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Volume 42, Issue 2
Banded $M$-Matrix Splitting Preconditioner for Riesz Space Fractional Reaction-Dispersion Equation

Shiping Tang, Aili Yang & Yujiang Wu

DOI: 10.4208/jcm.2203-m2020-0192

J. Comp. Math., 42 (2024), pp. 372-389.

Published online: 2024-01

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  • 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.

  • Keywords

Riesz space fractional equations, Toeplitz matrix, conjugate gradient method, Incomplete Cholesky decomposition, Banded $M$-matrix splitting.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-372, author = {Tang , ShipingYang , Aili and Wu , Yujiang}, title = {Banded $M$-Matrix Splitting Preconditioner for Riesz Space Fractional Reaction-Dispersion Equation }, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {2}, pages = {372--389}, 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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2203-m2020-0192}, url = {http://global-sci.org/intro/article_detail/jcm/22885.html} }
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 -

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.

Shiping Tang, Aili Yang & Yujiang Wu. (2024). Banded $M$-Matrix Splitting Preconditioner for Riesz Space Fractional Reaction-Dispersion Equation . Journal of Computational Mathematics. 42 (2). 372-389. doi:10.4208/jcm.2203-m2020-0192
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