High order difference schemes for nonlinear Riesz space variable-order fractional diffusion equations

Abstract

This article aims at studying new finite difference methods for one-dimensional and two-dimensional nonlinear Riesz space variable-order (VO) fractional diffusion equations. In the presented model, fractional derivatives are defined in the Riemann-Liouville type. Based on 4-point weighted-shifted-Grünwald-difference (4WSGD) operators for Riemann-Liouville constant-order fractional derivatives, which have a free parameter and have at least third order accuracy, we derive variable-order 4WSGD operators for space Riesz variable-order fractional derivatives. In order that the fully discrete schemes exhibit robust stability and can handle the nonlinear term efficiently, we employ the implicit Euler (IE) method to discretize the time derivative, which leads to IE-4WSGD schemes. The stability and convergence properties of the IE-4WSGD schemes are analyzed theoretically. Additionally, a parameter selection strategy is derived for 4WSGD schemes and banded preconditioners are put forward to accelerate the GMRES methods for solving the discretization linear systems. Numerical results demonstrate the effectiveness of the proposed IE-4WSGD schemes and preconditioners.