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Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation

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  • Ham, Seokjun
  • Kim, Junseok

Abstract

In this study, we present the stability analysis of a fully explicit finite difference method (FDM) for solving the Allen–Cahn (AC) equation. The AC equation is a second-order nonlinear partial differential equation (PDE), which describes the antiphase boundaries of the binary phase separation. In the presented stability analysis, we consider the explicit Euler method for the temporal derivative and second-order finite difference in the space direction. The explicit scheme is fast and accurate because it uses a small time step, however, it has a temporal step constraint. We analyze and compute that the explicit time step constraint formula guarantees the discrete maximum principle for the numerical solutions of the AC equation. The numerical stability of the explicit scheme automatically holds when we use the time satisfying the discrete maximum principle. The computational numerical experiments demonstrate the stability, discrete maximum principle, and accuracy of the explicit scheme for the constrained time step. Furthermore, it is shown that the time step obtained is not severe restriction when we consider the temporal accuracy.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:207:y:2023:i:c:p:453-465
    DOI: 10.1016/j.matcom.2023.01.016
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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.
    2. Jeong, Darae & Li, Yibao & Choi, Yongho & Lee, Chaeyoung & Yang, Junxiang & Kim, Junseok, 2021. "A practical adaptive grid method for the Allen–Cahn equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 573(C).
    3. Sungha Yoon & Darae Jeong & Chaeyoung Lee & Hyundong Kim & Sangkwon Kim & Hyun Geun Lee & Junseok Kim, 2020. "Fourier-Spectral Method for the Phase-Field Equations," Mathematics, MDPI, vol. 8(8), pages 1-36, August.
    4. Dongsun Lee & Seunggyu Lee, 2019. "Image Segmentation Based on Modified Fractional Allen–Cahn Equation," Mathematical Problems in Engineering, Hindawi, vol. 2019, pages 1-6, January.
    5. Montanelli, Hadrien & Bootland, Niall, 2020. "Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 307-327.
    6. 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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