An effective operator splitting scheme for general motion by mean curvature using a modified Allen–Cahn equation

Abstract

We present a fast and effective method for modeling general motion by mean curvature based on a modified Allen–Cahn equation. Employing the second-order operator time-splitting method, the original problem is discretized into three subproblems based on the different natures of each part of the model: the heat equation is solved by a Crank–Nicolson (CN) alternating direction implicit (ADI) finite difference scheme; the other two nonlinear equations have closed-form solutions and thus can be solved analytically. We demonstrate that the resulting scheme can preserve the maximum principle of the modified Allen–Cahn equation. Numerical experiments are presented to demonstrate the effectiveness of the proposed scheme.