Numerical specified homogenization of a discrete model with a local perturbation and application to traffic flow
Keywords:numerical specified homogenization, Hamilton-Jacobi equations, viscosity solutions, traffic flow, microscopic models, macroscopic models, convergence of numerical scheme
AbstractWe present techniques for the analysis and numerical analysis of non-local non-linear PDEs. We apply these techniques to an equation derived from the modelling of traffic flow. We introduce a macroscopic model in the form of a Hamilton--Jacobi equation with a junction condition. More precisely, the goal of this work is to obtain the 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. Previously we showed 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 article is to provide a numerical scheme able to obtain an approximation of this flux-limiter. We prove the convergence of this scheme and we give some numerical results.
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