TY - JOUR
T1 - Banded $M$-Matrix Splitting Preconditioner for Riesz Space Fractional Reaction-Dispersion Equation 
AU - Tang , Shiping
AU - Yang , Aili
AU - Wu , Yujiang
JO - Journal of Computational Mathematics
VL - 2
SP - 372
EP - 389
PY - 2024
DA - 2024/01
SN - 42
DO - http://doi.org/10.4208/jcm.2203-m2020-0192
UR - https://global-sci.org/intro/article_detail/jcm/22885.html
KW - Riesz space fractional equations, Toeplitz matrix, conjugate gradient method,
Incomplete Cholesky decomposition, Banded $M$-matrix splitting.
AB - <p style="text-align: justify;">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.</p>