A Splitting Preconditioner for Toeplitz-Like Linear Systems Arising from Fractional Diffusion Equations

In this paper, we study Toeplitz-like linear systems arising from time-dependent one-dimensional and two-dimensional Riesz space-fractional diffusion equations with variable diffusion coefficients. The coefficient matrix is a sum of a scalar identity matrix and a diagonal-times-Toeplitz matrix which allows fast matrix-vector multiplication in iterative solvers. We propose and develop a splitting preconditioner for this kind of matrix and analyze the spectra of the preconditioned matrix. Under mild conditions on variable diffusion coefficients, we show that the singular values of the preconditioned matrix are bounded above and below by positive constants which are independent of temporal and spatial discretization step-sizes. When the preconditioned conjugate gradient squared method is employed to solve such preconditioned linear systems, the method converges linearly within an iteration number independent of the discretization step-sizes. Numerical examples are given to illustrate the theoretical results and demonstrate that the performance of the proposed preconditioner is better than other tested solvers.