Pinning phenomena in numerical schemes of the Allen-Cahn equation

초록

In this study, we investigate the artificial pinning phenomena that emerge in numerical simulations of high-order Allen-Cahn (AC) equations. The AC equation is widely used to describe interface motion during phase separation processes. However, discretized numerical solutions may exhibit nonphysical behavior such as interface immobilization, or "pinning," particularly when the ratio between the interfacial thickness model parameter (epsilon) and the spatial numerical mesh size (h) is below a critical threshold. The present work systematically analyzes how this ratio, P = epsilon/h, influences interface dynamics and determines the critical values at which pinning begins to occur for various nonlinearity degrees (alpha) of the polynomial potential. A fully explicit finite difference method is used to simulate the AC equation in both two-and three-dimensional settings. A bisection algorithm is introduced to accurately identify the critical pinning threshold P* as a function of alpha. Numerical experiments demonstrate that as alpha increases, P* decreases, indicating that the solution becomes more sensitive to spatial discretization and requires finer grids to avoid artificial pinning. Additionally, computational results show that higher values of alpha yield thicker and smoother interface transitions, and the study also investigates the influence of these values on phase morphology under random initial conditions. The numerical results in this study provide quantitative guidance for selecting discretization parameters in phase-field simulations and emphasize the importance of balancing the interface resolution with model complexity. These insights contribute to the development of accurate and robust numerical schemes for simulating interfacial dynamics in complex physical systems.

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