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Volume 39, Issue 2
A Structure-Preserving Numerical Method for the Fourth-Order Geometric Evolution Equations for Planar Curves

Eiji Miyazaki, Tomoya Kemmochi, Tomohiro Sogabe & Shao-Liang Zhang

DOI: 10.4208/cmr.2022-0040

Commun. Math. Res., 39 (2023), pp. 296-330.

Published online: 2023-04

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  • 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.

  • Keywords

Geometric evolution equation, Willmore flow, Helfrich flow, numerical calculation, structure-preserving, discrete variational derivative method, tangential velocity.

  • AMS Subject Headings

37M15, 65M06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-39-296, author = {Miyazaki , EijiKemmochi , TomoyaSogabe , Tomohiro and Zhang , Shao-Liang}, title = {A Structure-Preserving Numerical Method for the Fourth-Order Geometric Evolution Equations for Planar Curves}, journal = {Communications in Mathematical Research }, year = {2023}, volume = {39}, number = {2}, pages = {296--330}, 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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2022-0040}, url = {http://global-sci.org/intro/article_detail/cmr/21549.html} }
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 -

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.

Miyazaki , EijiKemmochi , TomoyaSogabe , Tomohiro and Zhang , Shao-Liang. (2023). A Structure-Preserving Numerical Method for the Fourth-Order Geometric Evolution Equations for Planar Curves. Communications in Mathematical Research . 39 (2). 296-330. doi:10.4208/cmr.2022-0040
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