An unconditionally stable splitting method for the Allen–Cahn equation with logarithmic free energy

Jintae Park, Chaeyoung Lee, Yongho Choi, Hyun Geun Lee, Soobin Kwak, Youngjin Hwang, Junseok Kim

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)

Abstract

We present an unconditionally stable splitting method for the Allen–Cahn (AC) equation with logarithmic free energy which is more physically meaningful than the commonly used polynomial potentials. However, owing to the singularity of the logarithmic free energy, it is difficult to develop unconditionally stable computational methods for the AC equation with logarithmic potential. To overcome this difficulty, prior works added a stabilizing term to the logarithmic energy or used a regularized potential. In this study, the AC equation with logarithmic potential is solved by using an operator splitting method without adding a stabilizing term nor regularizing the logarithmic energy. The equation involving logarithmic free energy potential is solved using an interpolation method; the other diffusion equation is solved numerically by applying a finite difference method. Each solution algorithm is unconditionally stable, the proposed scheme is unconditionally stable. Various computational experiments demonstrate the performance of the proposed method.

Original languageEnglish
Article number18
JournalJournal of Engineering Mathematics
Volume132
Issue number1
DOIs
Publication statusPublished - 2022 Feb

Bibliographical note

Funding Information:
C. Lee was supported by the National Research Foundation(NRF), Korea, under project BK21 FOUR. Y. Choi was supported by National Research Foundation of Korea (NRF) grant funded by the Korea Government (NRF-2020R1C1C1A0101153712). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education(NRF-2019R1A2C1003053). The authors are grateful to the reviewers whose valuable suggestions and comments significantly improved the quality of this article.

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature B.V.

Keywords

  • Allen–Cahn equation
  • Flory–Huggins potential
  • Unconditionally stable scheme

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering

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