Banded M-Matrix Splitting Preconditioning Method For Riesz Space Fractional Reaction-dispersion Euqations

In recent decades,with the continuous exploration of physics,chemistry,biology and other fields,it has been found that fractional models have advantages that classic integer models cannot replace.Because the fractional diffusion equation can well describe the development of anomalous dispersion,it has attracted widespread attention from more and more scholars.This work focuses on preconditioning methods for high-order numerical schemes of Riesz space fractional order reaction-dispersion equations.First,based on the Crank-Nicolson and the weighted and shifted Gr¨unwald 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 used to test the feasibility and effectiveness of the proposed preconditioner.