Volume 11, Issue 4
A Separable Preconditioner for Time-Space Fractional Caputo-Riesz Diffusion Equations

Xuelei Lin, Michael K. Ng & Haiwei Sun

Numer. Math. Theor. Meth. Appl., 11 (2018), pp. 827-853.

Published online: 2018-06

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  • 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.

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@Article{NMTMA-11-827, author = {}, title = {A Separable Preconditioner for Time-Space Fractional Caputo-Riesz Diffusion Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {11}, number = {4}, pages = {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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2018.s09}, url = {http://global-sci.org/intro/article_detail/nmtma/12475.html} }
TY - JOUR T1 - A Separable Preconditioner for Time-Space Fractional Caputo-Riesz Diffusion Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 827 EP - 853 PY - 2018 DA - 2018/06 SN - 11 DO - http://doi.org/10.4208/nmtma.2018.s09 UR - https://global-sci.org/intro/article_detail/nmtma/12475.html KW - AB -

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.

Xuelei Lin, Michael K. Ng & Haiwei Sun. (2020). A Separable Preconditioner for Time-Space Fractional Caputo-Riesz Diffusion Equations. Numerical Mathematics: Theory, Methods and Applications. 11 (4). 827-853. doi:10.4208/nmtma.2018.s09
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