Unconditionally stable monte carlo simulation for solving the multi-dimensional Allen–Cahn equation

Youngjin Hwang, Ildoo Kim, Soobin Kwak, Seokjun Ham, Sangkwon Kim, Junseok Kim

Research output: Contribution to journalArticlepeer-review


In this study, we present an efficient and novel unconditionally stable Monte Carlo simulation (MCS) for solving the multi-dimensional Allen–Cahn (AC) equation, which can model the motion by mean curvature flow of a hypersurface. We use an operator splitting method, where the diffusion and nonlinear terms are solved separately. The diffusion term is calculated using MCS for the stochastic differential equation, while the nonlinear term is locally computed for each particle in a virtual grid. Several numerical experiments are presented to demonstrate the performance of the proposed algorithm. The computational results confirm that the proposed algorithm can solve the AC equation more efficiently as the dimension of space increases.

Original languageEnglish
Pages (from-to)5104-5123
Number of pages20
JournalElectronic Research Archive
Issue number8
Publication statusPublished - 2023

Bibliographical note

Funding Information:
The corresponding author (J. S. Kim) expresses thanks for the support from the BK21 FOUR program. The authors express their gratitude to the reviewers for their valuable and insightful feedback on the revised version of this article.

Publisher Copyright:
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)


  • Monte Carlo simulation
  • multi-dimensional Allen–Cahn equation
  • operator splitting method
  • unconditionally stable scheme

ASJC Scopus subject areas

  • General Mathematics


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