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

doi:10.3934/era.2023261

Electronic Research Archive

2023, Volume 31, Issue 8: 5104-5123. doi: 10.3934/era.2023261

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

Department of Mathematics, Korea University, Seoul 02841, Republic of Korea

Received: 13 April 2023 Revised: 27 June 2023 Accepted: 04 July 2023 Published: 17 July 2023

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.
- Monte Carlo simulation,
- operator splitting method,
- unconditionally stable scheme,
- multi-dimensional Allen–Cahn equation
Citation: Youngjin Hwang, Ildoo Kim, Soobin Kwak, Seokjun Ham, Sangkwon Kim, Junseok Kim. Unconditionally stable monte carlo simulation for solving the multi-dimensional Allen–Cahn equation[J]. Electronic Research Archive, 2023, 31(8): 5104-5123. doi: 10.3934/era.2023261

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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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