We consider the model problem where a curve in R3 moves according to the mean curvature flow (the curve shortening flow). We construct a semi-Lagrangian scheme based on the Feynman-Kac representation formula of the solutions of the related level set geometric equation. The first step is to obtain an approximation of the associated codimension-1 problem formulated by Ambrosio and Soner, where the squared distance from the initial curve is used as initial condition. Since the ε-sublevel of this evolution contains the curve, the next step is to extract the curve itself by following an optimal trajectory inside each ε-sublevel. We show that this procedure is robust and accurate as long as the "fattening" phenomenon does not occur. Moreover, it can still single out the physically meaningful solution when it occurs.

A semi-Lagrangian scheme for the curve shortening flow in codimension-2 / Carlini, E.; Falcone, M.; Ferretti, R.. - In: JOURNAL OF COMPUTATIONAL PHYSICS. - ISSN 0021-9991. - 225:2(2007), pp. 1388-1408. [10.1016/j.jcp.2007.01.028]

### A semi-Lagrangian scheme for the curve shortening flow in codimension-2

#### Abstract

We consider the model problem where a curve in R3 moves according to the mean curvature flow (the curve shortening flow). We construct a semi-Lagrangian scheme based on the Feynman-Kac representation formula of the solutions of the related level set geometric equation. The first step is to obtain an approximation of the associated codimension-1 problem formulated by Ambrosio and Soner, where the squared distance from the initial curve is used as initial condition. Since the ε-sublevel of this evolution contains the curve, the next step is to extract the curve itself by following an optimal trajectory inside each ε-sublevel. We show that this procedure is robust and accurate as long as the "fattening" phenomenon does not occur. Moreover, it can still single out the physically meaningful solution when it occurs.
##### Scheda breve Scheda completa
2007
Curve shortening; mean curvature motion; semi-Lagrangian scheme
01 Pubblicazione su rivista::01a Articolo in rivista
A semi-Lagrangian scheme for the curve shortening flow in codimension-2 / Carlini, E.; Falcone, M.; Ferretti, R.. - In: JOURNAL OF COMPUTATIONAL PHYSICS. - ISSN 0021-9991. - 225:2(2007), pp. 1388-1408. [10.1016/j.jcp.2007.01.028]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11573/1668559`
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