Quasi-Toeplitz Trigonometric Transform Splitting Methods for Spatial Fractional Diffusion Equations

The random walk model describing the super-diffusion competition phenomenon of particles can derive the spatial fractional diffusion equation. For irregular diffusion and super-diffusion phenomena, the use of such equation can obtain more accurate and realistic results, so it has a wide application background in practice. The implicit finite-difference method derived from the shifted Grunwald scheme is used to discretize the spatial fractional diffusion equation. The coefficient matrix of discrete system is in the form of the sum of a diagonal matrix and a Toeplitz matrix. In this paper, a preconditioner is constructed, which transforms the coefficient matrix into the form of an identity matrix plus a diagonal matrix multiplied by Toeplitz matrix. On this basis, a new quasi-Toeplitz trigonometric transform splitting iteration format (abbreviated as QTTTS method) is proposed. We theoretically verify the unconditional convergence of the new method, and obtain the effective optimal form of the iteration parameter. Finally, numerical simulation experiments also demonstrate the accurateness and efficiency of the new method.