Unconditionally convergent τ splitting iterative methods for variable coefficient Riesz space fractional diffusion equations

Abstract

In this paper, we consider fast solvers for discrete linear systems generated by Riesz space fractional diffusion equations. We extract a scalar matrix, a compensation matrix, and a $\tau$ matrix from the coefficient matrix, and use their sum to construct a class of $\tau$ splitting iterative methods. Additionally, we design a preconditioner for the conjugate gradient method. Theoretical analyses show that the proposed $\tau$ splitting iterative methods are unconditionally convergent with convergence rates independent of step-sizes. Numerical results are provided to demonstrate the effectiveness of the proposed iterative methods.