Stability analysis of a numerical method for the 3D high-order Allen–Cahn equation
The Allen–Cahn (AC) equation describes how phase separation occurs in binary alloy systems and the dynamics of interfaces between different phases. In the present study, we incorporated the function of high order polynomial potentials in the standard AC equation and present the stability condition f...
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| Format: | Article |
| Language: | English |
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AIP Publishing LLC
2025-01-01
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| Series: | AIP Advances |
| Online Access: | http://dx.doi.org/10.1063/5.0248165 |
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| author | Seokjun Ham Jyoti Jaeyong Choi Yunjae Nam Junseok Kim |
| author_facet | Seokjun Ham Jyoti Jaeyong Choi Yunjae Nam Junseok Kim |
| author_sort | Seokjun Ham |
| collection | DOAJ |
| description | The Allen–Cahn (AC) equation describes how phase separation occurs in binary alloy systems and the dynamics of interfaces between different phases. In the present study, we incorporated the function of high order polynomial potentials in the standard AC equation and present the stability condition for the numerical scheme that is used to solve the AC problem in three-dimensional space. We used a fully explicit Euler scheme for temporal derivatives and a second-order finite difference approach for spatial discretization. However, the explicit scheme is known for its speed and accuracy due to the use of small time steps, but it is subject to a temporal step size limitation. Here, we derived and validated a time step condition that satisfies the discrete maximum principle and assures the stability of the scheme. Several experiments are carried out under the constrained time step to ensure the accuracy of the explicit method, the stability of the scheme, and the discrete maximum principle. |
| format | Article |
| id | doaj-art-cf08a3953d7f4f89a6daf35a40384952 |
| institution | Kabale University |
| issn | 2158-3226 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | AIP Publishing LLC |
| record_format | Article |
| series | AIP Advances |
| spelling | doaj-art-cf08a3953d7f4f89a6daf35a403849522025-02-03T16:40:42ZengAIP Publishing LLCAIP Advances2158-32262025-01-01151015126015126-910.1063/5.0248165Stability analysis of a numerical method for the 3D high-order Allen–Cahn equationSeokjun Ham0Jyoti1Jaeyong Choi2Yunjae Nam3Junseok Kim4Department of Mathematics, Korea University, Seoul 02841, Republic of KoreaThe Institute of Basic Science, Korea University, Seoul 02841, Republic of KoreaDivision of Mathematics and Computer Science, University of Guam, Mangilao 96923, GuamProgram in Actuarial Science and Financial Engineering, Korea University, Seoul 02841, Republic of KoreaDepartment of Mathematics, Korea University, Seoul 02841, Republic of KoreaThe Allen–Cahn (AC) equation describes how phase separation occurs in binary alloy systems and the dynamics of interfaces between different phases. In the present study, we incorporated the function of high order polynomial potentials in the standard AC equation and present the stability condition for the numerical scheme that is used to solve the AC problem in three-dimensional space. We used a fully explicit Euler scheme for temporal derivatives and a second-order finite difference approach for spatial discretization. However, the explicit scheme is known for its speed and accuracy due to the use of small time steps, but it is subject to a temporal step size limitation. Here, we derived and validated a time step condition that satisfies the discrete maximum principle and assures the stability of the scheme. Several experiments are carried out under the constrained time step to ensure the accuracy of the explicit method, the stability of the scheme, and the discrete maximum principle.http://dx.doi.org/10.1063/5.0248165 |
| spellingShingle | Seokjun Ham Jyoti Jaeyong Choi Yunjae Nam Junseok Kim Stability analysis of a numerical method for the 3D high-order Allen–Cahn equation AIP Advances |
| title | Stability analysis of a numerical method for the 3D high-order Allen–Cahn equation |
| title_full | Stability analysis of a numerical method for the 3D high-order Allen–Cahn equation |
| title_fullStr | Stability analysis of a numerical method for the 3D high-order Allen–Cahn equation |
| title_full_unstemmed | Stability analysis of a numerical method for the 3D high-order Allen–Cahn equation |
| title_short | Stability analysis of a numerical method for the 3D high-order Allen–Cahn equation |
| title_sort | stability analysis of a numerical method for the 3d high order allen cahn equation |
| url | http://dx.doi.org/10.1063/5.0248165 |
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