Scaled diagonal-times-Toeplitz splitting iteration methods for solving discretized spatial fractional diffusion equations

In this paper, we firstly give a scaled diagonal-times-Toeplitz splitting (SDTS) iteration method for solving the discretization system from one-dimensional spatial fractional diffusion equations of variable coefficients. The SDTS iteration methods can be used to solve the one-dimensional spatial fractional diffusion equations of both anisotropic and isotropic, while the respectively scaled HSS (RSHSS) iteration method can be used only to solve anisotropic spatial fractional diffusion equations. Therefore, the SDTS iteration method is of wider range of applications than the RSHSS iteration method. Furthermore, the SDTS iteration method will naturally lead to a scaled diagonal-times-circulant splitting (SDCS) preconditioner. Theoretical analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Numerical experiments are provided, which demonstrate the feasibility and effectiveness of the SDCS preconditioned GMRES method for solving the proposed examples.