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September  2014, 9(3): 519-552. doi: 10.3934/nhm.2014.9.519

## An easy-to-use algorithm for simulating traffic flow on networks: Theoretical study

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

Received  May 2014 Revised  August 2014 Published  October 2014

In this paper we study a model for traffic flow on networks based on a hyperbolic system of conservation laws with discontinuous flux. Each equation describes the density evolution of vehicles having a common path along the network. In this formulation the junctions disappear since each path is considered as a single uninterrupted road.
We consider a Godunov-based approximation scheme for the system which is very easy to implement. Besides basic properties like the conservation of cars and positive bounded solutions, the scheme exhibits other nice properties, being able to select automatically a solution at network's nodes without requiring external procedures (e.g., maximization of the flux via a linear programming method). Moreover, the scheme can be interpreted as a discretization of the traffic models with buffer, although no buffer is introduced here.
Finally, we show how the scheme can be recast in the framework of the classical theory of traffic flow on networks, where a conservation law has to be solved on each arc of the network. This is achieved by solving the Riemann problem for a modified equation, and showing that its solution corresponds to the one computed by the numerical scheme.
Citation: Maya Briani, Emiliano Cristiani. An easy-to-use algorithm for simulating traffic flow on networks: Theoretical study. Networks & Heterogeneous Media, 2014, 9 (3) : 519-552. doi: 10.3934/nhm.2014.9.519
##### 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.  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.  doi: 10.1017/S0956792503005266.  Google Scholar [3] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford Lecture Series in Mathematics, (2000).   Google Scholar [4] A. Bressan and K. T. Nguyen, Conservation law models for traffic flow on a network of roads,, Preprint, (2014).   Google Scholar [5] G. Bretti, M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments,, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 379.  doi: 10.3934/dcdss.2014.7.379.  Google Scholar [6] R. Bürger and K. H. Karlsen, Conservation laws with discontinuous flux: A short introduction,, J. Eng. Math., 60 (2008), 241.  doi: 10.1007/s10665-008-9213-7.  Google Scholar [7] 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.  Google Scholar [8] 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.  doi: 10.1685/2010CAIM487.  Google Scholar [9] 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.  doi: 10.1016/0191-2615(94)90002-7.  Google Scholar [10] C. F. Daganzo, The cell transmission model, part II: Network traffic,, Transportation Research Part B, 29 (1995), 79.  doi: 10.1016/0191-2615(94)00022-R.  Google Scholar [11] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer,, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1915.  doi: 10.3934/dcds.2012.32.1915.  Google Scholar [12] M. Garavello and B. Piccoli, Source-destination flow on a road network,, Comm. Math. Sci., 3 (2005), 261.  doi: 10.4310/CMS.2005.v3.n3.a1.  Google Scholar [13] M. Garavello and B. Piccoli, Traffic Flow on Networks,, AIMS Series on Applied Mathematics, (2006).   Google Scholar [14] M. Garavello and B. Piccoli, A multibuffer model for LWR road networks,, Advances in Dynamic Network Modeling in Complex Transportation Systems, 2 (2013), 143.  doi: 10.1007/978-1-4614-6243-9_6.  Google Scholar [15] R. Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics and Traffic Flow,, Prentice-Hall, (1977).   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.  doi: 10.1016/j.trb.2009.10.005.  Google Scholar [17] M. Herty, C. Kirchner, S. Moutari and M. Rascle, Multicommodity flows on road networks,, Commun. Math. Sci., 6 (2008), 171.  doi: 10.4310/CMS.2008.v6.n1.a8.  Google Scholar [18] M. Herty, J.-P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow,, Netw. Heterog. Media, 4 (2009), 813.  doi: 10.3934/nhm.2009.4.813.  Google Scholar [19] 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.  doi: 10.1016/j.apnum.2006.04.003.  Google Scholar [20] M. Hilliges and W. Weidlich, A phenomenological model for dynamic traffic flow in networks,, Transportation Research Part B, 29 (1995), 407.  doi: 10.1016/0191-2615(95)00018-9.  Google Scholar [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.  doi: 10.1137/S0036141093243289.  Google Scholar [22] 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, (1996), 647.   Google Scholar [23] R. J. LeVeque, Numerical Methods for Conservation Laws,, Birkhäuser, (1992).  doi: 10.1007/978-3-0348-8629-1.  Google Scholar [24] 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.  doi: 10.1098/rspa.1955.0089.  Google Scholar [25] M. Mercier, Traffic flow modelling with junctions,, J. Math. Anal. Appl., 350 (2009), 369.  doi: 10.1016/j.jmaa.2008.09.040.  Google Scholar [26] P. I. Richards, Shock waves on the highway,, Operations Res., 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar [27] C. M. J. Tampère, R. Corthout, D. Cattrysse and L. H. Immers, A generic class of first order node models for dynamic macroscopic simulation of traffic flows,, Transportation Research Part B, 45 (2011), 289.   Google Scholar [28] J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux,, SIAM J. Numer. Anal., 38 (2000), 681.  doi: 10.1137/S0036142999363668.  Google Scholar [29] 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.  doi: 10.1016/S0965-8564(01)00042-8.  Google Scholar

show all references

##### 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.  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.  doi: 10.1017/S0956792503005266.  Google Scholar [3] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford Lecture Series in Mathematics, (2000).   Google Scholar [4] A. Bressan and K. T. Nguyen, Conservation law models for traffic flow on a network of roads,, Preprint, (2014).   Google Scholar [5] G. Bretti, M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments,, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 379.  doi: 10.3934/dcdss.2014.7.379.  Google Scholar [6] R. Bürger and K. H. Karlsen, Conservation laws with discontinuous flux: A short introduction,, J. Eng. Math., 60 (2008), 241.  doi: 10.1007/s10665-008-9213-7.  Google Scholar [7] 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.  Google Scholar [8] 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.  doi: 10.1685/2010CAIM487.  Google Scholar [9] 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.  doi: 10.1016/0191-2615(94)90002-7.  Google Scholar [10] C. F. Daganzo, The cell transmission model, part II: Network traffic,, Transportation Research Part B, 29 (1995), 79.  doi: 10.1016/0191-2615(94)00022-R.  Google Scholar [11] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer,, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1915.  doi: 10.3934/dcds.2012.32.1915.  Google Scholar [12] M. Garavello and B. Piccoli, Source-destination flow on a road network,, Comm. Math. Sci., 3 (2005), 261.  doi: 10.4310/CMS.2005.v3.n3.a1.  Google Scholar [13] M. Garavello and B. Piccoli, Traffic Flow on Networks,, AIMS Series on Applied Mathematics, (2006).   Google Scholar [14] M. Garavello and B. Piccoli, A multibuffer model for LWR road networks,, Advances in Dynamic Network Modeling in Complex Transportation Systems, 2 (2013), 143.  doi: 10.1007/978-1-4614-6243-9_6.  Google Scholar [15] R. Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics and Traffic Flow,, Prentice-Hall, (1977).   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.  doi: 10.1016/j.trb.2009.10.005.  Google Scholar [17] M. Herty, C. Kirchner, S. Moutari and M. Rascle, Multicommodity flows on road networks,, Commun. Math. Sci., 6 (2008), 171.  doi: 10.4310/CMS.2008.v6.n1.a8.  Google Scholar [18] M. Herty, J.-P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow,, Netw. Heterog. Media, 4 (2009), 813.  doi: 10.3934/nhm.2009.4.813.  Google Scholar [19] 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.  doi: 10.1016/j.apnum.2006.04.003.  Google Scholar [20] M. Hilliges and W. Weidlich, A phenomenological model for dynamic traffic flow in networks,, Transportation Research Part B, 29 (1995), 407.  doi: 10.1016/0191-2615(95)00018-9.  Google Scholar [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.  doi: 10.1137/S0036141093243289.  Google Scholar [22] 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, (1996), 647.   Google Scholar [23] R. J. LeVeque, Numerical Methods for Conservation Laws,, Birkhäuser, (1992).  doi: 10.1007/978-3-0348-8629-1.  Google Scholar [24] 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.  doi: 10.1098/rspa.1955.0089.  Google Scholar [25] M. Mercier, Traffic flow modelling with junctions,, J. Math. Anal. Appl., 350 (2009), 369.  doi: 10.1016/j.jmaa.2008.09.040.  Google Scholar [26] P. I. Richards, Shock waves on the highway,, Operations Res., 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar [27] C. M. J. Tampère, R. Corthout, D. Cattrysse and L. H. Immers, A generic class of first order node models for dynamic macroscopic simulation of traffic flows,, Transportation Research Part B, 45 (2011), 289.   Google Scholar [28] J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux,, SIAM J. Numer. Anal., 38 (2000), 681.  doi: 10.1137/S0036142999363668.  Google Scholar [29] 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.  doi: 10.1016/S0965-8564(01)00042-8.  Google Scholar
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