Stop-and-Go Waves in Car-Following Models

Waves, known as stop-and-go waves or phantom jams, can appear spontaneously in dense traffic. This causes a situation where drivers are faced with consecutive phases of acceleration and braking. Although waves are well understood in the setting of macroscopic models, the results for car-following models are not so numerous. Assuming string instability, this paper gives asymptotic estimates of the velocity and shape of these waves. It relies on the well-known saddle-point method in order to describe the trajectory of a vehicle caught in such a wave. Numerical experiments show that this method yields remarkably good estimates of the linearized model, even with a small number of vehicles, as well as a good estimate of the wave velocity.