Abstract
In this paper, a numerical method to solve the multi-dimensional spatial fractional Allen–Cahn equations has been investigated. After semi-discretizating the equations, a system of nonlinear ordinary differential equations with a Toeplitz structure is induced. We propose to split the Toeplitz matrix into the sum of a circulant matrix and a skew-circulant matrix, and apply the Strang splitting method. Such a two-level Strang splitting method will reduce the computational complexity to \({\mathcal {O}}(q\log q)\). Moreover, it preserves not only the discrete maximum principle unconditionally but also second-order convergence as well. By introducing a new modified energy formula, the energy dissipation property can be guaranteed. Finally, some numerical experiments are conducted to confirm the theories we put forward.
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This work is supported in part by research grants of the Science and Technology Development Fund, Macau SAR (File No. 0122/2020/A3), University of Macau (File No. MYRG2020-00224-FST).
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Cai, YY., Sun, HW. & Tam, SC. Numerical Study of a Fast Two-Level Strang Splitting Method for Spatial Fractional Allen–Cahn Equations. J Sci Comput 95, 71 (2023). https://doi.org/10.1007/s10915-023-02196-4
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DOI: https://doi.org/10.1007/s10915-023-02196-4