Existence and non-existence of semi-discrete shocks for a car-following model in traffic flow

Amin Ghorbel (1) , Régis Monneau (2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

We consider a (microscopic) car-following model in traffic flow that can be seen as a semi-discrete scheme (discretization in space only) of a (macroscopic) Hamilton-Jacobi equation. For this discrete model, and for general velocity laws satisfying a strict chord inequality, we construct travelling solutions that are naturally associated to "travelling shocks" for the conservation law derived from the Hamilton-Jacobi equation. These shocks can be interpreted as a phase transition between two states of different car densities. There is no smallness condition on the size of these shocks. This existence and uniqueness of the solution is done at the level of the Hamilton-Jacobi equation. A surprising non-existence result of semi-discrete shocks for this microscopic model is also presented in the case where a shock exists for the associated macroscopic model, but the velocity law V satisfies a non strict chord inequality.

Mots clés: car-following model; Hamilton-Jacobi equation; semi-discrete shock; traffic flow; viscosity solutions

Domaines: Equations aux dérivées partielles [math.AP]

Amin Ghorbel, Régis Monneau. Existence and non-existence of semi-discrete shocks for a car-following model in traffic flow. 2012. ⟨hal-00760671⟩