# American Institute of Mathematical Sciences

June  2014, 7(3): 379-394. doi: 10.3934/dcdss.2014.7.379

## An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments

 1 Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Via dei Taurini, 19 – 00185 Rome, Italy, Italy, Italy

Received  May 2013 Revised  July 2013 Published  January 2014

In this paper we propose a Godunov-based discretization of a hyperbolic system of conservation laws with discontinuous flux, modeling vehicular flow on a network. Each equation describes the density evolution of vehicles having a common path along the network. We show that the algorithm selects automatically an admissible solution at junctions, hence ad hoc external procedures (e.g., maximization of the flux via a linear programming method) usually employed in classical approaches are no needed. Since users have not to deal explicitly with vehicle dynamics at junction, the numerical code can be implemented in minutes. We perform a detailed numerical comparison with a Godunov-based scheme coming from the classical theory of traffic flow on networks which maximizes the flux at junctions.
Citation: Gabriella Bretti, Maya Briani, Emiliano Cristiani. An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 379-394. doi: 10.3934/dcdss.2014.7.379
##### References:
 [1] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Rational Mech. Anal., 201 (2011), 27-86. doi: 10.1007/s00205-010-0389-4. [2] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266. [3] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics, 20, Oxford University Press, New York, 2000. [4] G. Bretti, R. Natalini and B. Piccoli, A fluid-dynamic traffic model on road networks, Arch. Comput. Methods Eng., 14 (2007), 139-172. doi: 10.1007/s11831-007-9004-8. [5] M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: theoretical study, submitted, arXiv:1401.1651. [6] R. Bürger and K. H. Karlsen, Conservation laws with discontinuous flux: A short introduction, J. Eng. Math., 60 (2008), 241-247. doi: 10.1007/s10665-008-9213-7. [7] Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete and Continuous Dynamical Systems - Series B, 5 (2005), 599-630. doi: 10.3934/dcdsb.2005.5.599. [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] E. Cristiani, C. de Fabritiis and B. Piccoli, A fluid dynamic approach for traffic forecast from mobile sensor data, Commun. Appl. Ind. Math., 1 (2010), 54-71. doi: 10.1685/2010CAIM487. [10] C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transportation Research Part B, 28 (1994), 269-287. doi: 10.1016/0191-2615(94)90002-7. [11] C. F. Daganzo, The cell transmission model, part II: Network traffic, Transportation Research Part B, 29 (1995), 79-93. doi: 10.1016/0191-2615(94)00022-R. [12] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915. [13] M. Garavello and B. Piccoli, Source-destination flow on a road network, Comm. Math. Sci., 3 (2005), 261-283. doi: 10.4310/CMS.2005.v3.n3.a1. [14] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, 2006. [15] M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, Advances in Dynamic Network Modeling in Complex Transportation Systems, Complex Networks and Dynamic Systems, 2 (2013), 143-161. doi: 10.1007/978-1-4614-6243-9_6. [16] J. C. Herrera and A. M. Bayen, Incorporation of Lagrangian measurements in freeway traffic state estimation, Transportation Research Part B, 44 (2010), 460-481. doi: 10.1016/j.trb.2009.10.005. [17] M. Herty, J.-P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow, Netw. Heterog. Media, 4 (2009), 813-826. doi: 10.3934/nhm.2009.4.813. [18] M. Herty, M. Seaïd and A. K. Singh, A domain decomposition method for conservation laws with discontinuous flux function, Appl. Numer. Math., 57 (2007), 361-373. doi: 10.1016/j.apnum.2006.04.003. [19] M. Hilliges and W. Weidlich, A phenomenological model for dynamic traffic flow in networks, Transportation Research Part B, 29 (1995), 407-431. doi: 10.1016/0191-2615(95)00018-9. [20] 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. [21] J.-P. Lebacque, The Godunov scheme and what it means for first order traffic flow models, in Proc. of the 13th International Symposium on Transportation and Traffic Theory, Lyon, France (ed. J. B. Lesort), Elsevier, 1996, 647-677. [22] R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi: 10.1007/978-3-0348-8629-1. [23] M. J. Lighthill and G. B. Whitham, On kinetic waves. II. Theory of traffic flows on long crowded roads, Proc. Roy. Soc. Lond. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [24] M. Mercier, Traffic flow modelling with junctions, J. Math. Anal. Appl., 350 (2009), 369-383. doi: 10.1016/j.jmaa.2008.09.040. [25] P. I. Richards, Shock waves on the highway, Operation Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [26] J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698. doi: 10.1137/S0036142999363668. [27] G. C. K. Wong and S. C. Wong, A multi-class traffic flow model - an extension of LWR model with heterogeneous drivers, Transportation Research Part A, 36 (2002), 827-841. doi: 10.1016/S0965-8564(01)00042-8.

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##### References:
 [1] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Rational Mech. Anal., 201 (2011), 27-86. doi: 10.1007/s00205-010-0389-4. [2] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266. [3] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics, 20, Oxford University Press, New York, 2000. [4] G. Bretti, R. Natalini and B. Piccoli, A fluid-dynamic traffic model on road networks, Arch. Comput. Methods Eng., 14 (2007), 139-172. doi: 10.1007/s11831-007-9004-8. [5] M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: theoretical study, submitted, arXiv:1401.1651. [6] R. Bürger and K. H. Karlsen, Conservation laws with discontinuous flux: A short introduction, J. Eng. Math., 60 (2008), 241-247. doi: 10.1007/s10665-008-9213-7. [7] Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete and Continuous Dynamical Systems - Series B, 5 (2005), 599-630. doi: 10.3934/dcdsb.2005.5.599. [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] E. Cristiani, C. de Fabritiis and B. Piccoli, A fluid dynamic approach for traffic forecast from mobile sensor data, Commun. Appl. Ind. Math., 1 (2010), 54-71. doi: 10.1685/2010CAIM487. [10] C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transportation Research Part B, 28 (1994), 269-287. doi: 10.1016/0191-2615(94)90002-7. [11] C. F. Daganzo, The cell transmission model, part II: Network traffic, Transportation Research Part B, 29 (1995), 79-93. doi: 10.1016/0191-2615(94)00022-R. [12] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915. [13] M. Garavello and B. Piccoli, Source-destination flow on a road network, Comm. Math. Sci., 3 (2005), 261-283. doi: 10.4310/CMS.2005.v3.n3.a1. [14] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, 2006. [15] M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, Advances in Dynamic Network Modeling in Complex Transportation Systems, Complex Networks and Dynamic Systems, 2 (2013), 143-161. doi: 10.1007/978-1-4614-6243-9_6. [16] J. C. Herrera and A. M. Bayen, Incorporation of Lagrangian measurements in freeway traffic state estimation, Transportation Research Part B, 44 (2010), 460-481. doi: 10.1016/j.trb.2009.10.005. [17] M. Herty, J.-P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow, Netw. Heterog. Media, 4 (2009), 813-826. doi: 10.3934/nhm.2009.4.813. [18] M. Herty, M. Seaïd and A. K. Singh, A domain decomposition method for conservation laws with discontinuous flux function, Appl. Numer. Math., 57 (2007), 361-373. doi: 10.1016/j.apnum.2006.04.003. [19] M. Hilliges and W. Weidlich, A phenomenological model for dynamic traffic flow in networks, Transportation Research Part B, 29 (1995), 407-431. doi: 10.1016/0191-2615(95)00018-9. [20] 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. [21] J.-P. Lebacque, The Godunov scheme and what it means for first order traffic flow models, in Proc. of the 13th International Symposium on Transportation and Traffic Theory, Lyon, France (ed. J. B. Lesort), Elsevier, 1996, 647-677. [22] R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi: 10.1007/978-3-0348-8629-1. [23] M. J. Lighthill and G. B. Whitham, On kinetic waves. II. Theory of traffic flows on long crowded roads, Proc. Roy. Soc. Lond. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [24] M. Mercier, Traffic flow modelling with junctions, J. Math. Anal. Appl., 350 (2009), 369-383. doi: 10.1016/j.jmaa.2008.09.040. [25] P. I. Richards, Shock waves on the highway, Operation Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [26] J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698. doi: 10.1137/S0036142999363668. [27] G. C. K. Wong and S. C. Wong, A multi-class traffic flow model - an extension of LWR model with heterogeneous drivers, Transportation Research Part A, 36 (2002), 827-841. doi: 10.1016/S0965-8564(01)00042-8.
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