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A review of the numerical methods for solving the binary Allen–Cahn equation

Author

Listed:
  • Lee, Hyun Geun
  • Li, Yibao
  • Yang, Junxiang
  • Kwak, Soobin
  • Hwang, Youngjin
  • Ham, Seokjun
  • Kim, Hyundong
  • Jyoti,
  • Nam, Yunjae
  • Kim, Junseok

Abstract

In this review, we present an overview of numerical methods to solve the binary Allen–Cahn (AC) equation, which is extensively used to model phase separation processes in materials science. It describes the time-dependent evolution of interfaces between two phases and accounts for both local reaction kinetics and diffusion effects. This equation plays a critical role in understanding the behavior of interfaces. The AC equation has various applications across fields such as materials science, physics, and biology, where it helps to analyze and predict phenomena such as phase transitions, grain boundary motion, and pattern formation in complex systems. Its importance lies in its ability to model the dynamics of interfaces and help the study of pattern formation and phase transitions in diverse environments. We discuss various computational methodologies developed for this important mathematical model and describe their strengths, limitations, and applications in diverse scientific domains.

Suggested Citation

  • Lee, Hyun Geun & Li, Yibao & Yang, Junxiang & Kwak, Soobin & Hwang, Youngjin & Ham, Seokjun & Kim, Hyundong & Jyoti, & Nam, Yunjae & Kim, Junseok, 2025. "A review of the numerical methods for solving the binary Allen–Cahn equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 670(C).
    • Handle: RePEc:eee:phsmap:v:670:y:2025:i:c:s0378437125002778
    DOI: 10.1016/j.physa.2025.130625
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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. Tan, Zhijun & Yang, Junxiang & Chen, Jianjun & Kim, Junseok, 2023. "An efficient time-dependent auxiliary variable approach for the three-phase conservative Allen–Cahn fluids," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    4. 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.
    5. Youngjin Hwang & Chaeyoung Lee & Soobin Kwak & Yongho Choi & Seokjun Ham & Seungyoon Kang & Junxiang Yang & Junseok Kim & Francisco R. Villatoro, 2022. "Benchmark Problems for the Numerical Schemes of the Phase-Field Equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2022, pages 1-10, January.
    6. Yongho Kim & Gilnam Ryu & Yongho Choi, 2021. "Fast and Accurate Numerical Solution of Allen–Cahn Equation," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-12, December.
    7. 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.
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