Well-posedness and numerical analysis of a one-dimensional non-local transport equation modelling dislocations dynamics

Amin Ghorbel (1) , Régis Monneau (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

We consider a situation where dislocations are parallel lines moving in a single plane. For this simple geometry, dislocations dynamics is modelized by a one-dimensional non-local transport equation. We prove a result of existence and uniqueness for all time of the continuous viscosity solution for this equation. A finite difference scheme is proposed to approximate the continuous viscosity solution. We also prove an error estimate result between the continuous solution and the discrete solution and we provide some simulations.

Mots clés: convergence of numerical scheme; Dislocations dynamics; Peach-Koehler force; transport equation; eikonal equation; Hamilton-Jacobi equation; non-local equation; continuous viscosity solutions; convergence of numerical scheme.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Amin Ghorbel, Régis Monneau. Well-posedness and numerical analysis of a one-dimensional non-local transport equation modelling dislocations dynamics. Mathematics of Computation, 2010, 79 (271), pp.1535-1564. ⟨10.1090/S0025-5718-10-02326-4⟩. ⟨hal-00092875⟩