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Convergence of a large time-step scheme for Mean Curvature Motion

We analyse the properties of a semi-Lagrangian scheme for the approximation of the Mean Curvature Motion (MCM). This approximation is obtained coupling a stochastic method for the approximation of characteristics (to be understood in a generalized sense) with a local interpolation. The main feature of the scheme is that it can handle degeneracies, it is explicit and 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 the Barles-Souganidis theorem. The scheme is compared with similar existing schemes proposed by Crandall and Lions and, moree recently, by Kohn and Serfaty. Finally, several numerical test problems in 2D and 3D are presented.

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