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Volume 13, Issue 2
A Fast Two-Level Strang Splitting Method for Multi-Dimensional Spatial Fractional Allen-Cahn Equations with Discrete Maximum Principle

Yao-Yuan Cai, Zhi-Wei Fang, Hao Chen & Hai-Wei Sun

East Asian J. Appl. Math., 13 (2023), pp. 340-360.

Published online: 2023-04

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  • Abstract

Numerical solutions of the multi-dimensional spatial fractional Allen-Cahn equations are studied. After the semi-discretization of the spatial fractional Riesz derivative, a system of nonlinear ordinary differential equations with the Toeplitz structure is obtained. In order to reduce the computational complexity, a two-level Strang splitting method is proposed, where the Toeplitz matrix in the system is represented as the sum of circulant and skew-circulant matrices. Therefore, the method can be quickly implemented by the fast Fourier transform, avoiding expensive Toeplitz matrix exponential calculations. It is shown that the discrete maximum principle of the method is unconditionally preserved. Moreover, the analysis of errors in the infinite norm with second-order accuracy is carried out in both time and space. Numerical tests support the theoretical findings and show the efficiency of the method.

  • AMS Subject Headings

65F10, 65N22

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-13-340, author = {Cai , Yao-YuanFang , Zhi-WeiChen , Hao and Sun , Hai-Wei}, title = {A Fast Two-Level Strang Splitting Method for Multi-Dimensional Spatial Fractional Allen-Cahn Equations with Discrete Maximum Principle}, journal = {East Asian Journal on Applied Mathematics}, year = {2023}, volume = {13}, number = {2}, pages = {340--360}, abstract = {

Numerical solutions of the multi-dimensional spatial fractional Allen-Cahn equations are studied. After the semi-discretization of the spatial fractional Riesz derivative, a system of nonlinear ordinary differential equations with the Toeplitz structure is obtained. In order to reduce the computational complexity, a two-level Strang splitting method is proposed, where the Toeplitz matrix in the system is represented as the sum of circulant and skew-circulant matrices. Therefore, the method can be quickly implemented by the fast Fourier transform, avoiding expensive Toeplitz matrix exponential calculations. It is shown that the discrete maximum principle of the method is unconditionally preserved. Moreover, the analysis of errors in the infinite norm with second-order accuracy is carried out in both time and space. Numerical tests support the theoretical findings and show the efficiency of the method.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.2022-248.161022}, url = {http://global-sci.org/intro/article_detail/eajam/21652.html} }
TY - JOUR T1 - A Fast Two-Level Strang Splitting Method for Multi-Dimensional Spatial Fractional Allen-Cahn Equations with Discrete Maximum Principle AU - Cai , Yao-Yuan AU - Fang , Zhi-Wei AU - Chen , Hao AU - Sun , Hai-Wei JO - East Asian Journal on Applied Mathematics VL - 2 SP - 340 EP - 360 PY - 2023 DA - 2023/04 SN - 13 DO - http://doi.org/10.4208/eajam.2022-248.161022 UR - https://global-sci.org/intro/article_detail/eajam/21652.html KW - Two-level Strang splitting method, circulant and skew-circulant matrix splitting, discrete maximum principle, fast Fourier transform. AB -

Numerical solutions of the multi-dimensional spatial fractional Allen-Cahn equations are studied. After the semi-discretization of the spatial fractional Riesz derivative, a system of nonlinear ordinary differential equations with the Toeplitz structure is obtained. In order to reduce the computational complexity, a two-level Strang splitting method is proposed, where the Toeplitz matrix in the system is represented as the sum of circulant and skew-circulant matrices. Therefore, the method can be quickly implemented by the fast Fourier transform, avoiding expensive Toeplitz matrix exponential calculations. It is shown that the discrete maximum principle of the method is unconditionally preserved. Moreover, the analysis of errors in the infinite norm with second-order accuracy is carried out in both time and space. Numerical tests support the theoretical findings and show the efficiency of the method.

Yao-Yuan Cai, Zhi-Wei Fang, Hao Chen & Hai-Wei Sun. (2023). A Fast Two-Level Strang Splitting Method for Multi-Dimensional Spatial Fractional Allen-Cahn Equations with Discrete Maximum Principle. East Asian Journal on Applied Mathematics. 13 (2). 340-360. doi:10.4208/eajam.2022-248.161022
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