We consider the model problem where a curve in R^3 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 e-sublevel of this evolution contains the curve, the next step is to extract the curve itself by following an optimal trajectory inside each e-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.

Carlini, E., Falcone, M., Ferretti, R. (2007). A semi-Lagrangian scheme for the curve shortening flow in codimension 2. JOURNAL OF COMPUTATIONAL PHYSICS, 225, 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 R^3 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 e-sublevel of this evolution contains the curve, the next step is to extract the curve itself by following an optimal trajectory inside each e-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.
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2007
Carlini, E., Falcone, M., Ferretti, R. (2007). A semi-Lagrangian scheme for the curve shortening flow in codimension 2. JOURNAL OF COMPUTATIONAL PHYSICS, 225, 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/11590/142485`
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