We have presented a maximum principle preserving the unconditionally stable scheme for the Allen–Cahn (AC) equation with a high-order polynomial potential. The proposed method ensures the preservation of the maximum principle, a critical characteristic for accurately modeling phase transitions and maintaining physical consistency in simulations. The proposed method uses an operator splitting technique, a numerical approach that decomposes a complex problem into simpler subproblems, solved sequentially, to improve computational efficiency and stability. The operator splitting method applied to the AC equation yields one nonlinear equation and several linear equations. To solve the nonlinear equation, we applied the frozen coefficient method, which approximates variable coefficients in differential equations by treating them as constants within small regions, simplifies the problem, and enables more efficient numerical solutions. For several linear equations, which are diffusion equations, we applied a fully implicit finite difference scheme to obtain unconditional stability. By using these methods, we achieved unconditional stability for the AC equation. To validate the superior performance of the developed algorithm, we performed computational tests. Computational experiments demonstrated its unconditional stability, particularly in handling high-order polynomial potentials. Furthermore, we highlighted a distinctive feature of the AC equation in modeling phase separation under noisy data conditions.
Citation: Junseok Kim. Maximum principle preserving the unconditionally stable method for the Allen–Cahn equation with a high-order potential[J]. Electronic Research Archive, 2025, 33(1): 433-446. doi: 10.3934/era.2025021
We have presented a maximum principle preserving the unconditionally stable scheme for the Allen–Cahn (AC) equation with a high-order polynomial potential. The proposed method ensures the preservation of the maximum principle, a critical characteristic for accurately modeling phase transitions and maintaining physical consistency in simulations. The proposed method uses an operator splitting technique, a numerical approach that decomposes a complex problem into simpler subproblems, solved sequentially, to improve computational efficiency and stability. The operator splitting method applied to the AC equation yields one nonlinear equation and several linear equations. To solve the nonlinear equation, we applied the frozen coefficient method, which approximates variable coefficients in differential equations by treating them as constants within small regions, simplifies the problem, and enables more efficient numerical solutions. For several linear equations, which are diffusion equations, we applied a fully implicit finite difference scheme to obtain unconditional stability. By using these methods, we achieved unconditional stability for the AC equation. To validate the superior performance of the developed algorithm, we performed computational tests. Computational experiments demonstrated its unconditional stability, particularly in handling high-order polynomial potentials. Furthermore, we highlighted a distinctive feature of the AC equation in modeling phase separation under noisy data conditions.
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Junseok Kim. Maximum principle preserving the unconditionally stable method for the Allen–Cahn equation with a high-order potential[J]. Electronic Research Archive, 2025, 33(1): 433-446. doi: 10.3934/era.2025021
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