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, more recently, by Kohn and Serfaty. Finally, several numerical test problems in 2D and 3D are presented.
Carlini, E., Falcone, M., Ferretti, R. (2010). Convergence of a large time-step scheme for mean curvature motion. INTERFACES AND FREE BOUNDARIES, 12, 409-441 [10.4171/IFB/240].
Convergence of a large time-step scheme for mean curvature motion
FERRETTI, Roberto
2010-01-01
Abstract
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, more recently, by Kohn and Serfaty. Finally, several numerical test problems in 2D and 3D are presented.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.