A fully discrete curve-shortening polygonal evolution law for moving boundary problems

TL;DR

This paper introduces a fully discrete, implicit polygonal evolution law for numerically solving moving boundary problems like mean curvature flow, enabling larger time steps and accurate property preservation.

Contribution

It develops a novel fully implicit scheme for curve-shortening problems with a new implicit definition of tangent and normal vectors at vertices.

Findings

Allows larger time steps in simulations.

Accurately preserves area-related properties.

Effective for mean curvature and Hele-Shaw flows.

Abstract

We consider the numerical integration of moving boundary problems with the curve-shortening property, such as the mean curvature flow and Hele-Shaw flow. We propose a fully discrete curve-shortening polygonal evolution law. The proposed evolution law is fully implicit, and the key to the derivation is to devise the definitions of tangent and normal vectors and tangential and normal velocities at each vertex in an implicit manner. Numerical experiments show that the proposed method allows the use of relatively large time step sizes and also captures the area-preserving or dissipative property in good accuracy.