A compact finite difference method for reaction–diffusion problems using compact integration factor methods in high spatial dimensions

Abstract

This paper proposes and analyzes an efficient compact finite difference scheme for reaction-diffusion equation in high spatial dimensions. The scheme is based on a compact finite difference method (cFDM) for the spatial discretization. We prove that the proposed method is asymptotically stable for the linear case. By introducing the differentiation matrices, the semi-discrete reaction-diffusion equation can be rewritten as a system of nonlinear ordinary differential equations (ODEs) in matrices formulations. For the time discretization, we apply the compact implicit integration factor (cIIF) method which demands much less computational effort. This method combines the advantages of cFDM and cIIF methods to improve the accuracy without increasing the computational cost and reducing the stability range. Numerical examples are shown to demonstrate the accuracy, efficiency, and robustness of the method.

Stable finite difference method for fractional reaction–diffusion equations by compact implicit integration factor methods

“…It also should be noted that the ADI methods are limited to second-order accuracy in time. In this paper, an efficient compact implicit integration factor (cIIF) method [28,29] is developed. By introducing the compact representation for the matrix approximating the differential operator, the cIIF methods apply matrix exponential operations sequentially in every spatial direction.…”

Compact Difference Schemes Combined with Runge-Kutta Methods for Solving Unsteady Convection-Diffusion Problems

“…Discretization with a higher order in space variables is usually associated with large stencils, thereby increasing the band-width of the resulting matrix [5,6]. A class of compact finite difference approximations have been recently developed to solve convection-diffusion problems [7][8][9]. These compact finite difference schemes have fourth-order accuracy in space variables, but fail when accuracy in time variables is most needed [10,11].…”

Numerical Solution to Unsteady One-Dimensional Convection-Diffusion Problems Using Compact Difference Schemes Combined with Runge-Kutta Methods

“…Discretization with a higher order in space variables is usually associated with large stencils, thereby increasing the bandwidth of the resulting matrix [5,6]. A class of compact finite difference approximations has been recently developed to solve convectiondiffusion problems [7][8][9]. These compact finite difference schemes have fourth-order accuracy in space variables, but fail when accuracy in time variables is most needed [10,11].…”