Journal of Computational Mathematics Vol.42, No.2, 2024, 372–389. http://www.global-sci.org/jcm doi:10.4208/jcm.2203-m2020-0192 BANDED M -MATRIX SPLITTING PRECONDITIONER FOR RIESZ SPACE FRACTIONAL REACTION-DISPERSION EQUATION * Shiping Tang School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China Email: tangshp20@lzu.edu.cn Aili Yang 1) School of Mathematics and Statistics, Hainan Normal University, Haikou 570000, China Email: yangaili@lzu.edu.cn Yujiang Wu School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China Email: myjaw@lzu.edu.cn Abstract 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, Numer- ical examples are employed to test the robustness and the effectiveness of the proposed preconditioner. Mathematics subject classification: 65N15, 65N30. Key words: Riesz space fractional equations, Toeplitz matrix, conjugate gradient method, Incomplete Cholesky decomposition, Banded M-matrix splitting. 1. Introduction We consider the following initial-boundary problem of Riesz space fractional reaction-di- spersion equation (RSFRDE) [1]:          ∂u(x,t) ∂t = −Ku(x,t)+ K β ∂ β u(x,t) ∂ |x| β + f (x,t),x ∈ (a,b), t ∈ (0,T ], u(x, 0) = φ(x), x ∈ (a,b), u(x,t)= ψ(x,t), x ∈ R\(a,b),t ∈ [0,T ], (1.1) where 1 <β< 2 and the coefficients K,K β are positive constants, u(x,t) is an unknown function to be solved. In addition, f (x,t) is the source term, and the Riesz space fractional * Received July 20, 2020 / Revised version received February 14, 2022 / Accepted March 25, 2022 / Published online March 07, 2023 / 1) Corresponding author