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 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.