Year: 2018
Numerical Mathematics: Theory, Methods and Applications, Vol. 11 (2018), Iss. 4 : pp. 827–853
Abstract
In this paper, we study linear systems arising from time-space fractional Caputo-Riesz diffusion equations with time-dependent diffusion coefficients. The coefficient matrix is a summation of a block-lower-triangular-Toeplitz matrix (temporal component) and a block-diagonal-with-diagonal-times-Toeplitz-block matrix (spatial component). The main aim of this paper is to propose separable preconditioners for solving these linear systems, where a block ϵ-circulant preconditioner is used for the temporal component, while a block diagonal approximation is used for the spatial variable. The resulting preconditioner can be block-diagonalized in the temporal domain. Furthermore, the fast solvers can be employed to solve smaller linear systems in the spatial domain. Theoretically, we show that if the diffusion coefficient (temporal-dependent or spatial-dependent only) function is smooth enough, the singular values of the preconditioned matrix are bounded independent of discretization parameters. Numerical examples are tested to show the performance of proposed preconditioner.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.2018.s09
Numerical Mathematics: Theory, Methods and Applications, Vol. 11 (2018), Iss. 4 : pp. 827–853
Published online: 2018-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 27
-
A numerical framework for the approximate solution of fractional tumor-obesity model
Arshad, Sadia | Baleanu, Dumitru | Defterli, Ozlem |International Journal of Modeling, Simulation, and Scientific Computing, Vol. 10 (2019), Iss. 01 P.1941008
https://doi.org/10.1142/S1793962319410083 [Citations: 5] -
A parallel-in-time two-sided preconditioning for all-at-once system from a non-local evolutionary equation with weakly singular kernel
Lin, Xue-lei | Ng, Michael K. | Zhi, YajingJournal of Computational Physics, Vol. 434 (2021), Iss. P.110221
https://doi.org/10.1016/j.jcp.2021.110221 [Citations: 11] -
A RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation
Liu, Jianming | Li, Xinkai | Hu, XiulingJournal of Computational Physics, Vol. 384 (2019), Iss. P.222
https://doi.org/10.1016/j.jcp.2018.12.043 [Citations: 38] -
A single-sided all-at-once preconditioning for linear system from a non-local evolutionary equation with weakly singular kernels
Lin, Xuelei | Dong, Jiamei | Hon, SeanComputers & Mathematics with Applications, Vol. 169 (2024), Iss. P.1
https://doi.org/10.1016/j.camwa.2024.06.002 [Citations: 1] -
A Preconditioned MINRES Method for Block Lower Triangular Toeplitz Systems
Li, Congcong | Lin, Xuelei | Hon, Sean | Wu, Shu-LinJournal of Scientific Computing, Vol. 100 (2024), Iss. 3
https://doi.org/10.1007/s10915-024-02611-4 [Citations: 0] -
A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations
Liu, Jun | Wang, Xiang-Sheng | Wu, Shu-Lin | Zhou, TaoAdvances in Computational Mathematics, Vol. 48 (2022), Iss. 3
https://doi.org/10.1007/s10444-022-09928-4 [Citations: 5] -
A linearized finite difference scheme for time–space fractional nonlinear diffusion-wave equations with initial singularity
Elmahdi, Emadidin Gahalla Mohmed | Huang, JianfeiInternational Journal of Nonlinear Sciences and Numerical Simulation, Vol. 24 (2023), Iss. 5 P.1769
https://doi.org/10.1515/ijnsns-2021-0388 [Citations: 0] -
Fast preconditioned iterative methods for fractional Sturm–Liouville equations
Zhang, Lei | Zhang, Guo‐Feng | Liang, Zhao‐ZhengNumerical Methods for Partial Differential Equations, Vol. 37 (2021), Iss. 3 P.2278
https://doi.org/10.1002/num.22704 [Citations: 2] -
A Fast Block $\alpha$-Circulant Preconditoner for All-at-Once Systems From Wave Equations
Liu, Jun | Wu, Shu-LinSIAM Journal on Matrix Analysis and Applications, Vol. 41 (2020), Iss. 4 P.1912
https://doi.org/10.1137/19M1309869 [Citations: 30] -
Sine Transform Based Preconditioning for an Inverse Source Problem of Time-Space Fractional Diffusion Equations
Pang, Hong-Kui | Qin, Hai-Hua | Ni, ShuaiJournal of Scientific Computing, Vol. 100 (2024), Iss. 3
https://doi.org/10.1007/s10915-024-02634-x [Citations: 0] -
A parallel-in-time preconditioner for Crank–Nicolson discretization of a parabolic optimal control problem
Lin, Xue-Lei | Wu, Shu-LinJournal of Computational and Applied Mathematics, Vol. 451 (2024), Iss. P.116106
https://doi.org/10.1016/j.cam.2024.116106 [Citations: 0] -
High-order compact finite volume scheme for the 2D multi-term time fractional sub-diffusion equation
Su, Baojin | Jiang, ZiwenAdvances in Difference Equations, Vol. 2020 (2020), Iss. 1
https://doi.org/10.1186/s13662-020-03128-4 [Citations: 2]