Navier-Stokes-like equations for traffic flow

Abstract

The macroscopic traffic flow equations derived from the reduced Paveri-Fontana equation are closed starting with the maximization of the informational entropy. The homogeneous steady state taken as a reference is obtained for a specific model of the desired velocity and a kind of Chapman-Enskog method is developed to calculate the traffic pressure at the Navier-Stokes level. Numerical solution of the macroscopic traffic equations is obtained and its characteristics are analyzed.

Macroscopic Models in Traffic Flow

“…Also the macroscopic models can be proposed based on qualitative arguments, which take into account the general behavior of a compressible flow in a channel and apply some analogies to the motion of vehicles along a road [5,18,26,33]. There are numerous macroscopic models, some of them coming from a phenomenological approach [13] and also coming from a kinetic approach [9,22,25,30,32]. The common factor of macroscopic models being the description of the behavior of averaged quantities such as the density of vehicles, their average velocity and the velocity variance mainly.…”

The Onset of Traffic Phases in Highways: A Two Vehicle-Class Macroscopic Model

“…Other proposals are presented in [28,29]. In this manuscript we deal with the construction of a multiclass kinetic based macroscopic model assuming the agressive drivers model presented in [19]. This kind of models centers its attention in the space-time evolution of the one vehicle distribution function of i-class, f i (x, v i , t) where f i (x, v i , t)dxdv i gives us the number of vehicles of the i−class in position (x, x + dx) with speed (v i , v i + dv i ) at time t.…”

“…are parameters which measure a kind of collective relaxation time, they come from a BGK approximation method in the kinetic equation [31,19] and the quantities ǫ i , η i are determined by means of the compatibility conditions, Eqs. (12-13) and are given by…”

“…Some years latter the PaveriFontana kinetic equation was proposed in an extended phase space, where it is considered the drivers desired speed as an independent variable [15]. The ideas behind such kinetic equations were taken to develop macroscopic models as the proper averages over the distribution function [16,17,18,19,20,21,22]. Kinetic equations in traffic flow, as in any other subject, have mathematical characteristics which make them somewhat complicated.…”

“…The model is based on the Paveri-Fontana equation (PFE) which introduces a the new variable called desired speed, an average over the desired speed allow us to obtain a kinetic equation in a reduced phase space. As a consequence of the introduction of the aggressive drivers model [19,23,24], we can obtain the distribution function for the homogeneous steady state (equilibrium state) in an analytical way, and this is the basis for an approximate method to go far from such state. As a result, a first order deviation in the average speed gradient is obtained for the distribution function of each class, which is taken to obtain a closure relation.…”