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

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.