TY - JOUR
T1 - A Structure-Preserving Numerical Method for the Fourth-Order Geometric Evolution Equations for Planar Curves
AU - Miyazaki , Eiji
AU - Kemmochi , Tomoya
AU - Sogabe , Tomohiro
AU - Zhang , Shao-Liang
JO - Communications in Mathematical Research 
VL - 2
SP - 296
EP - 330
PY - 2023
DA - 2023/04
SN - 39
DO - http://doi.org/10.4208/cmr.2022-0040
UR - https://global-sci.org/intro/article_detail/cmr/21549.html
KW - Geometric evolution equation, Willmore flow, Helfrich flow, numerical calculation, structure-preserving, discrete variational derivative method, tangential velocity.
AB - <p style="text-align: justify;">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.</p>