Journal of Computational Physics
Numerical Discretization of Boundary Conditions for First Order Hamilton--Jacobi Equations
SIAM Journal on Numerical Analysis
Computing the Effective Hamiltonian Using a Variational Approach
SIAM Journal on Control and Optimization
Convergence of a first order scheme for a non-local Eikonal equation
Applied Numerical Mathematics - Numerical methods for viscosity solutions and applications
An approximation scheme for the effective Hamiltonian and applications
Applied Numerical Mathematics - Numerical methods for viscosity solutions and applications
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In this paper we are interested in the collective motion of dislocations defects in crystals. Mathematically, we study the homogenisation of a non-local Hamilton-Jacobi equation. We prove some qualitative properties on the effective Hamiltonian and we 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 provide numerical computations of the effective Hamiltonian for several models corresponding to the dynamics of dislocations where no theoretical analysis is available.