A new diagonal and Toeplitz splitting preconditioning method for solving time-dependent Riesz space-fractional diffusion equations

The initial-boundary value problem of the Riesz space-fractional diffusions equation is an important class of equations arising in many application fields. In this paper, we apply the Grunwald-Letnikov type formulas to discretize the time-dependent Riesz space-fractional diffusion equations, and obtain a system of linear equations from the discretization results. A new diagonal and Toeplitz splitting (NDTS) iteration method is constructed for this linear system. Based on the NDTS iteration method, an NDTS tau preconditioner is proposed and the generalized minimal residual (GMRES) method combined with this preconditioner is applied to solve the linear system. We theoretically show that the eigenvalues of the NDTS tau preconditioned matrix are clustered. Numerical experiments illustrate the efficiency of the proposed method.