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Energy dissipative numerical schemes for gradient flows of planar curves

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Abstract

In this paper, we develop an energy dissipative numerical scheme for gradient flows of planar curves, such as the curvature flow and the elastic flow. Our study presents a general framework for solving such equations. To discretize the time variable, we use a similar approach to the discrete partial derivative method, which is a structure-preserving method for gradient flows of graphs. For the approximation of curves, we use B-spline curves. Owing to the smoothness of B-spline functions, we can directly address higher order derivatives. Moreover, since B-spline curves require few degrees of freedom, we can reduce the computational cost. In the last part of the paper, we present some numerical examples of the elastic flow, which exhibit topology-changing solutions and more complicated evolution. Videos illustrating our method are available on YouTube.

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Notes

  1. URL: https://www.youtube.com/playlist?list=PLMF3dSqWEii69loXvCtgDCI4PYijq_4L3.

  2. URL: https://www.youtube.com/playlist?list=PLMF3dSqWEii69loXvCtgDCI4PYijq_4L3.

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Acknowledgements

The author would like to thank Prof. Yoshihiro Tonegawa and Dr. Takahito Kashiwabara for bringing this topic to my attention and encouraging me through valuable discussions. Also, the author would like to thank the anonymous reviewer for valuable comments and suggestions to improve the quality of the paper. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan, and by JSPS KAKENHI Grant Number 15J07471, Japan.

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  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Tomoya Kemmochi

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  1. Tomoya Kemmochi
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Correspondence to Tomoya Kemmochi.

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Communicated by Jan Nordström.

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Kemmochi, T. Energy dissipative numerical schemes for gradient flows of planar curves. Bit Numer Math 57, 991–1017 (2017). https://doi.org/10.1007/s10543-017-0685-6

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  • DOI: https://doi.org/10.1007/s10543-017-0685-6

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