Convergent scheme for a non-local transport system modeling dislocations dynamics

Abstract

In this paper, we consider a diagonal hyperbolic, not necessarily strictly hyperbolic, nonlinear and non-local system of transport equations that arises in the theory of dislocations densities dynamics. Under certain monotony assumptions on the initial data, global existence and uniqueness results of Hloc1 solutions for this system was proved in Hajj (2007). We propose a natural implicit scheme verifying a gradient L2 estimate at the discrete level compatible with that obtained in the continuous case in Hajj (2007). Which leads us to prove the convergence of the numerical solution to the continuous one. To our knowledge, this is the first convergent scheme proposed for this model.