The Allen–Cahn model with a time-dependent parameter for motion by mean curvature up to the singularity

Abstract

In this article, we investigate the temporal evolution of arbitrary, simple, closed two-dimensional (2D) and three-dimensional (3D) interfaces under motion driven by mean curvature up to a singularity. To facilitate this investigation, we propose a novel Allen–Cahn (AC) model with a time-dependent interfacial thickness parameter. The original AC equation was developed to model the phase separation of a binary mixture. It is well known that a level set or interface of the solution of the AC equation obeys the dynamics of motion by curvature as the interfacial thickness parameter approaches zero. Generally, it is difficult to find a closed-form analytic solution of the AC equation with any initial condition. Therefore, we need to estimate the solution of the AC equation through computational approaches such as finite difference method (FDM), finite element method (FEM), Fourier-spectral method (FSM), and finite volume method (FVM). Any simple, closed curves and convex surfaces eventually shrink to a point due to motion by mean curvature. Therefore, it becomes necessary to use adaptive mesh techniques to resolve this small size problem. However, even though we use adaptive mesh techniques, we still confront the relatively thick interfacial transition layer when the curves or interfaces become very small. To avoid this problem, we can start with a very small mesh size for a small value of the interfacial parameter, which results in an extremely high computational cost even when using adaptive mesh techniques. To resolve these issues, we present the AC equation with a time-dependent interfacial parameter and develop an adaptive mesh refinement system. To show the superior performance of the proposed mathematical equation and its computational algorithm, we present various numerical experiments and investigate the motion by mean curvature up to the singularity of curves in 2D space and interfaces in 3D space.