Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations
Guillaume Costeseque Jean-Patrick Lebacque
Discrete & Continuous Dynamical Systems - S 2014, 7(3): 411-433 doi: 10.3934/dcdss.2014.7.411
In this paper, we consider a numerical scheme to solve first order Hamilton-Jacobi (HJ) equations posed on a junction. The main mathematical properties of the scheme are first recalled and then we give a traffic flow interpretation of the key elements. The scheme formulation is also adapted to compute the vehicles densities on a junction. The equivalent scheme for densities recovers the well-known Godunov scheme outside the junction point. We give two numerical illustrations for a merge and a diverge which are the two main types of traffic junctions. Some extensions to the junction model are finally discussed.
keywords: traffic numerical scheme Hamilton-Jacobi equations. Junction
Special issue on Mathematics of Traffic Flow Modeling, Estimation and Control
Alexandre M. Bayen Hélène Frankowska Jean-Patrick Lebacque Benedetto Piccoli H. Michael Zhang
Networks & Heterogeneous Media 2013, 8(3): i-ii doi: 10.3934/nhm.2013.8.3i
This Special Issue gathers contributions, most of which were presented at the Workshop ``Mathematics of Traffic Flow Modeling, Estimation and Control", organized at the Institute for Pure and Applied Mathematics of the University of California Los Angeles on December 7--9 2011.

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A dynamical two-dimensional traffic model in an anisotropic network
Tibye Saumtally Jean-Patrick Lebacque Habib Haj-Salem
Networks & Heterogeneous Media 2013, 8(3): 663-684 doi: 10.3934/nhm.2013.8.663
The aim of this paper is to build a dynamical traffic model in a dense urban area. The main contribution of this article is to take into account the four possible directions of traffic flows with flow vectors of dimension $4$ and not $2$ as in fluid mechanic on a plan. Traffic flows are viewed as confrontation results between users demands and a travel supply of the network. The model gathers elements of intersection theory and two-dimensional continuum networks.
keywords: partial differential equations dynamical model. users' demand network supply Traffic network

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