Specified homogenization of a discrete traffic model leading to an effective junction condition

Nicolas Forcadel; Wilfredo Salazar; Mamdouh Zaydan

doi:10.3934/cpaa.2018104

Article Contents

2018, Volume 17, Issue 5: 2173-2206. Doi: 10.3934/cpaa.2018104

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Specified homogenization of a discrete traffic model leading to an effective junction condition

Normandie Univ, INSA de Rouen, LMI (EA 3226 - FR CNRS 3335), 76000 Rouen, France, 685 Avenue de l'Université, 76801 St Etienne du Rouvray cedex

* Corresponding author
* Corresponding author

Received: May 31, 2017

Revised: December 31, 2017

Published: April 2018

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Abstract

In this paper, we focus on deriving traffic flow macroscopic models from microscopic models containing a local perturbation such as a traffic light. At the microscopic scale, we consider a first order model of the form "follow the leader" i.e. the velocity of each vehicle depends on the distance to the vehicle in front of it. We consider a local perturbation located at the origin that slows down the vehicles. At the macroscopic scale, we obtain an explicit Hamilton-Jacobi equation left and right of the origin and a junction condition at the origin (in the sense of [25]) which keeps the memory of the local perturbation. As it turns out, the macroscopic model is equivalent to a LWR model, with a flux limiting condition at the junction. Finally, we also present qualitative properties concerning the flux limiter at the junction.

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Mathematics Subject Classification: 35D40, 90B20, 35B27, 35F20, 45K05.

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Figure 1. Schematic representation of the microscopic model

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Figure 2. Schematic representation of the macroscopic model

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Figure 3. Schematic representation of the optimal velocity function $V$

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Figure 4. Schematic representation of $\bar{H}$

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Figure 5. Schematic representation of the function $\rho$

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Figure 6. Schematic representation of the function $\rho$

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