An explicit hybrid finite difference scheme for the Allen–Cahn equation

Darae Jeong, Junseok Kim

Research output: Contribution to journalArticlepeer-review

40 Citations (Scopus)


In this paper, we propose an explicit hybrid numerical method for solving the Allen–Cahn equation, which models antiphase domain coarsening process in a binary mixture. The proposed method is based on an operator splitting method. First, we solve the linear diffusion part using the explicit Euler method. Second, we solve the nonlinear term using the closed-form analytical solution. We show the stability condition of the proposed numerical scheme. We also show the pointwise boundedness of the numerical solution for the Allen–Cahn equation under a solvability condition. Numerical experiments such as linear stability analysis, traveling wave, motion by mean curvature, image segmentation, and crystal growth are presented to demonstrate the performance of the proposed method.

Original languageEnglish
Pages (from-to)247-255
Number of pages9
JournalJournal of Computational and Applied Mathematics
Publication statusPublished - 2018 Oct 1

Bibliographical note

Funding Information:
The first author (D. Jeong) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) ( NRF-2017R1E1A1A03070953 ). The corresponding author (Junseok Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( NRF-2016R1D1A1B03933243 ). The authors greatly appreciate the reviewers for their constructive comments and suggestions, which have improved the quality of this paper.

Publisher Copyright:
© 2018 Elsevier B.V.


  • Allen–Cahn equation
  • Explicit hybrid method
  • Finite difference method
  • Operator splitting method

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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