A structure-preserving numerical method for the fourth-order geometric evolution equations for planar curves

TL;DR

This paper introduces a structure-preserving numerical method for fourth-order geometric evolution equations of planar curves, effectively maintaining key properties and preventing vertex concentration through a modified tangential velocity approach.

Contribution

The study develops a novel discrete variational derivative method with a modified tangential velocity to ensure stability and property preservation in simulating geometric flows.

Findings

Accurately captures properties of Willmore and Helfrich flows

Prevents vertex concentration in numerical simulations

Demonstrates high accuracy in numerical experiments

Abstract

For fourth-order geometric evolution equations for planar curves with the dissipation of the bending energy, including the Willmore and the Helfrich flows, we consider a numerical approach. In this study, we construct a structure-preserving method based on a discrete variational derivative method. Furthermore, to prevent the vertex concentration that may lead to numerical instability, we discretely introduce Deckelnick's tangential velocity. Here, a modification term is introduced in the process of adding tangential velocity. This modified term enables the method to reproduce the equations' properties while preventing vertex concentration. Numerical experiments demonstrate that the proposed approach captures the equations' properties with high accuracy and avoids the concentration of vertices.