Maximum Principle-Preserving Computational Algorithm for the 3D High-Order Allen–Cahn Equation
We propose an unconditionally stable computational algorithm that preserves the maximum principle for the three-dimensional (3D) high-order Allen–Cahn (AC) equation. The presented algorithm applies an operator-splitting technique that decomposes the original equation into nonlinear and linear diffus...
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MDPI AG
2025-03-01
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| author | Junseok Kim Youngjin Hwang |
| author_facet | Junseok Kim Youngjin Hwang |
| author_sort | Junseok Kim |
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| description | We propose an unconditionally stable computational algorithm that preserves the maximum principle for the three-dimensional (3D) high-order Allen–Cahn (AC) equation. The presented algorithm applies an operator-splitting technique that decomposes the original equation into nonlinear and linear diffusion equations. To guarantee the unconditional stability of the numerical solution, we solve the nonlinear equation using the frozen coefficient technique, which simplifies computations by approximating variable coefficients by constants within small regions. For the linear equation, we use an implicit finite difference scheme under the operator-splitting method. To validate the efficiency of the proposed algorithm, we conducted several computational tests. The numerical results confirm that the scheme achieves unconditional stability even for large time step sizes and high-order polynomial potential. In addition, we analyze motion by mean curvature in three-dimensional space and show that the numerical solutions closely match the analytical solutions. Finally, the robustness of the method is evaluated under noisy data conditions, and its ability to accurately classify complex data structures is demonstrated. These results confirm the efficiency and reliability of the proposed computational algorithm for simulating phase-field models with a high-order polynomial potential. |
| format | Article |
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| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-03-01 |
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| series | Mathematics |
| spelling | doaj-art-d87e5cedec6042d6bfd629860946ff812025-08-20T02:09:11ZengMDPI AGMathematics2227-73902025-03-01137108510.3390/math13071085Maximum Principle-Preserving Computational Algorithm for the 3D High-Order Allen–Cahn EquationJunseok Kim0Youngjin Hwang1Department of Mathematics, Korea University, Seoul 02841, Republic of KoreaDepartment of Mathematics, Korea University, Seoul 02841, Republic of KoreaWe propose an unconditionally stable computational algorithm that preserves the maximum principle for the three-dimensional (3D) high-order Allen–Cahn (AC) equation. The presented algorithm applies an operator-splitting technique that decomposes the original equation into nonlinear and linear diffusion equations. To guarantee the unconditional stability of the numerical solution, we solve the nonlinear equation using the frozen coefficient technique, which simplifies computations by approximating variable coefficients by constants within small regions. For the linear equation, we use an implicit finite difference scheme under the operator-splitting method. To validate the efficiency of the proposed algorithm, we conducted several computational tests. The numerical results confirm that the scheme achieves unconditional stability even for large time step sizes and high-order polynomial potential. In addition, we analyze motion by mean curvature in three-dimensional space and show that the numerical solutions closely match the analytical solutions. Finally, the robustness of the method is evaluated under noisy data conditions, and its ability to accurately classify complex data structures is demonstrated. These results confirm the efficiency and reliability of the proposed computational algorithm for simulating phase-field models with a high-order polynomial potential.https://www.mdpi.com/2227-7390/13/7/1085frozen coefficient methodoperator-splitting methoddata classification |
| spellingShingle | Junseok Kim Youngjin Hwang Maximum Principle-Preserving Computational Algorithm for the 3D High-Order Allen–Cahn Equation Mathematics frozen coefficient method operator-splitting method data classification |
| title | Maximum Principle-Preserving Computational Algorithm for the 3D High-Order Allen–Cahn Equation |
| title_full | Maximum Principle-Preserving Computational Algorithm for the 3D High-Order Allen–Cahn Equation |
| title_fullStr | Maximum Principle-Preserving Computational Algorithm for the 3D High-Order Allen–Cahn Equation |
| title_full_unstemmed | Maximum Principle-Preserving Computational Algorithm for the 3D High-Order Allen–Cahn Equation |
| title_short | Maximum Principle-Preserving Computational Algorithm for the 3D High-Order Allen–Cahn Equation |
| title_sort | maximum principle preserving computational algorithm for the 3d high order allen cahn equation |
| topic | frozen coefficient method operator-splitting method data classification |
| url | https://www.mdpi.com/2227-7390/13/7/1085 |
| work_keys_str_mv | AT junseokkim maximumprinciplepreservingcomputationalalgorithmforthe3dhighorderallencahnequation AT youngjinhwang maximumprinciplepreservingcomputationalalgorithmforthe3dhighorderallencahnequation |