Banded $M$-Matrix Splitting Preconditioner for Riesz Space Fractional Reaction-Dispersion Equation (Journal of Computational Mathematics, 2024)
Based on the Crank-Nicolson and the weighted and shifted Grünwald operators, wepresent an implicit difference scheme for the Riesz space fractional reaction-dispersionequations and also analyze the stability and the convergence of this implicit differencescheme. However, after estimating the condition number of the coefficient matrix of thediscretized scheme, we find that this coefficient matrix is ill-conditioned when the spatialmesh-size is sufficiently small. To overcome this deficiency, we further develop an effectivebanded $M$-matrix splitting preconditioner for the coefficient matrix. Some properties ofthis preconditioner together with its preconditioning effect are discussed. Finally, Numerical examples are employed to test the robustness and the effectiveness of the proposedpreconditioner.