# American Institute of Mathematical Sciences

September  2013, 8(3): 685-705. doi: 10.3934/nhm.2013.8.685

## Numerical discretization of Hamilton--Jacobi equations on networks

 1 University of Mannheim, School of Business Informatics and Mathematics, A5-6, 68131 Mannheim, Germany, Germany 2 RWTH Aachen University, IGPM, Templergraben 55, 52056 Aachen

Received  November 2012 Revised  June 2013 Published  October 2013

We discuss a numerical discretization of Hamilton--Jacobi equations on networks. The latter arise for example as reformulation of the Lighthill--Whitham--Richards traffic flow model. We present coupling conditions for the Hamilton--Jacobi equations and derive a suitable numerical algorithm. Numerical computations of travel times in a round-about are given.
Citation: Simone Göttlich, Ute Ziegler, Michael Herty. Numerical discretization of Hamilton--Jacobi equations on networks. Networks and Heterogeneous Media, 2013, 8 (3) : 685-705. doi: 10.3934/nhm.2013.8.685
##### References:
 [1] A. M. Bayen and C. G. Claudel, Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations, SIAM J. Control Optim., 49 (2011), 383-402. doi: 10.1137/090778754. [2] A. Bressan and K. Han, Optima and equilibria for a model of traffic flow, SIAM J. Math. Anal., 43 (2011), 2384-2417. doi: 10.1137/110825145. [3] G. Bretti, R. Natalini and B. Piccoli, Numerical approximations of a traffic flow model on networks, Netw. Heterog. Media, 1 (2006), 57-84. doi: 10.3934/nhm.2006.1.57. [4] G. Bretti and B. Piccoli, A tracking algorithm for car paths on road networks, SIAM J. Appl. Dyn. Syst., 7 (2008), 510-531. doi: 10.1137/070697768. [5] Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 599-630. doi: 10.3934/dcdsb.2005.5.599. [6] 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. [7] R. Corthout, G. Flötteröd, F. Viti and C. M. J. Tampère, Non-unique flows in macroscopic first-order intersection models, Transportation Res. Part B, 46 (2012), 343-359. doi: 10.1016/j.trb.2011.10.011. [8] C. F. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions, Transportation Res. Part B, 39 (2005), 187-196. doi: 10.1016/j.trb.2004.04.003. [9] C. F. Daganzo, The cell transmission model, part II: Network traffic, Transportation Res. Part B, 29 (1995), 79-93. doi: 10.1016/0191-2615(94)00022-R. [10] C. F. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications, Networks and Heterogeneous Media, 1 (2006), 601-619. doi: 10.3934/nhm.2006.1.601. [11] G. B. Dantzig, "Linear Programming and Extensions," Princeton University Press, Princeton, N.J. 1963 xvi+625 pp. [12] C. D'Apice, R. Manzo and B. Piccoli, A fluid dynamic model for telecommunication networks with sources and destinations, SIAM J. Appl. Math., 68 (2008), 981-1003. doi: 10.1137/060674132. [13] C. D'Apice, R. Manzo and L. Rarità, Splitting of traffic flows to control congestion in special events, Int. J. Math. Math. Sci., (2011), Art. ID 563171, 18 pages. doi: 10.1155/2011/563171. [14] G. Flötteröd and J. Rohde, Operational macroscopic modeling of complex urban intersections, Transportation Res. Part B: Methodological, 45 (2011), 903-922 . [15] M. Garavello and B. Piccoli, "Traffic Flow on Networks," AIMS Series on Applied Mathematics, 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. xvi+243 pp. [16] M. Garavello and B. Piccoli, Source-destination flow on a road network, Commun. Math. Sci., 3 (2005), 261-283. [17] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, (Russian) Mat. Sb. (N. S.), 47 (1959), 271-306. [18] B. Haut and G. Bastin, A second order model of road junctions in fluid models of traffic networks, Netw. Heterog. Media, 2 (2007), 227-253. doi: 10.3934/nhm.2007.2.227. [19] M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM Journal on Scientific Computing, 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X. [20] M. Herty and M. Rascle, Coupling conditions for a class of "second-order'' models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616. doi: 10.1137/05062617X. [21] H. Holden and N. H. 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. [22] C. Imbert, R. Monneau and H. Zidnani, A Hamilton-Jacobi approach to junction problems and application to traffic flow, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002. [23] A. Kurganov and E. Tadmor, New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations, J. Comput. Phys., 160 (2000), 720-742. doi: 10.1006/jcph.2000.6485. [24] J.-P. Lebacque and M. Khoshyaran, First order macroscopic traffic flow models for networks in the context of dynamic assignment, Transportation Planning Applied Optimization, 64 (2004), 119-140. doi: 10.1007/0-306-48220-7_8. [25] R. J. LeVeque, "Numerical Methods for Conservation Laws," Second edition. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-8629-1. [26] M. J. Lighthill and G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, Proc. Royal Society London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [27] Y. Makigami, G. F. Newell and R. Rothery, Three-dimensional representation of traffic flow, Transportation Science, 5 (1971), 302-313. doi: 10.1287/trsc.5.3.302. [28] P. Mazarè, A. Dehwah, C. Claudel and A. Bayen, Analytical and grid-free solutions to the lighthill-whitham-richards traffic flow model, Transportation Res. Part B: Methodological, 45 (2011), 1727-1748. [29] K. Moskowitz, Discussion of freeway level of service as influenced by volume and capacity characteristics, Highway Research Record, 99 (1965), 43-44. [30] G. F. Newell, A simplified theory of kinematic waves in highway traffic: (I) general theory; (ii) queuing at freeway bottlenecks; (iii) multi-destination flow, Transportation Res. Part B, 27 (1993), 281-313. [31] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.

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##### References:
 [1] A. M. Bayen and C. G. Claudel, Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations, SIAM J. Control Optim., 49 (2011), 383-402. doi: 10.1137/090778754. [2] A. Bressan and K. Han, Optima and equilibria for a model of traffic flow, SIAM J. Math. Anal., 43 (2011), 2384-2417. doi: 10.1137/110825145. [3] G. Bretti, R. Natalini and B. Piccoli, Numerical approximations of a traffic flow model on networks, Netw. Heterog. Media, 1 (2006), 57-84. doi: 10.3934/nhm.2006.1.57. [4] G. Bretti and B. Piccoli, A tracking algorithm for car paths on road networks, SIAM J. Appl. Dyn. Syst., 7 (2008), 510-531. doi: 10.1137/070697768. [5] Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 599-630. doi: 10.3934/dcdsb.2005.5.599. [6] 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. [7] R. Corthout, G. Flötteröd, F. Viti and C. M. J. Tampère, Non-unique flows in macroscopic first-order intersection models, Transportation Res. Part B, 46 (2012), 343-359. doi: 10.1016/j.trb.2011.10.011. [8] C. F. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions, Transportation Res. Part B, 39 (2005), 187-196. doi: 10.1016/j.trb.2004.04.003. [9] C. F. Daganzo, The cell transmission model, part II: Network traffic, Transportation Res. Part B, 29 (1995), 79-93. doi: 10.1016/0191-2615(94)00022-R. [10] C. F. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications, Networks and Heterogeneous Media, 1 (2006), 601-619. doi: 10.3934/nhm.2006.1.601. [11] G. B. Dantzig, "Linear Programming and Extensions," Princeton University Press, Princeton, N.J. 1963 xvi+625 pp. [12] C. D'Apice, R. Manzo and B. Piccoli, A fluid dynamic model for telecommunication networks with sources and destinations, SIAM J. Appl. Math., 68 (2008), 981-1003. doi: 10.1137/060674132. [13] C. D'Apice, R. Manzo and L. Rarità, Splitting of traffic flows to control congestion in special events, Int. J. Math. Math. Sci., (2011), Art. ID 563171, 18 pages. doi: 10.1155/2011/563171. [14] G. Flötteröd and J. Rohde, Operational macroscopic modeling of complex urban intersections, Transportation Res. Part B: Methodological, 45 (2011), 903-922 . [15] M. Garavello and B. Piccoli, "Traffic Flow on Networks," AIMS Series on Applied Mathematics, 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. xvi+243 pp. [16] M. Garavello and B. Piccoli, Source-destination flow on a road network, Commun. Math. Sci., 3 (2005), 261-283. [17] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, (Russian) Mat. Sb. (N. S.), 47 (1959), 271-306. [18] B. Haut and G. Bastin, A second order model of road junctions in fluid models of traffic networks, Netw. Heterog. Media, 2 (2007), 227-253. doi: 10.3934/nhm.2007.2.227. [19] M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM Journal on Scientific Computing, 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X. [20] M. Herty and M. Rascle, Coupling conditions for a class of "second-order'' models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616. doi: 10.1137/05062617X. [21] H. Holden and N. H. 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. [22] C. Imbert, R. Monneau and H. Zidnani, A Hamilton-Jacobi approach to junction problems and application to traffic flow, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002. [23] A. Kurganov and E. Tadmor, New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations, J. Comput. Phys., 160 (2000), 720-742. doi: 10.1006/jcph.2000.6485. [24] J.-P. Lebacque and M. Khoshyaran, First order macroscopic traffic flow models for networks in the context of dynamic assignment, Transportation Planning Applied Optimization, 64 (2004), 119-140. doi: 10.1007/0-306-48220-7_8. [25] R. J. LeVeque, "Numerical Methods for Conservation Laws," Second edition. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-8629-1. [26] M. J. Lighthill and G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, Proc. Royal Society London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [27] Y. Makigami, G. F. Newell and R. Rothery, Three-dimensional representation of traffic flow, Transportation Science, 5 (1971), 302-313. doi: 10.1287/trsc.5.3.302. [28] P. Mazarè, A. Dehwah, C. Claudel and A. Bayen, Analytical and grid-free solutions to the lighthill-whitham-richards traffic flow model, Transportation Res. Part B: Methodological, 45 (2011), 1727-1748. [29] K. Moskowitz, Discussion of freeway level of service as influenced by volume and capacity characteristics, Highway Research Record, 99 (1965), 43-44. [30] G. F. Newell, A simplified theory of kinematic waves in highway traffic: (I) general theory; (ii) queuing at freeway bottlenecks; (iii) multi-destination flow, Transportation Res. Part B, 27 (1993), 281-313. [31] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.
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