Existence and Nonexistence of Semidiscrete Shocks for a Car-Following Model in Traffic Flow

Abstract

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 solutions that are naturally associated to travelling 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.