Fast and efficient numerical method for solving the Allen–Cahn equation on the cubic surface

Youngjin Hwang, Junxiang Yang, Gyeongyu Lee, Seokjun Ham, Seungyoon Kang, Soobin Kwak and Junseok Kim

Mathematics and Computers in Simulation (MATCOM), 2024, vol. 215, issue C, 338-356

Abstract: In this study, we present a fast and efficient finite difference method (FDM) for solving the Allen–Cahn (AC) equation on the cubic surface. The proposed method applies appropriate boundary conditions in the two-dimensional (2D) space to calculate numerical solutions on cubic surfaces, which is relatively simpler than a direct computation in the three-dimensional (3D) space. To numerically solve the AC equation on the cubic surface, we first unfold the cubic surface domain in the 3D space into the 2D space, and then apply the FDM on the six planar sub-domains with appropriate boundary conditions. The proposed method solves the AC equation using an operator splitting method that splits the AC equation into the linear and nonlinear terms. To demonstrate that the proposed algorithm satisfies the properties of the AC equation on the cubic surface, we perform the numerical experiments such as convergence test, total energy decrease, and maximum principle.

Keywords: Cubic surface; Finite difference method; Diffusion equation; Allen–Cahn equation (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:215:y:2024:i:c:p:338-356

DOI: 10.1016/j.matcom.2023.07.024

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