An $n$ -populations model for traffic flow

Abstract

We present an n-population generalization of the Lighthill-Whitham and Richards traffic flow model. This model is analytically interesting because of several non-standard features. For instance, it leads to non-classical shocks and enjoys an unexpected stability in spite of the presence of umbilic points. Furthermore, while satisfying all the minimal 'common sense' requirements, it also allows for a description of phenomena often neglected by other models, such as overtaking.or more complicated relations, for instance making the distinction between light and congested traffic.One might question, a priori, the validity of (1.6). In general, density-speed relations may well be different from one class to the other. In a merely numerical study, the special form (1.6) of speed laws can obviously be abandoned. However, the system (1.3)-(1.4) is too general for any analytical study because even its hyperbolicity is unclear. To reach significant analytical results, we assume (1.6) throughout the paper.Concerning the specific choice of ψ, (1.7) has the advantage of making the flux of the system (1.3)-(1.6) quadratic. As we shall demonstrate below, this special and nevertheless meaningful feature leaves the possibility of a much richer mathematical analysis of (1.3)-(1.6). From the modelling point of view, this very specific choice is questionable, but it will not lead to any qualitative inconsistency.Anyway, we shall also point out properties of (1.3)-(1.6) that are valid for any ψ. For a discussion of numerical results with the other choice (1.8), we refer to Wong & Wong [30].Interestingly, both from the applied point of view and the mathematical point of view, we shall show that (1.3)-(1.6) is endowed with a far from standard structure. In particular, due to the presence of an umbilic point, system (1.3)-(1.6) does not fit into any of the recent theoretical results [4,8,26]. Besides, (1.3) comprehends an n × n analog of the Keyfitz-Kranzer system [17], and leads to the presence of over-compressive shocks as well.In spite of that, as the study below and several numerical experiments suggest, we expect it to be well-posed.The next section deals with the general n-population model, while § 3 specializes to the case n = 2. Some technical details are deferred to the final Appendix.

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