Fast evolution numerical method for the Allen–Cahn equation

Junxiang Yang, Yibao Li, Chaeyoung Lee, Yongho Choi, Junseok Kim

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1 Citation (Scopus)


We present a fast evolution numerical algorithm for solving the Allen–Cahn (AC) equations. One of efficient computational techniques for the AC equation is the operator splitting method. We split the AC equation into the linear heat and nonlinear equations; and then solve the linear part using the Fourier spectral method and the nonlinear part using an analytic closed-form solution. These steps are unconditionally stable. However, if a large time step is used, then the nonlinear part dominates the evolution and results in a sharp interfacial transition layer. To overcome these problems, we propose a time rescaling method to the nonlinear part of the AC equation. Computational tests verify the performance of the proposed method which makes the evolution fast and interfacial transition layer be uniform.

Original languageEnglish
Article number102430
JournalJournal of King Saud University - Science
Issue number1
Publication statusPublished - 2023 Jan

Bibliographical note

Funding Information:
J. Yang is supported by the National Natural Science Foundation of China (No. 12201657), the China Postdoctoral Science Foundation (No. 2022M713639), and the 2022 International Postdoctoral Exchange Fellowship Program (Talent-Introduction Program) (No. YJ20220221). C. Lee was supported by the National Research Foundation(NRF), Korea, under project BK21 FOUR. The corresponding author (J.S. Kim) was supported by the Brain Korea 21 FOUR from the Ministry of Education of the Republic of Korea. Y. Choi was supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF-2020R1C1C1A0101153712). The authors thank the reviewers for the constructive and helpful comments on the revision of this article.

Publisher Copyright:
© 2022 The Author(s)


  • Allen–Cahn equation
  • Fast evolution scheme
  • Operator splitting method

ASJC Scopus subject areas

  • General


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