in space is proposed for two-dimensional fractional percolation equations. Using the Fourier transform, a general approximation for the mixed fractional derivatives is analyzed. An approach based on the classical Crank-Nicolson scheme combined with the Richardson extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Consistency, stability and convergence of the method are established. Numerical experiments illustrating the effectiveness of the theoretical analysis are provided.