Fast compact implicit integration factor method with non-uniform meshes for the two-dimensional nonlinear Riesz space-fractional reaction-diffusion equation

In this paper, we propose a fast compact implicit integration factor (FcIIF) method with non-uniform time meshes for solving the two-dimensional nonlinear Riesz space-fractional reaction-diffusion equation. The weighted and shifted Gruwald-Letnikov (WSGD) approximation is employed to the spatial discretization of the equation, and a system of nonlinear ordinary differential equations (ODEs) in matrix form is obtained. Since the cIIF method can provide excellent stability properties with good efficiency by decoupling the treatment of the diffusion and reaction terms, a fast cIIF (FcIIF) method with non-uniform time meshes is developed to solve the resultant nonlinear system of ODEs. Compared with the cIIF method, the proposed FcIIF method avoids the direct calculation of dense exponential matrices and requires less computational cost. The stability, accuracy and effectiveness of the proposed method are verified by the linear stability analysis and various numerical experiments. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.