Nonlinear Waves in Mathematical Models of Traffic Flow

Michel Rascle
Université de Nice Sophia Antipolis

I will discuss the relations between discrete ODE models of traffic flow (e.g. Follow the Leader) and their PDE continuous version, if any. In particular, I will discuss the roles of both the “convective” and the relaxation parts of classical ODE models: e.g. Bando’s OVM (Optimal Velocity) is a pure relaxation (discrete) model, whereas the Aw-Rascle (AR) class of (fluid) models is purely convective.

I will mainly focus on two classes of mixed models, both in the context of a small, but nonzero relaxation time: the first is the class of "(ARG)" models, mainly developed by J. Greenberg, which adds a relaxation term to (AR), with a weak form of stability condition. (SC), satisfied both for small and large densities, but not for intermediate regimes. I will recall why this model allows for periodic solutions: Traveling Waves - Shocks and seems to be a nice compromise between too much stability, like (AR) and not enough, e.g. crashes, like (OVM).

I will then discuss recent efforts, in connection with J. Greenberg, to analyze with the same ideas the Intelligent Driver Model (IDM) of Helbing and Treiber, which at leading order can be revisited as a discretization of a variant of (ARG) model, though with significant differences as to higher order terms. I will discuss how apparently minor and reasonable modifications of these “convective” terms can lead to qualitative changes in the solutions.

Keywords: Traffic Flow, Nonlinear Waves

* Michel Rascle, Lab. J.A. Dieudonné, CNRS and U. of Nice, Parc Valrose, 06108 Nice Cedex, France Email:, Tel.:(33) (0)4 92 07 62 35

Presentation (PDF File)

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