Homogenization of a discrete model for a bifurcation and application to traffic flow

Abstract

The goal of this paper is to derive rigorously a macroscopic traffic flow model, for a simple bifurcation, from a microscopic model. At the microscopic scale, we consider a first order model of the form “follow the leader” i.e. the velocity of each vehicle depends on the distance to the vehicle in front of it. We consider the case of a very simple bifurcation in which one road separates into two and one vehicle over two goes to the right and the other goes to the left. At the bifurcation, we then have to add a phase of transition because the vehicle in front will change. Moreover, we assume that the velocity on each road can be different. At the macroscopic scale, we obtain an explicit Hamilton-Jacobi equation on each road and a junction condition (in the sense of [1]) located at the bifurcation. From this case of a simple bifurcation, we then extend to more general scenarios. For instance, the case of a different distribution of the vehicles at the bifurcation or even to consider more than two outgoing roads. For these extensions we only present the results and explain how to adapt the proofs from the case of a simple bifurcation. Up to our knowledge, this is the first rigorous result for micro-macro passage on a junction.