An Alternating Direction Implicit Method for Mean Curvature Flows

TL;DR

This paper introduces an ADI-based Cartesian grid method for simulating mean curvature flows in 2D and 3D, enabling stable, efficient evolution of hypersurfaces without restrictive time step constraints.

Contribution

The paper presents a novel ADI-based semi-implicit method that simplifies mean curvature flow simulation on Cartesian grids by decomposing hypersurfaces into overlapping subsets.

Findings

The method effectively removes stiffness, allowing larger time steps.

Numerical examples validate the accuracy and stability of the approach.

The approach works in both two and three space dimensions.

Abstract

This paper is concerned with the mean curvature flow, which describes the dynamics of a hypersurface whose normal velocity is determined by local mean curvature. We present a Cartesian grid-based method for solving mean curvature flows in two and three space dimensions. The present method embeds a closed hypersurface into a fixed Cartesian grid and decomposes it into multiple overlapping subsets. For each subset, extra tangential velocities are introduced such that marker points on the hypersurface only moves along grid lines. By utilizing an alternating direction implicit (ADI)-type time integration method, the subsets are evolved alternately by solving scalar parabolic partial differential equations on planar domains. The method removes the stiffness using a semi-implicit scheme and has no high-order stability constraint on time step size. Numerical examples in two and three space dimensions are presented to validate the proposed method.