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 makesit not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.
A convergent scheme for a non local Hamilton-Jacobi equation modelling dislocation dynamic / O., Alvarez; Carlini, Elisabetta; R., Monneau; E., Rouy. - In: NUMERISCHE MATHEMATIK. - ISSN 0029-599X. - 104:(2006), pp. 413-444. [10.1007/s00211-006-0030-5]
A convergent scheme for a non local Hamilton-Jacobi equation modelling dislocation dynamic
CARLINI, Elisabetta;
2006
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 makesit not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.