A Numerical Study for the Homogenization of One-Dimensional Models describing the Motion of Dislocations

In this paper we are interested in the collective motion of dislocations defects in crystals. Mathematically we study the homogenization of a non-local Hamilton-Jacobi equation. We prove some qualitative properties on the effective hamiltonian. We also provide a numerical scheme which is proved to be monotone under some suitable CFL conditions. Using this scheme, we compute numerically the effective hamiltonian. Furthermore we also provide numerical computations of the effective hamiltonian for several models corresponding to the dynamics of dislocations where no theoretical analysis is available