Nonstandard Finite Difference Method for Nonlinear Riesz Space Fractional Reaction-Diffusion Equation

Abstract

In this paper, a modified nonstandard finite difference method for the two-dimensional Riesz space fractional reaction-diffusion equations is developed. The space fractional derivative is discretized by the shifted Grünwald-Letnikov method and the nonlinear reaction term is approximated by Taylor formula instead of Micken's. Multigrid method is introduced to reduce the computation time of the traditional Gauss-Seidal method. The stability and convergence of the nonstandard implicit difference scheme are strictly proved. The method is extended to simulate the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis.