Gas-kinetic derivation of Navier-Stokes-like traffic equations

TL;DR

This paper derives a consistent macroscopic traffic model from basic vehicle interaction laws, resulting in Navier-Stokes-like equations that improve upon previous models and address their criticisms.

Contribution

It constructs a gas-kinetic traffic equation from fundamental principles and derives Navier-Stokes-like traffic equations using Chapman-Enskog expansion.

Findings

Derivation of equilibrium relations for fundamental diagram and variance-density.

Development of a closed system of macroscopic traffic equations.

Model correction for finite vehicle space requirements.

Abstract

Macroscopic traffic models have recently been severely criticized to base on lax analogies only and to have a number of deficiencies. Therefore, this paper shows how to construct a logically consistent fluid-dynamic traffic model from basic laws for the acceleration and interaction of vehicles. These considerations lead to the gas-kinetic traffic equation of Paveri-Fontana. Its stationary and spatially homogeneous solution implies equilibrium relations for the `fundamental diagram', the variance-density relation, and other quantities which are partly difficult to determine empirically. Paveri-Fontana's traffic equation allows the derivation of macroscopic moment equations which build a system of non-closed equations. This system can be closed by the well proved method of Chapman and Enskog which leads to Euler-like traffic equations in zeroth-order approximation and to Navier-Stokes-like traffic equations in first-order approximation. The latter are finally corrected for the finite space requirements of vehicles. It is shown that the resulting model is able to withstand the above mentioned criticism.