Stability analysis of a numerical method for the 3D high-order Allen–Cahn equation

• Published in: AIP advances Vol. 15; no. 1; pp. 015126 - 015126-9
• Main Authors: Ham, Seokjun, Jyoti, Choi, Jaeyong, Nam, Yunjae, Kim, Junseok
• Format: Journal Article
• Language: English
• Published: Melville American Institute of Physics 01.01.2025; AIP Publishing LLC
• Subjects: Alloy systems; Binary alloys; Binary systems; Interface stability; Mathematical analysis; Maximum principle; Numerical methods; Phase separation; Polynomials; Stability analysis
• ISSN: 2158-3226; 2158-3226
• DOI: 10.1063/5.0248165

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.