IRK-WSGD methods for space fractional diffusion equations

Abstract

In this paper, we develop high order numerical schemes for the solution of the initial-boundary value problem of one-dimensional and two-dimensional space fractional diffusion equations of orders belonging to the interval (1,2). Firstly, certain weighted and shifted Grünwald difference (WSGD) operator is used to approximate space Riemann-Liouville fractional derivatives, resulting in a linear system of ordinary differential equations (ODEs). Then an implicit Runge-Kutta (IRK) method is applied to discretize the resulted ODEs. Thus, we get an IRK-WSGD method for the fractional diffusion equation. We prove that under certain hypotheses, the proposed IRK-WSGD schemes are stable and have temporally fourth order accuracy and spatially second/third order accuracy. Preconditioning for discretization linear systems is discussed. Numerical experiments are presented to illustrate the accuracy and efficiency of the method.