Efficient ADI schemes and preconditioning for a class of high-dimensional spatial fractional diffusion equations with variable diffusion coefficients

Abstract

In this paper, alternating direction implicit (ADI) finite difference method and preconditioned Krylov subspace method are combined to solve a class of high-dimensional spatial fractional diffusion equations with variable diffusion coefficients. We prove the unconditional stability and convergence rate of the ADI finite difference method provided that the diffusion coefficients satisfy the given conditions. For the linear system in each spatial direction, we establish a circulant approximate inverse preconditioner to accelerate the Krylov subspace method. In addition, we also use matrix-free algorithms and fast Fourier transforms (FFT) to speed up the solution of linear systems. Numerical experiments show the utility of the ADI method and the preconditioner.