An Explicit Adaptive Finite Difference Method for the Cahn–Hilliard Equation

Seokjun Ham, Yibao Li, Darae Jeong, Chaeyoung Lee, Soobin Kwak, Youngjin Hwang, Junseok Kim

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


In this study, we propose an explicit adaptive finite difference method (FDM) for the Cahn–Hilliard (CH) equation which describes the process of phase separation. The CH equation has been successfully utilized to model and simulate diverse field applications such as complex interfacial fluid flows and materials science. To numerically solve the CH equation fast and efficiently, we use the FDM and time-adaptive narrow-band domain. For the adaptive grid, we define a narrow-band domain including the interfacial transition layer of the phase field based on an undivided finite difference and solve the numerical scheme on the narrow-band domain. The proposed numerical scheme is based on an alternating direction explicit (ADE) method. To make the scheme conservative, we apply a mass correction algorithm after each temporal iteration step. To demonstrate the superior performance of the proposed adaptive FDM for the CH equation, we present two- and three-dimensional numerical experiments and compare them with those of other previous methods.

Original languageEnglish
Article number80
JournalJournal of Nonlinear Science
Issue number6
Publication statusPublished - 2022 Dec

Bibliographical note

Funding Information:
The first author (S. Ham) was supported by the National Research Foundation (NRF), Korea, under project BK21 FOUR. Y.B. Li is supported by the Fundamental Research Funds for the Central Universities (No.XTR042019005). The corresponding author (J.S. Kim) was supported by Korea University Grant. The authors are grateful to the reviewers for their suggestions and comments on the revision of this article.

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.


  • Adaptive finite difference scheme
  • Cahn–Hilliard equation
  • Stable numerical method

ASJC Scopus subject areas

  • Modelling and Simulation
  • General Engineering
  • Applied Mathematics


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