A simple and efficient numerical method for the Allen–Cahn equation on effective symmetric triangular meshes

Youngjin Hwang; Seokjun Ham; Chaeyoung Lee; Gyeonggyu Lee; Seungyoon Kang; Junseok Kim; Youngjin Hwang; Seokjun Ham; Chaeyoung Lee; Gyeonggyu Lee; Seungyoon Kang; Junseok Kim

doi:10.3934/era.2023233

Electronic Research Archive

2023, Volume 31, Issue 8: 4557-4578. doi: 10.3934/era.2023233

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A simple and efficient numerical method for the Allen–Cahn equation on effective symmetric triangular meshes

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

Academic Editor: Alicia Cordero

Received: 01 May 2023 Revised: 06 June 2023 Accepted: 13 June 2023 Published: 20 June 2023

In this paper, we propose a novel, simple, efficient, and explicit numerical method for the Allen–Cahn (AC) equation on effective symmetric triangular meshes. First, we compute the net vector of all vectors starting from each node point to its one-ring neighbor vertices and virtually adjust the neighbor vertices so that the net vector is zero. Then, we define the values at the virtually adjusted nodes using linear and quadratic interpolations. Finally, we define a discrete Laplace operator on triangular meshes. We perform several computational experiments to demonstrate the performance of the proposed numerical method for the Laplace operator, the diffusion equation, and the AC equation on triangular meshes.
- diffusion equation,
- discrete Laplace operator,
- triangular meshes,
- Allen–Cahn equation
Citation: Youngjin Hwang, Seokjun Ham, Chaeyoung Lee, Gyeonggyu Lee, Seungyoon Kang, Junseok Kim. A simple and efficient numerical method for the Allen–Cahn equation on effective symmetric triangular meshes[J]. Electronic Research Archive, 2023, 31(8): 4557-4578. doi: 10.3934/era.2023233

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Abstract

In this paper, we propose a novel, simple, efficient, and explicit numerical method for the Allen–Cahn (AC) equation on effective symmetric triangular meshes. First, we compute the net vector of all vectors starting from each node point to its one-ring neighbor vertices and virtually adjust the neighbor vertices so that the net vector is zero. Then, we define the values at the virtually adjusted nodes using linear and quadratic interpolations. Finally, we define a discrete Laplace operator on triangular meshes. We perform several computational experiments to demonstrate the performance of the proposed numerical method for the Laplace operator, the diffusion equation, and the AC equation on triangular meshes.

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