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On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick
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Elliott, Charles M. and Fritz, Hans (2017) On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick. IMA Journal of Numerical Analysis, 37 (2). pp. 543-603. doi:10.1093/imanum/drw020
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Official URL: http://dx.doi.org/10.1093/imanum/drw020
Abstract
In this paper we discuss novel numerical schemes for the computation of the curve shortening and mean curvature flows that are based on special reparametrizations. The main idea is to use special solutions to the harmonic map heat flow in order to reparametrize the equations of motion. This idea is widely known from the Ricci flow as the DeTurck trick. By introducing a variable time scale for the harmonic map heat flow, we obtain families of numerical schemes for the reparametrized flows. For the curve shortening flow this family unveils a surprising geometric connection between the numerical schemes in [5] and [9]. For the mean curvature flow we obtain families of schemes with good mesh properties similar to those in [3]. We prove error estimates for the semi-discrete scheme of the curve shortening flow. The behaviour of the fully-discrete schemes with respect to the redistribution of mesh points is studied in numerical experiments. We also discuss possible generalizations of our ideas to other extrinsic flows.
| Item Type: | Journal Article | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||||
| Divisions: | Faculty of Science > Mathematics | ||||||||||
| Library of Congress Subject Headings (LCSH): | Ricci flow, Equations of motion, Harmonic maps | ||||||||||
| Journal or Publication Title: | IMA Journal of Numerical Analysis | ||||||||||
| Publisher: | Oxford University Press | ||||||||||
| ISSN: | 0272-4979 | ||||||||||
| Official Date: | 1 April 2017 | ||||||||||
| Dates: |
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| Volume: | 37 | ||||||||||
| Number: | 2 | ||||||||||
| Page Range: | pp. 543-603 | ||||||||||
| DOI: | 10.1093/imanum/drw020 | ||||||||||
| Status: | Peer Reviewed | ||||||||||
| Publication Status: | Published | ||||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||||
| Funder: | Alexander von Humboldt-Stiftung (AvHS) |
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