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

doi:10.3934/era.2023261

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 journal › Article › peer-review

11 Citations (Scopus)

Abstract

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 language	English
Pages (from-to)	5104-5123
Number of pages	20
Journal	Electronic Research Archive
Volume	31
Issue number	8
DOIs	https://doi.org/10.3934/era.2023261
Publication status	Published - 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)

Keywords

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

ASJC Scopus subject areas

General Mathematics

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10.3934/era.2023261

Cite this

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title = "Unconditionally stable monte carlo simulation for solving the multi-dimensional Allen–Cahn equation",

abstract = "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.",

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AU - Hwang, Youngjin

AU - Kim, Ildoo

AU - Kwak, Soobin

AU - Ham, Seokjun

AU - Kim, Sangkwon

AU - Kim, Junseok

N1 - 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)

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - Monte Carlo simulation

KW - multi-dimensional Allen–Cahn equation

KW - operator splitting method

KW - unconditionally stable scheme

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DO - 10.3934/era.2023261

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VL - 31

SP - 5104

EP - 5123

JO - Electronic Research Archive

JF - Electronic Research Archive

IS - 8

ER -

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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