On Approximations of the Curve Shortening Flow and of the Mean Curvature Flow based on the DeTurck trick

TL;DR

This paper introduces novel numerical schemes for curve shortening and mean curvature flows using reparametrizations inspired by the DeTurck trick, leading to improved mesh properties and error estimates.

Contribution

It develops new reparametrized numerical schemes based on harmonic map heat flow for curvature flows, revealing geometric connections and enhancing mesh quality.

Findings

Error estimates for semi-discrete schemes of curve shortening flow

Numerical experiments on mesh redistribution behavior

Families of schemes with improved mesh properties

Abstract

In this paper we discuss novel numerical schemes for the computation of the curve shortening and mean curvature flows that are based on special reparametrizations. The main idea is to use special solutions to the harmonic map heat flow in order to reparametrize the equations of motion. This idea is widely known from the Ricci flow as the DeTurck trick. By introducing a variable time scale for the harmonic map heat flow, we obtain families of numerical schemes for the reparametrized flows. For the curve shortening flow this family unveils a surprising geometric connection between the numerical schemes in [5] and [9]. For the mean curvature flow we obtain families of schemes with good mesh properties similar to those in [3]. We prove error estimates for the semi-discrete scheme of the curve shortening flow. The behaviour of the fully-discrete schemes with respect to the redistribution of mesh points is studied in numerical experiments. We also discuss possible generalizations of our ideas to other extrinsic flows.