# The BGK approximation of kinetic models for traffic

## TL;DR

This paper introduces a modified BGK kinetic model for traffic flow that achieves conditional stability and captures stop-and-go waves, bridging microscopic and macroscopic traffic models.

## Contribution

A new BGK-type kinetic model is derived that stabilizes solutions in congested traffic and reproduces realistic wave phenomena, enhancing traffic flow modeling.

## Key findings

- The modified model reproduces stop-and-go waves as backward propagating signals.
- It provides a mesoscopic description linking microscopic and macroscopic models.
- The model achieves conditional stability in congested traffic regimes.

## Abstract

We study spatially non-homogeneous kinetic models for vehicular traffic flow. Classical formulations, as for instance the BGK equation, lead to unconditionally unstable solutions in the congested regime of traffic. We address this issue by deriving a modified formulation of the BGK-type equation. The new kinetic model allows to reproduce conditionally stable non-equilibrium phenomena in traffic flow. In particular, stop and go waves appear as bounded backward propagating signals occurring in bounded regimes of the density where the model is unstable. The BGK-type model introduced here also offers the mesoscopic description between the microscopic follow-the-leader model and the macroscopic Aw-Rascle and Zhang model.