On a degenerate second order traffic model: Existence of discrete evolutions, deterministic many-particle limit and first order approximation

We propose and analyze a new microscopic second-order follow-the-leader-type scheme to describe traffic flows. The main novelty of this model consists of multiplying the second-order term by a nonlinear function of the global density, with the intent of considering the attentiveness of the drivers depending on the amount of congestion. Such term makes the system highly degenerate; indeed, coherently with the modellistic viewpoint, we allow for the nonlinearity to vanish as soon as consecutive vehicles are very close to each other. We first show existence of solutions to the degenerate discrete system. We then perform a rigorous discrete-to-continuum limit, as the number of vehicles grows larger and larger, by making use of suitable piece-wise constant approximations of the relevant macroscopic variables. The resulting continuum system turns out to be described by a degenerate pressure-less Euler-type equation, and we discuss how this could be considered an alternative to the groundbreaking Aw-Rascle-Zhang traffic model. Finally, we study the singular limit to first-order dynamics in the spirit of a vanishing-inertia argument. This eventually validates the use of first-order macroscopic models with nonlinear mobility to describe a congested traffic stream.