The Riemann Problem at a Junction for a Phase Transition Traffic Model

Abstract

We extend the Phase Transition model for traffic proposed in [8], by Colombo, Marcellini, and Rascle to the network case. More precisely, we consider the Riemann problem for such a system at a general junction with n incoming and m outgoing roads. We propose a Riemann solver at the junction which conserves both the number of cars and the maximal speed of each vehicle, which is a key feature of the Phase Transition model. For special junctions, we prove that the Riemann solver is well defined.

A Riemann solver at a junction compatible with a homogenization limit

“…see Figure 1. As in [6,9], we assume the following hypotheses. Figure 1: The free phase F and the congested phase C resulting from (2.1) in the coordinates, from left to right, (ρ, η) and (ρ, ρv).…”

“…With a limit-average procedure, we are able to find the relation between the incoming maximal speeds and the outgoing one. In this way, the maximal outgoing speed turns out to be a convex combination of the n incoming ones and it satisfies the corresponding condition prescribed by the Riemann solver in [9].…”

“…This paper deals with the Phase Transition traffic model, proposed by Colombo, Marcellini, and Rascle in [6], at a junction with n ≥ 2 incoming roads and only one outgoing road. The aim is to give a mathematical derivation for the solution of the Riemann problem at the crossroad proposed in [9]. We obtain the justification for such solution by using a homogenization procedure.…”

“…The extension of the Phase Transition model to the case of networks is considered in [9]. The key point for extending a model to a network consists in providing a concept of solution at nodes.…”

“…A reasonable Riemann solver has to satisfy the mass conservation, a consistency condition; it should produce waves with negative speed in the incoming edges and with positive speed in the outgoing ones. A Riemann solver satisfying such properties is proposed in [9]. In particular, it prescribes that the maximal speed in the outgoing road is a convex combination of the maximal speed in the incoming arcs.…”