Fast Iterative Solvers for Linear Systems Arising from Time-Dependent Space-Fractional Diffusion Equations

In this paper, we study the linear systems arising from the discretization of time-dependent space-fractional diffusion equations. By using a finite difference discretization scheme for the time derivative and a finite volume discretization scheme for the space-fractional derivative, Toeplitz-like linear systems are obtained. We propose using the approximate inverse-circulant preconditioner to deal with such Toeplitz-like matrices, and we show that the spectra of the corresponding preconditioned matrices are clustered around 1. Experimental results on time-dependent and space-fractional diffusion equations are presented to demonstrate that the preconditioned Krylov subspace methods converge very quickly.