Finite difference method

About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.

An explicit fourth‐order compact finite difference scheme for three‐dimensional convection–diffusion equation

Abstract: We present an explicit fourth-order compact finite difference scheme for approximating the three-dimensional convection diffusion equation with variable coefficients. This 19-point formula is defined on a uniform cubic grid. We compare the advantages and implementation costs of the new scheme with the standard 7-point scheme in the context of basic iterative methods. Numerical examples are used to verify the fourth-order convergence rate of the scheme and to show that the Gauss Seidel iterative method converges for large values of the convection coefficients. Some algebraic properties of the coefficient matrices arising from different discretization schemes are compared. We also comment on the potential use of the fourth-order compact scheme with multilevel iterative methods

The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media

Abstract: A technique based on the combination of Fourier pseudospectral method and the perfectly matched layer (PML) is developed to simulate transient acoustic wave propagation in multidimensional, inhomogeneous, absorptive media. Instead of the finite difference approximation in the conventional finite-difference time-domain (FDTD) method, this technique uses trigonometric functions, through an FFT (fast Fourier transform) algorithm, to represent the spatial derivatives in partial differential equations. Traditionally the Fourier pseudospectral method is used only for spatially periodic problems because the use of FFT implies periodicity. In order to overcome this limitation, the perfectly matched layer is used to attenuate the waves from other periods, thus allowing the method to be applicable to unbounded media. This new algorithm, referred to as the pseudospectral time-domain (PSTD) algorithm, is developed to solve large-scale problems for acoustic waves. It has an infinite order of accuracy in the spatial derivatives, and thus requires much fewer unknowns than the conventional FDTD method. Numerical results confirms the efficacy of the PSTD method.

Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping

Abstract: SUMMARY We investigate the stability of some high-order finite element methods, namely the spectral element method and the interior-penalty discontinuous Galerkin method (IP-DGM), for acoustic or elastic wave propagation that have become increasingly popular in the recent past. We consider the Lax-Wendroff method (LWM) for time stepping and show that it allows for a larger time step than the classical leap-frog finite difference method, with higher-order accuracy. In particular the fourth-order LWM allows for a time step 73 per cent larger than that of the leap-frog method; the computational cost is approximately double per time step, but the larger time step partially compensates for this additional cost. Necessary, but not sufficient, stability conditions are given for the mentioned methods for orders up to 10 in space and time. The stability conditions for IP-DGM are approximately 20 and 60 per cent more restrictive than those for SEM in the acoustic and elastic cases, respectively.