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A second-order maximum bound principle preserving operator splitting method for the Allen–Cahn equation with applications in multi-phase systems

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  • Xiao, Xufeng
  • Feng, Xinlong

Abstract

In this paper, a highly efficient space–time operator splitting finite element method is presented to solve the two- and three-dimensional Allen–Cahn equations. The main advantage of the proposed method is that it reduces the high storage requirements and complexity of the high-dimensional computation by splitting the high-dimensional problem into a series of one-dimensional subproblems. The proposed method is space–time second-order and can be performed in parallel. Moreover, a bound preserving least-distance modification technique is developed to force the discrete maximum bound principle in solving each one-dimensional subproblem. Finally, numerical simulations including the two- and multi-phase separations, mean curvature flows and dendritic crystal growth in two and three dimensions are provided to demonstrate the validity and accuracy of the proposed method.

Suggested Citation

  • Xiao, Xufeng & Feng, Xinlong, 2022. "A second-order maximum bound principle preserving operator splitting method for the Allen–Cahn equation with applications in multi-phase systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 36-58.
    • Handle: RePEc:eee:matcom:v:202:y:2022:i:c:p:36-58
    DOI: 10.1016/j.matcom.2022.05.024
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    References listed on IDEAS

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    1. Choi, Jeong-Whan & Lee, Hyun Geun & Jeong, Darae & Kim, Junseok, 2009. "An unconditionally gradient stable numerical method for solving the Allen–Cahn equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(9), pages 1791-1803.
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    Cited by:

    1. Ham, Seokjun & Kim, Junseok, 2023. "Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 453-465.
    2. Hwang, Youngjin & Yang, Junxiang & Lee, Gyeongyu & Ham, Seokjun & Kang, Seungyoon & Kwak, Soobin & Kim, Junseok, 2024. "Fast and efficient numerical method for solving the Allen–Cahn equation on the cubic surface," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 215(C), pages 338-356.

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