An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface
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 numeri...
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AIMS Press
2024-12-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241641 |
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| author | Youngjin Hwang Jyoti Soobin Kwak Hyundong Kim Junseok Kim |
| author_facet | Youngjin Hwang Jyoti Soobin Kwak Hyundong Kim Junseok Kim |
| author_sort | Youngjin Hwang |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-22df03cfdd5f4ec2be348c1ae8f4d981 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
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| series | AIMS Mathematics |
| spelling | doaj-art-22df03cfdd5f4ec2be348c1ae8f4d9812025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912344473446510.3934/math.20241641An explicit numerical method for the conservative Allen–Cahn equation on a cubic surfaceYoungjin Hwang0Jyoti1Soobin Kwak2Hyundong Kim3Junseok Kim4Department of Mathematics, Korea University, Seoul, 02841, Republic of KoreaThe Institute of Basic Science, Korea University, Seoul, 02841, Republic of KoreaDepartment of Mathematics, Korea University, Seoul, 02841, Republic of KoreaDepartment of Mathematics and Physics, Gangneung-Wonju National University, Gangneung 25457, Republic of KoreaDepartment of Mathematics, Korea University, Seoul, 02841, Republic of KoreaWe 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.https://www.aimspress.com/article/doi/10.3934/math.20241641finite difference schemecubic domainlagrange multiplierexplicit schemeconservative ac equation |
| spellingShingle | Youngjin Hwang Jyoti Soobin Kwak Hyundong Kim Junseok Kim An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface AIMS Mathematics finite difference scheme cubic domain lagrange multiplier explicit scheme conservative ac equation |
| title | An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface |
| title_full | An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface |
| title_fullStr | An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface |
| title_full_unstemmed | An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface |
| title_short | An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface |
| title_sort | explicit numerical method for the conservative allen cahn equation on a cubic surface |
| topic | finite difference scheme cubic domain lagrange multiplier explicit scheme conservative ac equation |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241641 |
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