A traffic-flow model with constraints for the modeling of traffic jams

Recently, Berthelin et al.,(4) introduced a traffic-flow model describing the formation and the dynamics of traffic jams. This model which consists of a Constrained Pressureless Gas Dynamics system assumes that the maximal density constraint is independent of the velocity. However, in practice, the distribution of vehicles on a highway depends on their velocity. In this paper, a more realistic model namely the Second Order Model with Constraints (in short SOMC) is proposed, derived from the Aw and Rascle model,(1) which takes into account this feature. Moreover, when the maximal density constraint is saturated, the SOMC model "relaxes" to the Lighthill and Whitham model.(20) An existence result of weak solutions for this model by means of cluster dynamics is proved in order to construct a sequence of approximations, and the associated Riemann problem is solved completely.