Banded $M$-Matrix Splitting Preconditioner for Riesz Space Fractional Reaction-Dispersion Equation

Abstract

Based on the Crank-Nicolson and the weighted and shifted Grünwald operators, we present an implicit difference scheme for the Riesz space fractional reaction-dispersion equations and also analyze the stability and the convergence of this implicit difference scheme. However, after estimating the condition number of the coefficient matrix of the discretized scheme, we find that this coefficient matrix is ill-conditioned when the spatial mesh-size is sufficiently small. To overcome this deficiency, we further develop an effective banded $M$-matrix splitting preconditioner for the coefficient matrix. Some properties of this preconditioner together with its preconditioning effect are discussed. Finally, Numerical examples are employed to test the robustness and the effectiveness of the proposed preconditioner.