Motion by Mean Curvature with Constraints Using a Modified Allen-Cahn Equation

Abstract

In this article, we present a simple and accurate computational scheme for motion by mean curvature with constraints using a modified Allen-Cahn (AC) equation. The modified AC equation contains a nonlinear source term which enforces the constraints such as volume and average mean curvature. We use a linear convex splitting-type method with Fourier spectral method to numerically solve the modified AC equation. We perform several characteristic computational tests to demonstrate the efficiency and accuracy of the proposed method. The computational results confirm the robust and high performance of the proposed algorithm.