First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow

Abstract

This article deals with a review and critical analysis of first order hydrodynamic models of vehicular traffic flow obtained by the closure of the mass conservation equation. The closure is obtained by phenomenological models suitable to relate the local mean velocity to local density profiles. Various models are described and critically analyzed in the deterministic and stochastic case. The analysis is developed in view of applications of the models to traffic flow simulations for networks of roads. Some research perspectives are derived from the above analysis and proposed in the last part of the paper.

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

“…Numerous mathematical models have been proposed for one-directional flows of vehicular traffic; reviews of this topic are given in the monographs by Helbing [62], Kerner [63] and Garavello and Piccoli [64], as well as in the articles by Bellomo et al [65,66,67]. These and other works vividly illustrate that the number of balance equations (for the car density, velocity, and possibly other flow variables) that form a time-dependent model based on partial differential equations, as well as the algebraic structure of these equations, is a topic of current research.…”

On the coupling of steady and adaptive velocity grids in vehicular traffic modelling

“…The selection of the velocity-density closure relation, namely the functional dependence of the equilibrium velocity v e = v e (ρ) on the local density ρ, plays a key role in the macroscopic modelling of traffic flow [13,14], but is, nonetheless, subject to experimental investigation [12]. Generally, most authors working on traffic modelling plug suitable relations of the type v e (ρ) directly into their models; see the review [15].…”

Conservation laws on complex networks

“…An extension to multipopulation can be found; see [7]. We refer the reader to [6,24,29] for a general presentation of the various models.…”