A preconditioner based on sine transform for space fractional diffusion equations

Abstract

The spatial fractional diffusion equations are discretized by the finite difference method and linear systems are obtained. Transforming the coefficient matrices of these linear systems to obtain the structure of diagonal-plus-Toeplitz. In order to solve this class of linear systems, we construct a two-parameter DTS iteration method on the basis of the DTS iteration method and give its asymptotic convergence conditions. On this basis, we introduce a two-parameter sine-transform-based splitting preconditioner to improve the convergence rate of the generalized minimal residual (GMRES) iteration method, and theoretically prove that the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Finally, numerical experiments demonstrate the efficiency of the new method.