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

We consider a situation where dislocations are parallel lines moving in a single plane. For this simple geometry, dislocations dynamics is modeled 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.