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), 1017-1034. doi: 10.1080/03605307908820117.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. Costeseque, J.-P. Lebacque and R. Monneau, A convergent scheme for Hamilton-Jacobi equations on a junction: Application to traffic, submitted, (2013). Google Scholar

[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.  Google Scholar

[12]

G. Flötteröd and J. Rohde, Operational macroscopic modeling of complex urban intersections, Transport. Res. B, 45 (2011), 903-922. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. Göttlich, M. Herty and U. Ziegler, Numerical discretization of Hamilton-Jacobi equations on networks, Networks and Heterogeneous Media, 8 (2013), 685-705. Google Scholar

[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). Google Scholar

[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.  Google Scholar

[20]

C. Imbert and R. Monneau, The vertex test function for Hamilton-Jacobi equations on networks, preprint, arXiv:1306.2428, (2013). Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[24]

J. P. Lebacque, Semi-macroscopic simulation of urban traffic, in Proc. of the Int. 84 Minneapolis Summer Conference. AMSE, 4, (1984), 273-291. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[30]

P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. Costeseque, J.-P. Lebacque and R. Monneau, A convergent scheme for Hamilton-Jacobi equations on a junction: Application to traffic, submitted, (2013). Google Scholar

[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.  Google Scholar

[12]

G. Flötteröd and J. Rohde, Operational macroscopic modeling of complex urban intersections, Transport. Res. B, 45 (2011), 903-922. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. Göttlich, M. Herty and U. Ziegler, Numerical discretization of Hamilton-Jacobi equations on networks, Networks and Heterogeneous Media, 8 (2013), 685-705. Google Scholar

[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). Google Scholar

[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.  Google Scholar

[20]

C. Imbert and R. Monneau, The vertex test function for Hamilton-Jacobi equations on networks, preprint, arXiv:1306.2428, (2013). Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[24]

J. P. Lebacque, Semi-macroscopic simulation of urban traffic, in Proc. of the Int. 84 Minneapolis Summer Conference. AMSE, 4, (1984), 273-291. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[30]

P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar

[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.  Google Scholar

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