Macroscopic limits of non-local kinetic descriptions of vehicular traffic

TL;DR

This paper derives macroscopic traffic models from microscopic vehicle dynamics using kinetic equations, showing how different particle interactions lead to known macroscopic models like Aw-Rasque-Zhang.

Contribution

It establishes a rigorous link between non-local particle models and macroscopic traffic equations, including the first and second order models, through hydrodynamic limits.

Findings

Optimal speed dynamics produce a non-local first order model.

Follow-the-leader dynamics lead to the Aw-Rasque-Zhang model.

Numerical simulations confirm the particle-macroscopic correspondence.

Abstract

We study the derivation of macroscopic traffic models out of optimal speed and follow-the-leader particle dynamics as hydrodynamic limits of non-local Povzner-type kinetic equations. As a first step, we show that optimal speed vehicle dynamics produce a first order macroscopic model with non-local flux. Next, we show that non-local follow-the-leader vehicle dynamics have a universal macroscopic counterpart in the second order Aw-Rascle-Zhang traffic model, at least when the non-locality of the interactions is sufficiently small. Finally, we show that the same qualitative result holds also for a general class of follow-the-leader dynamics based on the headway of the vehicles rather than on their speed. We also investigate the correspondence between the solutions to particle models and their macroscopic limits by means of numerical simulations.