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.