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March  2018, 38(3): 1103-1125. doi: 10.3934/dcds.2018046

Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces

 1 Graduate School of Maritime Sciences, Kobe University, Higashinada, Kobe 658-0022, Japan 2 Yasuna Machine Designing, Hojo-Umehara, Himeji 670-0945, Japan

*Corresponding author

Received  December 2016 Revised  September 2017 Published  December 2017

Fund Project: The first author is supported by JSPS KAKENHI Grant Numbers JP17K05364 and JP252470080.

We consider a threshold-type algorithm for curvature-dependent motions (CDM for short) of hypersurfaces. This algorithm was numerically studied by Kimura - Notsu [13], Esedoḡlu - Ruuth - Tsai [7] and Mohammad - Švadlenka [16], where they used the signed distance function as the level set function for CDM. The convergence of this algorithm and its optimal rate have been considered in Ishii - Kimura [12]. In this paper we give different approaches to the optimal rate of convergence to the smooth and compact CDM from [12]. As for the optimality, we give a more precise estimate than that in [12].

Citation: Katsuyuki Ishii, Takahiro Izumi. Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1103-1125. doi: 10.3934/dcds.2018046
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References:
Graphs of $w_k$, $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{w}_k$ and $\bar{w}_k$ (thick curves)
Graphs of of $w^k$, $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{w}^k$ and $\bar{w}^k$ (thick curves)
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