# American Institute of Mathematical Sciences

September  2013, 8(3): 663-684. doi: 10.3934/nhm.2013.8.663

## A dynamical two-dimensional traffic model in an anisotropic network

 1 Université Paris Est, GRETTIA, Ifsttar, 14-20 boulevard Newton, Cité Descartes Champs sur Marne, 77447 Marne la Vallée Cedex 2, France 2 Ifsttar, COSYS-GRETTIA, 14-20 boulevard Newton, Cité Descartes Champs sur Marne, 77447 Marne la Vallée Cedex 2, France

Received  August 2012 Revised  July 2013 Published  October 2013

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.
Citation: Tibye Saumtally, Jean-Patrick Lebacque, Habib Haj-Salem. A dynamical two-dimensional traffic model in an anisotropic network. Networks and Heterogeneous Media, 2013, 8 (3) : 663-684. doi: 10.3934/nhm.2013.8.663
##### References:
 [1] Jean-Bernard Baillon and Guillaume Carlier, From discrete to continuous Wardrop equilibria, Networks and Heterogeneous Media, 7 (2012), 219-241. doi: 10.3934/nhm.2012.7.219. [2] Giuseppe Maria Coclite and Benedetto Piccoli, "Traffic Flow on a Road Network," Technical Report, SISSA, 2002. [3] Giuseppe Maria Coclite, Mauro Garavello and Benedetto Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. [4] Nikolas Geroliminis and Carlos Daganzo, Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings, Transportation Research B, 42 (2008), 759-770. doi: 10.1016/j.trb.2008.02.002. [5] Helge Holden and Nils Henrik Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289. [6] Yangbeibei Ji, Winnie Daamen, Serge Hoogendoorn, Sascha Hoogendoorn-Lanser and Xiaoyu Qian, Investigating the shape of the macroscopic fundamental diagram using simulation data, Transportation Research Record: Journal of the Transportation Research Board, Transportation Research Board of the National Academies, Washington D. C., 2161 (2010), 40-48. doi: 10.3141/2161-05. [7] Jean-Patrick Lebacque, The Godunov scheme and what it means for first order traffic flow models, Transportation and Traffic Theory. Proceedings of the 13th ISTTT (J. B. Lesort Editor), Elsevier (1996), 647-677. [8] Jean-Patrick Lebacque and Megan Khoshyaran, Macroscopic flow models: Intersection modeling, network modeling, Transportation planning: The state of the art. Kluwer Academic Press, 6th Meeting of the Euro Working Group on Transportation (2002), 119-139. [9] Jean-Patrick Lebacque and Megan Khoshyaran, First-order macroscopic traffic flow models: Intersection modeling, network modeling, Transportation and Traffic Theory. Flow, Dynamics and Human Interaction, 16th International Symposium on Transportation and Traffic Theory (2005), 365-386. [10] M. H. Lighthill and G. B. Whitham, On kinematic waves II: A theory of traffic flow on long crowded roads, Proceedings of the Royal Society, London, A, 229 (1955), 317-345. [11] Luis Miguel Romero Prez and Francisco García Benítez, Traffic flow continuum modeling by hypersingular boundary integral equations, International Journal for Numerical Methods in Engineering, 82 (2010), 47-63. doi: 10.1002/nme.2754. [12] Paul I. Richards, Shock-waves on the highway, Operations Research, 4 (1956), 42-51. [13] Azuma Taguchi and Masao Iri, Continuum approximation to dense networks and its application to the analysis of urban road networks. Applications, Mathematical Programming Study, 20 (1982), 178-217. [14] John Glen Wardrop, Some theorical aspects of road traffic research, Proceedings of the Institute of Civil Engineers, II (1952), 328-378. [15] S. C. Wong, Multi-commodity traffic assignement by continuum approximation of network flow with variable demand, Transportation Research B, 32 (1998), 567-581. [16] Hai Yang, Sam Yagar and Yasunori Iida, Traffic assignment in congested discrete/ continuous transportation system, Transportation Research B, 28 (1994), 161-174. doi: 10.1016/0191-2615(94)90023-X.

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##### References:
 [1] Jean-Bernard Baillon and Guillaume Carlier, From discrete to continuous Wardrop equilibria, Networks and Heterogeneous Media, 7 (2012), 219-241. doi: 10.3934/nhm.2012.7.219. [2] Giuseppe Maria Coclite and Benedetto Piccoli, "Traffic Flow on a Road Network," Technical Report, SISSA, 2002. [3] Giuseppe Maria Coclite, Mauro Garavello and Benedetto Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. [4] Nikolas Geroliminis and Carlos Daganzo, Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings, Transportation Research B, 42 (2008), 759-770. doi: 10.1016/j.trb.2008.02.002. [5] Helge Holden and Nils Henrik Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289. [6] Yangbeibei Ji, Winnie Daamen, Serge Hoogendoorn, Sascha Hoogendoorn-Lanser and Xiaoyu Qian, Investigating the shape of the macroscopic fundamental diagram using simulation data, Transportation Research Record: Journal of the Transportation Research Board, Transportation Research Board of the National Academies, Washington D. C., 2161 (2010), 40-48. doi: 10.3141/2161-05. [7] Jean-Patrick Lebacque, The Godunov scheme and what it means for first order traffic flow models, Transportation and Traffic Theory. Proceedings of the 13th ISTTT (J. B. Lesort Editor), Elsevier (1996), 647-677. [8] Jean-Patrick Lebacque and Megan Khoshyaran, Macroscopic flow models: Intersection modeling, network modeling, Transportation planning: The state of the art. Kluwer Academic Press, 6th Meeting of the Euro Working Group on Transportation (2002), 119-139. [9] Jean-Patrick Lebacque and Megan Khoshyaran, First-order macroscopic traffic flow models: Intersection modeling, network modeling, Transportation and Traffic Theory. Flow, Dynamics and Human Interaction, 16th International Symposium on Transportation and Traffic Theory (2005), 365-386. [10] M. H. Lighthill and G. B. Whitham, On kinematic waves II: A theory of traffic flow on long crowded roads, Proceedings of the Royal Society, London, A, 229 (1955), 317-345. [11] Luis Miguel Romero Prez and Francisco García Benítez, Traffic flow continuum modeling by hypersingular boundary integral equations, International Journal for Numerical Methods in Engineering, 82 (2010), 47-63. doi: 10.1002/nme.2754. [12] Paul I. Richards, Shock-waves on the highway, Operations Research, 4 (1956), 42-51. [13] Azuma Taguchi and Masao Iri, Continuum approximation to dense networks and its application to the analysis of urban road networks. Applications, Mathematical Programming Study, 20 (1982), 178-217. [14] John Glen Wardrop, Some theorical aspects of road traffic research, Proceedings of the Institute of Civil Engineers, II (1952), 328-378. [15] S. C. Wong, Multi-commodity traffic assignement by continuum approximation of network flow with variable demand, Transportation Research B, 32 (1998), 567-581. [16] Hai Yang, Sam Yagar and Yasunori Iida, Traffic assignment in congested discrete/ continuous transportation system, Transportation Research B, 28 (1994), 161-174. doi: 10.1016/0191-2615(94)90023-X.
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