Stability Analysis for Maximum Principle Preserving Explicit Isotropic Schemes of the Allen-Cahn Equation

Abstract

In practical applications, the Allen-Cahn (AC) equation is commonly used to model microstructure evolutions, including alloy solidification, crystal growth, fingerprint image restoration, and image segmentation. However, when we discretize the AC equation with a conventional finite difference scheme, the directional bias in error terms introduces anisotropy into the numerical results, affecting interface dynamics. To address this issue, we use two- and three-dimensional isotropic finite difference schemes to solve the AC equation. Stability of the proposed algorithm is verified by deriving the time step constraints in both 2D and 3D domains. To demonstrate the sharp estimation of the stability constraints, we conducted several numerical experiments and found the maximum principle is guaranteed under the analyzed time-step constraint.