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Fast and efficient numerical method for solving the Allen–Cahn equation on the cubic surface

Author

Listed:
  • Hwang, Youngjin
  • Yang, Junxiang
  • Lee, Gyeongyu
  • Ham, Seokjun
  • Kang, Seungyoon
  • Kwak, Soobin
  • Kim, Junseok

Abstract

In this study, we present a fast and efficient finite difference method (FDM) for solving the Allen–Cahn (AC) equation on the cubic surface. The proposed method applies appropriate boundary conditions in the two-dimensional (2D) space to calculate numerical solutions on cubic surfaces, which is relatively simpler than a direct computation in the three-dimensional (3D) space. To numerically solve the AC equation on the cubic surface, we first unfold the cubic surface domain in the 3D space into the 2D space, and then apply the FDM on the six planar sub-domains with appropriate boundary conditions. The proposed method solves the AC equation using an operator splitting method that splits the AC equation into the linear and nonlinear terms. To demonstrate that the proposed algorithm satisfies the properties of the AC equation on the cubic surface, we perform the numerical experiments such as convergence test, total energy decrease, and maximum principle.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:215:y:2024:i:c:p:338-356
    DOI: 10.1016/j.matcom.2023.07.024
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    References listed on IDEAS

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    1. 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.
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