# American Institute of Mathematical Sciences

June  2014, 7(3): 411-433. doi: 10.3934/dcdss.2014.7.411

## Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations

 1 Université Paris-Est, Ecole des Ponts ParisTech, CERMICS & IFSTTAR, GRETTIA, 6 & 8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 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

Received  June 2013 Revised  October 2013 Published  January 2014

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.
Citation: Guillaume Costeseque, Jean-Patrick Lebacque. Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 411-433. doi: 10.3934/dcdss.2014.7.411
##### References:
 [1] C. Bardos, A. Y. Le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117. [2] H. Bar-Gera and S. Ahn, Empirical macroscopic evaluation of freeway merge-ratios, Transport. Res. C, 18 (2010), 457-470. doi: 10.1016/j.trc.2009.09.002. [3] G. Bretti, R. Natalini and B. Piccoli, Fast algorithms for the approximation of a fluid-dynamic model on networks, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 427-448. doi: 10.3934/dcdsb.2006.6.427. [4] G. Bretti, R. Natalini and B. Piccoli, Numerical approximations of a traffic flow model on networks, Networks and Heterogeneous Media, 1 (2006), 57-84. doi: 10.3934/nhm.2006.1.57. [5] C. Buisson, J. P. Lebacque and J. B. Lesort, STRADA: A discretized macroscopic model of vehicular traffic flow in complex networks based on the Godunov scheme, in Proceedings of the IEEE-SMC IMACS'96 Multiconference, Symposium on Modelling, Analysis and Simulation, 2, 1996, 976-981. [6] M. J. Cassidy and S. Ahn, Driver turn-taking behavior in congested freeway merges, Transportation Research Record, Journal of the Transportation Research Board, 1934 (2005), 140-147. doi: 10.3141/1934-15. [7] C. Claudel and A. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory, IEEE Transactions on Automatic Control, 55 (2010), 1142-1157. doi: 10.1109/TAC.2010.2041976. [8] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. [9] R. Corthout, G. Flötteröd, F. Viti and C. M. J. Tampère, Non-unique flows in macroscopic first-order intersection models, Transport. Res. B, 46 (2012), 343-359. doi: 10.1016/j.trb.2011.10.011. [10] G. Costeseque, J.-P. Lebacque and R. Monneau, A convergent scheme for Hamilton-Jacobi equations on a junction: Application to traffic, submitted, (2013). [11] C. F. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications, Networks and Heterogeneous Media, AIMS, 1 (2006), 601-619. doi: 10.3934/nhm.2006.1.601. [12] G. Flötteröd and J. Rohde, Operational macroscopic modeling of complex urban intersections, Transport. Res. B, 45 (2011), 903-922. [13] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. [14] N. H. Gartner, C. J. Messer and A. K. Rathi, Revised Monograph of Traffic Flow Theory, Online publication of the Transportation Research Board, FHWA, 2001. Available from: http://www.tfhrc.gov/its/tft/tft.htm. [15] J. Gibb, A model of traffic flow capacity constraint through nodes for dynamic network loading with queue spillback, Transportation Research Record: Journal of the Transportation Research Board, (2011), 113-122. doi: 10.3141/2263-13. [16] S. K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Math. Sb., 47 (1959), 271-290. [17] S. Göttlich, M. Herty and U. Ziegler, Numerical discretization of Hamilton-Jacobi equations on networks, Networks and Heterogeneous Media, 8 (2013), 685-705. [18] K. Han, B. Piccoli, T. L. Friesz and T. Yao, A continuous-time link-based kinematic wave model for dynamic traffic networks, preprint, arXiv:1208.5141, (2012). [19] H. Holden and N. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 4 (1995), 999-1017. doi: 10.1137/S0036141093243289. [20] C. Imbert and R. Monneau, The vertex test function for Hamilton-Jacobi equations on networks, preprint, arXiv:1306.2428, (2013). [21] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002. [22] M. M. Khoshyaran and J. P. Lebacque, Internal state models for intersections in macroscopic traffic flow models, Accepted in Proceedings of Traffic and Granular Flow 09, (2009). [23] J. A. Laval and L. Leclercq, The Hamilton-Jacobi partial differential equation and the three representations of traffic flow, Transport. Res. B, 52 (2013), 17-30. doi: 10.1016/j.trb.2013.02.008. [24] J. P. Lebacque, Semi-macroscopic simulation of urban traffic, in Proc. of the Int. 84 Minneapolis Summer Conference. AMSE, 4, (1984), 273-291. [25] J. P. Lebacque, The Godunov scheme and what it means for first order traffic flow models, in 13th ISTTT Symposium, Elsevier, (ed., J. B. Lesort), New York, 1996, 647-678. [26] J. P. Lebacque and M. M. Khoshyaran, Macroscopic flow models (First order macroscopic traffic flow models for networks in the context of dynamic assignment), in Transportation planning, the state of the art, Kluwer Academic Press, (eds. M. Patriksson et M. Labbé), 2002, 119-140. doi: 10.1007/0-306-48220-7_8. [27] J. P. Lebacque and M. M. Koshyaran, First-order macroscopic traffic flow models: intersection modeling, network modeling, in Proceedings of the 16th International Symposium on the Transportation and Traffic Theory, College Park, Maryland, USA, Elsevier, Oxford, (ed., H. S. Mahmassani), 2005, 365-386. [28] M. J. Lighthill and G. B. Whitham, On kinetic waves. II. Theory of traffic flows on long crowded roads, Proc. Roy. Soc. London Ser. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [29] G. F. Newell, A simplified theory of kinematic waves in highway traffic, (i) General theory, (ii) Queueing at freeway bottlenecks, (iii) Multi-destination flows, Transport. Res. B, 4 (1993), 281-313. [30] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [31] C. Tampere, R. Corthout, D. Cattrysse and L. Immers, A generic class of first order node models for dynamic macroscopic simulations of traffic flows, Transport. Res. B, 45 (2011), 289-309. doi: 10.1016/j.trb.2010.06.004.

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##### References:
 [1] C. Bardos, A. Y. Le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117. [2] H. Bar-Gera and S. Ahn, Empirical macroscopic evaluation of freeway merge-ratios, Transport. Res. C, 18 (2010), 457-470. doi: 10.1016/j.trc.2009.09.002. [3] G. Bretti, R. Natalini and B. Piccoli, Fast algorithms for the approximation of a fluid-dynamic model on networks, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 427-448. doi: 10.3934/dcdsb.2006.6.427. [4] G. Bretti, R. Natalini and B. Piccoli, Numerical approximations of a traffic flow model on networks, Networks and Heterogeneous Media, 1 (2006), 57-84. doi: 10.3934/nhm.2006.1.57. [5] C. Buisson, J. P. Lebacque and J. B. Lesort, STRADA: A discretized macroscopic model of vehicular traffic flow in complex networks based on the Godunov scheme, in Proceedings of the IEEE-SMC IMACS'96 Multiconference, Symposium on Modelling, Analysis and Simulation, 2, 1996, 976-981. [6] M. J. Cassidy and S. Ahn, Driver turn-taking behavior in congested freeway merges, Transportation Research Record, Journal of the Transportation Research Board, 1934 (2005), 140-147. doi: 10.3141/1934-15. [7] C. Claudel and A. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory, IEEE Transactions on Automatic Control, 55 (2010), 1142-1157. doi: 10.1109/TAC.2010.2041976. [8] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. [9] R. Corthout, G. Flötteröd, F. Viti and C. M. J. Tampère, Non-unique flows in macroscopic first-order intersection models, Transport. Res. B, 46 (2012), 343-359. doi: 10.1016/j.trb.2011.10.011. [10] G. Costeseque, J.-P. Lebacque and R. Monneau, A convergent scheme for Hamilton-Jacobi equations on a junction: Application to traffic, submitted, (2013). [11] C. F. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications, Networks and Heterogeneous Media, AIMS, 1 (2006), 601-619. doi: 10.3934/nhm.2006.1.601. [12] G. Flötteröd and J. Rohde, Operational macroscopic modeling of complex urban intersections, Transport. Res. B, 45 (2011), 903-922. [13] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. [14] N. H. Gartner, C. J. Messer and A. K. Rathi, Revised Monograph of Traffic Flow Theory, Online publication of the Transportation Research Board, FHWA, 2001. Available from: http://www.tfhrc.gov/its/tft/tft.htm. [15] J. Gibb, A model of traffic flow capacity constraint through nodes for dynamic network loading with queue spillback, Transportation Research Record: Journal of the Transportation Research Board, (2011), 113-122. doi: 10.3141/2263-13. [16] S. K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Math. Sb., 47 (1959), 271-290. [17] S. Göttlich, M. Herty and U. Ziegler, Numerical discretization of Hamilton-Jacobi equations on networks, Networks and Heterogeneous Media, 8 (2013), 685-705. [18] K. Han, B. Piccoli, T. L. Friesz and T. Yao, A continuous-time link-based kinematic wave model for dynamic traffic networks, preprint, arXiv:1208.5141, (2012). [19] H. Holden and N. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 4 (1995), 999-1017. doi: 10.1137/S0036141093243289. [20] C. Imbert and R. Monneau, The vertex test function for Hamilton-Jacobi equations on networks, preprint, arXiv:1306.2428, (2013). [21] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002. [22] M. M. Khoshyaran and J. P. Lebacque, Internal state models for intersections in macroscopic traffic flow models, Accepted in Proceedings of Traffic and Granular Flow 09, (2009). [23] J. A. Laval and L. Leclercq, The Hamilton-Jacobi partial differential equation and the three representations of traffic flow, Transport. Res. B, 52 (2013), 17-30. doi: 10.1016/j.trb.2013.02.008. [24] J. P. Lebacque, Semi-macroscopic simulation of urban traffic, in Proc. of the Int. 84 Minneapolis Summer Conference. AMSE, 4, (1984), 273-291. [25] J. P. Lebacque, The Godunov scheme and what it means for first order traffic flow models, in 13th ISTTT Symposium, Elsevier, (ed., J. B. Lesort), New York, 1996, 647-678. [26] J. P. Lebacque and M. M. Khoshyaran, Macroscopic flow models (First order macroscopic traffic flow models for networks in the context of dynamic assignment), in Transportation planning, the state of the art, Kluwer Academic Press, (eds. M. Patriksson et M. Labbé), 2002, 119-140. doi: 10.1007/0-306-48220-7_8. [27] J. P. Lebacque and M. M. Koshyaran, First-order macroscopic traffic flow models: intersection modeling, network modeling, in Proceedings of the 16th International Symposium on the Transportation and Traffic Theory, College Park, Maryland, USA, Elsevier, Oxford, (ed., H. S. Mahmassani), 2005, 365-386. [28] M. J. Lighthill and G. B. Whitham, On kinetic waves. II. Theory of traffic flows on long crowded roads, Proc. Roy. Soc. London Ser. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [29] G. F. Newell, A simplified theory of kinematic waves in highway traffic, (i) General theory, (ii) Queueing at freeway bottlenecks, (iii) Multi-destination flows, Transport. Res. B, 4 (1993), 281-313. [30] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [31] C. Tampere, R. Corthout, D. Cattrysse and L. Immers, A generic class of first order node models for dynamic macroscopic simulations of traffic flows, Transport. Res. B, 45 (2011), 289-309. doi: 10.1016/j.trb.2010.06.004.
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