On kk-step CSCS-based polynomial preconditioners for Toeplitz linear systems with application to fractional diffusion equations

The implicit finite difference scheme with the shifted Grüwald formula for discretizing the fractional diffusion equations (FDEs) often results in the ill-conditioned non-Hermitian Toeplitz systems. In the present paper, we consider to solve such Toeplitz systems by exploiting the preconditioned GMRES method. A kk-step polynomial preconditioner is designed based on the circulant and skew-circulant splitting (CSCS) iteration method proposed by Ng (2003). Theoretical and experimental results involving numerical solutions of FDEs demonstrate that the proposed kk-step preconditioner is efficient to accelerate the GMRES solver for non-Hermitian Toeplitz systems.