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

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 problemformulated by Ambrosio and Soner, where the squared distance from the initial curve is used as initial condition. Since the epsilon-sublevel of this evolution contains the curve, the next step is to extract the curve itself by following an optimal trajectory inside each epsilon-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.