# 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 & 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. doi: 10.1080/03605307908820117. [2] H. Bar-Gera and S. Ahn, Empirical macroscopic evaluation of freeway merge-ratios,, Transport. Res. C, 18 (2010), 457. 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. 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. 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, 2 (1996), 976. [6] M. J. Cassidy and S. Ahn, Driver turn-taking behavior in congested freeway merges,, Transportation Research Record, 1934 (2005), 140. 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. 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. 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. 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, 1 (2006), 601. 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. [13] M. Garavello and B. Piccoli, Traffic Flow on Networks,, AIMS Series on Applied Mathematics, (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, (2001). [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. 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. [17] S. Göttlich, M. Herty and U. Ziegler, Numerical discretization of Hamilton-Jacobi equations on networks,, Networks and Heterogeneous Media, 8 (2013), 685. [18] K. Han, B. Piccoli, T. L. Friesz and T. Yao, A continuous-time link-based kinematic wave model for dynamic traffic networks,, preprint, (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. doi: 10.1137/S0036141093243289. [20] C. Imbert and R. Monneau, The vertex test function for Hamilton-Jacobi equations on networks,, preprint, (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. 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. 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. [25] J. P. Lebacque, The Godunov scheme and what it means for first order traffic flow models,, in 13th ISTTT Symposium, (1996), 647. [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, (2002), 119. 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, (2005), 365. [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. 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. [30] P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42. 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. 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. doi: 10.1080/03605307908820117. [2] H. Bar-Gera and S. Ahn, Empirical macroscopic evaluation of freeway merge-ratios,, Transport. Res. C, 18 (2010), 457. 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. 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. 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, 2 (1996), 976. [6] M. J. Cassidy and S. Ahn, Driver turn-taking behavior in congested freeway merges,, Transportation Research Record, 1934 (2005), 140. 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. 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. 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. 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, 1 (2006), 601. 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. [13] M. Garavello and B. Piccoli, Traffic Flow on Networks,, AIMS Series on Applied Mathematics, (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, (2001). [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. 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. [17] S. Göttlich, M. Herty and U. Ziegler, Numerical discretization of Hamilton-Jacobi equations on networks,, Networks and Heterogeneous Media, 8 (2013), 685. [18] K. Han, B. Piccoli, T. L. Friesz and T. Yao, A continuous-time link-based kinematic wave model for dynamic traffic networks,, preprint, (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. doi: 10.1137/S0036141093243289. [20] C. Imbert and R. Monneau, The vertex test function for Hamilton-Jacobi equations on networks,, preprint, (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. 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. 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. [25] J. P. Lebacque, The Godunov scheme and what it means for first order traffic flow models,, in 13th ISTTT Symposium, (1996), 647. [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, (2002), 119. 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, (2005), 365. [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. 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. [30] P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42. 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. doi: 10.1016/j.trb.2010.06.004.
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