An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface

Youngjin Hwang; Jyoti; Soobin Kwak; Hyundong Kim; Junseok Kim; Youngjin Hwang; Jyoti; Soobin Kwak; Hyundong Kim; Junseok Kim

doi:10.3934/math.20241641

AIMS Mathematics

2024, Volume 9, Issue 12: 34447-34465. doi: 10.3934/math.20241641

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An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface

1.
Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea
2.
The Institute of Basic Science, Korea University, Seoul, 02841, Republic of Korea
3.
Department of Mathematics and Physics, Gangneung-Wonju National University, Gangneung 25457, Republic of Korea
4.
Institute for Smart Infrastructure, Gangneung-Wonju National University, Gangneung 25457, Republic of Korea

Received: 08 October 2024 Revised: 27 November 2024 Accepted: 04 December 2024 Published: 06 December 2024
MSC : 80M20, 39A14, 65M06

We introduced a fully explicit finite difference method (FDM) designed for numerically solving the conservative Allen–Cahn equation (CAC) on a cubic surface. In this context, the cubic surface refers to the combined areas of the six square faces that enclose the volume of a cube. The proposed numerical solution approach is structured into two sequential steps. First, the Allen–Cahn (AC) equation was solved by applying the fully explicit FDM, which is computationally efficient. Following this, the conservation term is resolved using the updated solution from the AC equation to ensure consistency with the underlying conservation principles. To evaluate the effectiveness of the proposed scheme, computational tests are performed to verify that the resulting numerical solution of the CAC equation successfully conserves the discrete mass. Additionally, the solution is examined for its ability to exhibit the property of constrained motion by mass conserving mean curvature, a critical characteristic of the CAC equation. These two properties are fundamental to the integrity and accuracy of the CAC equation.
- finite difference scheme,
- cubic domain,
- Lagrange multiplier,
- explicit scheme,
- conservative AC equation
Citation: Youngjin Hwang, Jyoti, Soobin Kwak, Hyundong Kim, Junseok Kim. An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface[J]. AIMS Mathematics, 2024, 9(12): 34447-34465. doi: 10.3934/math.20241641

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Abstract

We introduced a fully explicit finite difference method (FDM) designed for numerically solving the conservative Allen–Cahn equation (CAC) on a cubic surface. In this context, the cubic surface refers to the combined areas of the six square faces that enclose the volume of a cube. The proposed numerical solution approach is structured into two sequential steps. First, the Allen–Cahn (AC) equation was solved by applying the fully explicit FDM, which is computationally efficient. Following this, the conservation term is resolved using the updated solution from the AC equation to ensure consistency with the underlying conservation principles. To evaluate the effectiveness of the proposed scheme, computational tests are performed to verify that the resulting numerical solution of the CAC equation successfully conserves the discrete mass. Additionally, the solution is examined for its ability to exhibit the property of constrained motion by mass conserving mean curvature, a critical characteristic of the CAC equation. These two properties are fundamental to the integrity and accuracy of the CAC equation.

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