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Second-Order Semi-Lagrangian Exponential Time Differencing Method with Enhanced Error Estimate for the Convective Allen–Cahn Equation

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Abstract

The convective Allen–Cahn (CAC) equation has been widely used for simulating multiphase flows of incompressible fluids, which contains an extra convective term but still maintains the same maximum bound principle (MBP) as the classic Allen–Cahn equation. Based on the operator splitting approach, we propose a second-order semi-Lagrangian exponential time differencing method for solving the CAC equation, that preserves the discrete MBP unconditionally. In our scheme, the AC equation part is first spatially discretized via the central finite difference scheme, then it is efficiently solved by using the exponential time differencing method with FFT-based fast implementation. The transport equation part is computed by combining the semi-Lagrangian approach with a cut-off post-processing within the finite difference framework. MBP stability and convergence analysis of our fully discretized scheme are presented. In particular, we conduct an improved error estimation for the semi-Lagrangian method with variable velocity, so that the error of our scheme is not spoiled by the reciprocal of the time step size. Extensive numerical tests in two and three dimensions are also carried out to validate the theoretical results and demonstrate the performance of our scheme.

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Acknowledgements

J. Li was partially supported by Lanzhou University Double World-Class Project. R. Lan is partially supported by Shandong Excellent Young Scientists Program (Overseas) under the grant 2023HWYQ-064 and OUC Youth Talents Project. Y. Cai was partially supported by National Natural Science Foundation of China grant 12171041. L. Ju was partially supported by U.S. National Science Foundation grant DMS-2109633.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    Jingwei Li

  2. School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, Shandong, China

    Rihui Lan

  3. Laboratory of Mathematics and Complex Systems and School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Yongyong Cai

  4. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Lili Ju

  5. Department of Scientific Computing, Florida State University, Tallahassee, FL, 32306, USA

    Xiaoqiang Wang

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  1. Jingwei Li
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Li, J., Lan, R., Cai, Y. et al. Second-Order Semi-Lagrangian Exponential Time Differencing Method with Enhanced Error Estimate for the Convective Allen–Cahn Equation. J Sci Comput 97, 7 (2023). https://doi.org/10.1007/s10915-023-02316-0

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