# 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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: theoretical study,, submitted, ().   Google Scholar [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.  Google Scholar [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.  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] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, 2006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [22] R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi: 10.1007/978-3-0348-8629-1.  Google Scholar [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.  Google Scholar [24] M. Mercier, Traffic flow modelling with junctions, J. Math. Anal. Appl., 350 (2009), 369-383. doi: 10.1016/j.jmaa.2008.09.040.  Google Scholar [25] P. I. Richards, Shock waves on the highway, Operation Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: theoretical study,, submitted, ().   Google Scholar [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.  Google Scholar [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.  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] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, 2006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [22] R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi: 10.1007/978-3-0348-8629-1.  Google Scholar [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.  Google Scholar [24] M. Mercier, Traffic flow modelling with junctions, J. Math. Anal. Appl., 350 (2009), 369-383. doi: 10.1016/j.jmaa.2008.09.040.  Google Scholar [25] P. I. Richards, Shock waves on the highway, Operation Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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