Florent Berthelin

A second order traffic-flow model with constraint on the velocity for the Modeling of Traffic Jams

In this talk, we derive a second order model, called the Second Order Model with Constraints (SOMC),  from the Aw-Rascle model through a singular limit. We prove an existence result of weak solutions  for such a model and discuss the associated Riemann problem. In contrast with a previous model where we assumed that the maximal density is constant (therefore independent of the velocity), here, we take into account the dependence of the maximal density constraint on the velocity. This consideration leads to a more realistic formulation, since it is well known that in practice, the distribution of vehicles on a highway, depends on their velocity. Furthermore, the particularity of the model we propose here, is its double-sided behaviour. Indeed, when the density constraint is saturated i.e., the maximal density is attained, for a given velocity, the SOMC model behaves like the Lighthill-Whitham first order model, whereas in the free flow our model behaves like the pressureless gas model. Moreover, even in the Riemann problem, the interaction between two constant states in either regime can produce new states in the other regime: in other words the two regimes are intimately coupled and thus cannot ignore each other.