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

TLDR

Macroscopic traffic models have recently been severely criticized for relying on lax analogies and exhibiting several deficiencies. This paper constructs a logically consistent fluid‑dynamic traffic model from basic laws governing vehicle acceleration and interaction. By employing Paveri‑Fontana’s gas‑kinetic traffic equation, the authors derive macroscopic moment equations and close the resulting non‑closed system with the Chapman–Enskog method, yielding Euler‑like and Navier‑Stokes‑like traffic equations. The resulting Navier‑Stokes‑like equations reproduce equilibrium relations such as the fundamental diagram and variance‑density relation, are corrected for vehicle space, and successfully address the earlier criticisms of macroscopic traffic models. © 1996 The American Physical Society.

Abstract

Macroscopic traffic models have recently been severely criticized as based on lax analogies only and having 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 that are partly difficult to determine empirically. Paveri-Fontana's traffic equation allows the derivation of macroscopic moment equations that build a system of nonclosed 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. \textcopyright{} 1996 The American Physical Society.