A Structure-Preserving Numerical Method for the Fourth-Order Geometric Evolution Equations for Planar Curves (Communications in Mathematical Research, 2023)
For fourth-order geometric evolution equations for planar curveswith the dissipation of the bending energy, including the Willmore and theHelfrich flows, we consider a numerical approach. In this study, we constructa 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’ propertieswhile preventing vertex concentration. Numerical experiments demonstratethat the proposed approach captures the equations’ properties with high accuracy and avoids the concentration of vertices.
Keywords: Geometric evolution equation; Willmore flow; Helfrich flow; numerical calculation; structure-preserving; discrete variational derivative method; tangential velocity.