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Second-order maximum principle preserving Strang’s splitting schemes for anisotropic fractional Allen-Cahn equations

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Abstract

In this paper, we exploit the Strang splitting technique for solving the multidimensional Allen-Cahn equations with anisotropic spatial fractional Riesz derivatives. Fully discrete numerical methods are proposed using exponential Strang’s splitting schemes for the time integration with finite difference discretization in space. It is proved that the proposed methods can preserve the discrete maximum principle unconditionally. Furthermore, the fully discrete methods are theoretically confirmed to be convergent with second-order accuracy in both of time and space. In practical implementation, the proposed algorithms require to compute the matrix exponential associated with only one-dimensional discretized matrices that possess Toeplitz structure. Meanwhile, a fast algorithm is further developed for evaluating the product of the Toeplitz matrix exponential with a vector. Numerical examples are presented to verify the theoretical analysis and demonstrate the efficiency of the proposed methods.

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Acknowledgements

The first author was partially supported by the National Natural Science Foundation of China (Grant No.11971085), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (No. KJQN202000543), and the Program of Chongqing Innovation Research Group Project in University (No. CXQT19018). The second author was partially supported by research grants of the Science and Technology Development Fund, Macau SAR (file no. 0118/2018/A3), and MYRG2020-00224-FST from University of Macau.

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  1. College of Mathematics Science, Chongqing Normal University, Chongqing, China

    Hao Chen

  2. Department of Mathematics, University of Macau, Macao, China

    Hai-Wei Sun

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  1. Hao Chen
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Correspondence to Hai-Wei Sun.

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Chen, H., Sun, HW. Second-order maximum principle preserving Strang’s splitting schemes for anisotropic fractional Allen-Cahn equations. Numer Algor 90, 749–771 (2022). https://doi.org/10.1007/s11075-021-01207-5

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Mathematics Subject Classification (2010)