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We study the relation between the average traffic flow and the vehicle density on road networks that we call 2D-traffic fundamental diagram. We show that this diagram presents mainly four phases. We analyze different cases. First, the case…

Optimization and Control · Mathematics 2013-02-06 Nadir Farhi , Maurice Goursat , Jean-Pierre Quadrat

Most car-following models show a transition from laminar to ``congested'' flow and vice versa. Deterministic models often have a density range where a disturbance needs a sufficiently large critical amplitude to move the flow from the…

Statistical Mechanics · Physics 2007-05-23 Kai Nagel , Christopher Kayatz , Peter Wagner

In this paper we study a phase transition model for vehicular traffic flows. Two phases are taken into account, according to whether the traffic is light or heavy. We assume that the two phases have a non-empty intersection, the so called…

Analysis of PDEs · Mathematics 2018-04-23 Mohamed Benyahia , Carlotta Donadello , Nikodem Dymski , Massimiliano D. Rosini

Kinetic models for vehicular traffic are reviewed and considered from the point of view of deriving macroscopic equations. A derivation of the associated macroscopic traffic flow equations leads to different types of equations: in certain…

Numerical Analysis · Mathematics 2015-03-13 Raul Borsche , Axel Klar , Mark Kimathi

This paper deals with the Aw-Rascle-Zhang model for traffic flow on uni-directional road networks. For the conservation of the mass and the generalized momentum, we construct weak solutions for Riemann problems at the junctions. We…

Numerical Analysis · Mathematics 2020-05-26 Simone Göttlich , Michael Herty , Salissou Moutari , Jennifer Weißen

Based on experimental traffic data obtained from German and US highways, we propose a novel two-dimensional first-order macroscopic traffic flow model. The goal is to reproduce a detailed description of traffic dynamics for the real road…

Physics and Society · Physics 2017-11-10 Michael Herty , Adrian Fazekas , Giuseppe Visconti

We propose a model describing the traffic flow on a road with variable widths in this paper. The model, which is modified the Aw-Rascle model, is not conservative because of the source term. We obtain the elementary waves of the new traffic…

Analysis of PDEs · Mathematics 2021-08-16 Wancheng Sheng , Qinglong Zhang

The Biham-Middleton-Levine traffic model is perhaps the simplest system exhibiting phase transitions and self-organization. Moreover, it is an underpinning to extensive modern studies of traffic flow. The general belief is that the system…

Statistical Mechanics · Physics 2007-05-23 R. M. D'Souza

We study the derivation of generic high order macroscopic traffic models from a follow-the-leader particle description via a kinetic approach. First, we recover a third order traffic model as the hydrodynamic limit of an Enskog-type kinetic…

Physics and Society · Physics 2021-04-09 Felisia Angela Chiarello , Benedetto Piccoli , Andrea Tosin

Recently we proposed an extension to the traffic model of Aw, Rascle and Greenberg. The extended traffic model can be written as a hyperbolic system of balance laws and numerically reproduces the reverse $\lambda$ shape of the fundamental…

Physics and Society · Physics 2009-11-11 Florian Siebel , Wolfram Mauser

We propose and analyse a new microscopic second order Follow-the-Leader type scheme to describe traffic flows. The main novelty of this model consists in multiplying the second order term by a nonlinear function of the global density, with…

Analysis of PDEs · Mathematics 2025-03-04 Dario Mazzoleni , Emanuela Radici , Filippo Riva

Traffic waves can rise even from single lane car-following behaviour. To better understand and mitigate traffic waves, it is necessary to use analytical tools like mathematical models, data analysis, and micro-simulations that can capture…

Physics and Society · Physics 2023-10-10 Nour Khoudari , Rabie Ramadan , Megan Ross , Benjamin Seibold

This article deals with macroscopic traffic flow models on a road network. More precisely, we consider coupling conditions at junctions for the Aw-Rascle-Zhang second order model consisting of a hyperbolic system of two conservation laws.…

Numerical Analysis · Mathematics 2018-04-23 Oliver Kolb , Guillaume Costeseque , Paola Goatin , Simone Göttlich

In the present paper a review and numerical comparison of a special class of multi-phase traffic theories based on microscopic, kinetic and macroscopic traffic models is given. Macroscopic traffic equations with multi-valued fundamental…

Numerical Analysis · Mathematics 2015-03-20 Raul Borsche , Mark Kimathi , Axel Klar

In this paper we present a theoretical analysis of a recently proposed two-dimensional Cellular Automata model for traffic flow in cities with the novel ingredient of turning capability. Numerical simulations of this model show that there…

comp-gas · Physics 2009-10-22 J. M. Molera , F. C. Martinez , J. A. Cuesta , R. Brito

Statistical mechanics of a small system of cars on a single-lane road is developed. The system is not characterized by a Hamiltonian but by a conditional probability of a velocity of a car for the given velocity and distance of the car…

Physics and Society · Physics 2009-11-13 Anton Šurda

Analytical investigations are made on BML two-dimensional traffic flow model with alternative movement and exclude-volume effect. Several exact results are obtained, including the upper critical density above which there are only jamming…

Statistical Mechanics · Physics 2007-05-23 Y. Shi

Traffic flow at low densities (free traffic) is characterized by a quasi-one-dimensional relation between traffic flow and vehicle density, while no such fundamental diagram exists for `synchronized' congested traffic flow. Instead, a…

Statistical Mechanics · Physics 2009-11-07 Dirk Helbing , Davide Batic , Martin Schoenhof , Martin Treiber

A simple algorithm for constructing an effective traffic model is presented. The algorithm uses statistically well-defined quantities extracted from the flow-density plot, and the resulting effective model naturally captures and predicts…

Adaptation and Self-Organizing Systems · Physics 2014-12-16 Bo Yang , Christopher Monterola

In the Biham-Middleton-Levine traffic model cars are placed with some density p on a two dimensional torus, and move according to a (simple) set of predefined rules. Computer simulations show this system exhibits many interesting phenomena:…

Probability · Mathematics 2007-09-11 Itai Benjamini , Ori Gurel-Gurevich , Roey Izkovsky