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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 and 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), 27-86. 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]

A. Bressan and K. T. Nguyen, Conservation law models for traffic flow on a network of roads, Preprint, 2014.

[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), 379-394. doi: 10.3934/dcdss.2014.7.379.

[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]

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.

[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), 54-71. doi: 10.1685/2010CAIM487.

[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), 269-287. doi: 10.1016/0191-2615(94)90002-7.

[10]

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.

[11]

M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915.

[12]

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.

[13]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, 2006.

[14]

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.

[15]

R. Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics and Traffic Flow, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, USA, 1977.

[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, C. Kirchner, S. Moutari and M. Rascle, Multicommodity flows on road networks, Commun. Math. Sci., 6 (2008), 171-187. doi: 10.4310/CMS.2008.v6.n1.a8.

[18]

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.

[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), 361-373. doi: 10.1016/j.apnum.2006.04.003.

[20]

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.

[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]

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), (1996), 647-677.

[23]

R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi: 10.1007/978-3-0348-8629-1.

[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), 317-345. doi: 10.1098/rspa.1955.0089.

[25]

M. Mercier, Traffic flow modelling with junctions, J. Math. Anal. Appl., 350 (2009), 369-383. doi: 10.1016/j.jmaa.2008.09.040.

[26]

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

[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), 289-309.

[28]

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.

[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), 827-841. doi: 10.1016/S0965-8564(01)00042-8.

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), 27-86. 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]

A. Bressan and K. T. Nguyen, Conservation law models for traffic flow on a network of roads, Preprint, 2014.

[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), 379-394. doi: 10.3934/dcdss.2014.7.379.

[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]

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.

[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), 54-71. doi: 10.1685/2010CAIM487.

[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), 269-287. doi: 10.1016/0191-2615(94)90002-7.

[10]

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.

[11]

M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915.

[12]

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.

[13]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, 2006.

[14]

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.

[15]

R. Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics and Traffic Flow, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, USA, 1977.

[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, C. Kirchner, S. Moutari and M. Rascle, Multicommodity flows on road networks, Commun. Math. Sci., 6 (2008), 171-187. doi: 10.4310/CMS.2008.v6.n1.a8.

[18]

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.

[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), 361-373. doi: 10.1016/j.apnum.2006.04.003.

[20]

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.

[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]

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), (1996), 647-677.

[23]

R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi: 10.1007/978-3-0348-8629-1.

[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), 317-345. doi: 10.1098/rspa.1955.0089.

[25]

M. Mercier, Traffic flow modelling with junctions, J. Math. Anal. Appl., 350 (2009), 369-383. doi: 10.1016/j.jmaa.2008.09.040.

[26]

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

[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), 289-309.

[28]

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.

[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), 827-841. doi: 10.1016/S0965-8564(01)00042-8.

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