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

Authors:
E. Carlini;M. Falcone;R. Ferretti
Affiliations:
Dipartimento di Matematica, Universití di Roma "La Sapienza", P.le Aldo Moro 2, I-00185 Roma, Italy;Dipartimento di Matematica, Universití di Roma "La Sapienza", P.le Aldo Moro 2, I-00185 Roma, Italy;Dipartimento di Matematica, Universití di Roma Tre, Largo S. Leonardo Murialdo 1, I-00146 Roma, Italy
Venue:
Journal of Computational Physics
Year:
2007

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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.