Convergence of a large time-step scheme for mean curvature motion

Abstract

We analyse the properties of a semi-Lagrangian scheme for the approximation of the Mean Curvature Motion (MCM). This approximation is obtained by coupling a stochastic method for the approximation of characteristics (to be understood in a generalized sense) with a local interpolation. The main features of the scheme are that it can handle degeneracies, it is explicit and it allows for large time steps. We also propose a modified version of this scheme, for which monotonicity and consistency can be proved. Then convergence to the viscosity solution of the MCM equation follows by an extension of the Barles–Souganidis theorem. The scheme is also compared with similar existing schemes proposed by Crandall and Lions and, more recently, by Kohn and Serfaty. Finally, several numerical test problems in 2D and 3D are presented.