A convergent scheme for a non local Hamilton Jacobi equation modelling dislocation dynamics

Authors:
O. Alvarez;E. Carlini;R. Monneau;E. Rouy
Affiliations:
UMR 60-85, Université de Rouen, 76821, Mont-Saint Aignan Cedex, France;Dipartimento di Matematica, Università di Roma “La Sapienza”, P. Aldo Moro 2, 00185, Rome, Italy;CERMICS, ENPC 6 et 8 avenue Blaise Pascal, Citè Descartes, Champs sur Marne, 77455, Marne la Vallèe Cedex 2, France;Departement de Mathematiques, Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69134, Ecully Cedex, France
Venue:
Numerische Mathematik
Year:
2006

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Abstract

We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makes it not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.