Numerical specified homogenization of a discrete model with a local perturbation and application to traffic flow

The goal of this work is to present a numerical homogenization of a non-local PDE deriving from a first order discrete model for traffic flow that simulates the presence of a local perturbation. In a previous work, we have shown that the solution of the discrete microscopic model converges to the (unique) solution of a Hamilton-Jacobi equation posed on a network and with a junction condition (it can be seen as a flux limiter that keeps the memory of the local perturbation). The goal of this paper is to provide a numerical scheme able to provide an approximation of this flux-limiter. We will show the convergence of this scheme and we will provide some numerical results