Second Order Threshold Dynamics Schemes for Two Phase Motion by Mean Curvature

TL;DR

This paper introduces two second order accurate threshold dynamics schemes for two-phase motion by mean curvature, improving accuracy over existing first order methods and providing rigorous stability guarantees.

Contribution

It presents the first rigorous second order accurate algorithms for two-phase threshold dynamics, with one achieving dimension-independent consistency and the other offering unconditional energy stability.

Findings

First scheme consistent to second order in any dimension.

Second scheme achieves second order accuracy in 2D with unconditional stability.

Both algorithms improve upon the accuracy of existing first order methods.

Abstract

The threshold dynamics algorithm of Merriman, Bence, and Osher is only first order accurate in the two-phase setting. Its accuracy degrades further to half order in the multi-phase setting, a shortcoming it has in common with other related, more recent algorithms such as the equal surface tension version of the Voronoi implicit interface method. As a first, rigorous step in addressing this shortcoming, we present two different second order accurate versions of two-phase threshold dynamics. Unlike in previous efforts in this direction, we present careful consistency calculations for both of our algorithms. The first algorithm is consistent with its limit (motion by mean curvature) up to second order in any space dimension. The second achieves second order accuracy only in dimension two, but comes with a rigorous stability guarantee (unconditional energy stability) in any dimension -- a first for high order schemes of its type.