September  2016, 11(3): 395-413. doi: 10.3934/nhm.2016002

On the micro-to-macro limit for first-order traffic flow models on networks

1. 

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

2. 

Dipartimento di Matematica "G. Castelnuovo", Sapienza, Università di Roma, Rome, Italy

Received  May 2015 Revised  August 2015 Published  August 2016

Connections between microscopic follow-the-leader and macroscopic fluid-dynamics traffic flow models are already well understood in the case of vehicles moving on a single road. Analogous connections in the case of road networks are instead lacking. This is probably due to the fact that macroscopic traffic models on networks are in general ill-posed, since the conservation of the mass is not sufficient alone to characterize a unique solution at junctions. This ambiguity makes more difficult to find the right limit of the microscopic model, which, in turn, can be defined in different ways near the junctions. In this paper we show that a natural extension of the first-order follow-the-leader model on networks corresponds, as the number of vehicles tends to infinity, to the LWR-based multi-path model introduced in [4,5].
Citation: Emiliano Cristiani, Smita Sahu. On the micro-to-macro limit for first-order traffic flow models on networks. Networks and Heterogeneous Media, 2016, 11 (3) : 395-413. doi: 10.3934/nhm.2016002
References:
[1]

A. Aw, A. Klar, M. Rascle and T. Materne, Derivation of continuum flow traffic models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955.

[2]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677.

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, volume 20 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2000.

[4]

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.

[5]

M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Theoretical study, Netw. Heterog. Media, 9 (2014), 519-552. doi: 10.3934/nhm.2014.9.519.

[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. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, Rend. Sem. Mat. Univ. Padova, 131 (2014), 217-235. doi: 10.4171/RSMUP/131-13.

[8]

G. Costeseque, Analyse et modelisation du trafic routier: Passage du microscopique au macroscopique, Master thesis, Ecole des Ponts Paris-Tech, 2011.

[9]

E. Cristiani and F. S. Priuli, A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks, Netw. Heterog. Media, 10 (2015), 857-876. doi: 10.3934/nhm.2015.10.857.

[10]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Rational Mech. Anal., 217 (2015), 831-871. doi: 10.1007/s00205-015-0843-4.

[11]

M. Fellendorf and P. Vortisch, Microscopic traffic flow simulator VISSIM, In: J. Barceló (Ed.), Fundamentals of traffic simulation, International Series in Operations Research & Management Science, 145 (2010), 63-93. doi: 10.1007/978-1-4419-6142-6_2.

[12]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to traffic flow on road networks, Math. Models Methods Appl. Sci., 25 (2015), 423-461. doi: 10.1142/S0218202515400023.

[13]

L. Fermo and A. Tosin, Fundamental diagrams for kinetic equations of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 449-462. doi: 10.3934/dcdss.2014.7.449.

[14]

N. Forcadel and W. Salazar, A junction condition by specified homogenization of a discrete model with a local perturbation and application to traffic flow, preprint, HAL-01097085, 2014.

[15]

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.

[16]

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.

[17]

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

[18]

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.

[19]

J. M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM J. Appl. Math., 62 (2001), 729-745. doi: 10.1137/S0036139900378657.

[20]

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

[21]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.

[22]

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

[23]

M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM J. Sci. Comput., 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X.

[24]

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.

[25]

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.

[26]

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.

[27]

S. Moutari and M. Rascle, A hybrid Lagrangian model based on the Aw-Rascle traffic flow model, SIAM J. Appl. Math., 68 (2007), 413-436. doi: 10.1137/060678415.

[28]

L. A. Pipes, An operational analysis of traffic dynamics, J. Appl. Phys., 24 (1953), 274-281. doi: 10.1063/1.1721265.

[29]

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

[30]

E. Rossi, A justification of a LWR model based on a follow the leader description, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 579-591. doi: 10.3934/dcdss.2014.7.579.

[31]

J. Shen and X. Jin, Detailed traffic animation for urban road networks, Graphical Models, 74 (2012), 265-282. doi: 10.1016/j.gmod.2012.04.002.

show all references

References:
[1]

A. Aw, A. Klar, M. Rascle and T. Materne, Derivation of continuum flow traffic models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955.

[2]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677.

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, volume 20 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2000.

[4]

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.

[5]

M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Theoretical study, Netw. Heterog. Media, 9 (2014), 519-552. doi: 10.3934/nhm.2014.9.519.

[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. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, Rend. Sem. Mat. Univ. Padova, 131 (2014), 217-235. doi: 10.4171/RSMUP/131-13.

[8]

G. Costeseque, Analyse et modelisation du trafic routier: Passage du microscopique au macroscopique, Master thesis, Ecole des Ponts Paris-Tech, 2011.

[9]

E. Cristiani and F. S. Priuli, A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks, Netw. Heterog. Media, 10 (2015), 857-876. doi: 10.3934/nhm.2015.10.857.

[10]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Rational Mech. Anal., 217 (2015), 831-871. doi: 10.1007/s00205-015-0843-4.

[11]

M. Fellendorf and P. Vortisch, Microscopic traffic flow simulator VISSIM, In: J. Barceló (Ed.), Fundamentals of traffic simulation, International Series in Operations Research & Management Science, 145 (2010), 63-93. doi: 10.1007/978-1-4419-6142-6_2.

[12]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to traffic flow on road networks, Math. Models Methods Appl. Sci., 25 (2015), 423-461. doi: 10.1142/S0218202515400023.

[13]

L. Fermo and A. Tosin, Fundamental diagrams for kinetic equations of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 449-462. doi: 10.3934/dcdss.2014.7.449.

[14]

N. Forcadel and W. Salazar, A junction condition by specified homogenization of a discrete model with a local perturbation and application to traffic flow, preprint, HAL-01097085, 2014.

[15]

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.

[16]

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.

[17]

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

[18]

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.

[19]

J. M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM J. Appl. Math., 62 (2001), 729-745. doi: 10.1137/S0036139900378657.

[20]

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

[21]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.

[22]

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

[23]

M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM J. Sci. Comput., 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X.

[24]

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.

[25]

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.

[26]

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.

[27]

S. Moutari and M. Rascle, A hybrid Lagrangian model based on the Aw-Rascle traffic flow model, SIAM J. Appl. Math., 68 (2007), 413-436. doi: 10.1137/060678415.

[28]

L. A. Pipes, An operational analysis of traffic dynamics, J. Appl. Phys., 24 (1953), 274-281. doi: 10.1063/1.1721265.

[29]

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

[30]

E. Rossi, A justification of a LWR model based on a follow the leader description, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 579-591. doi: 10.3934/dcdss.2014.7.579.

[31]

J. Shen and X. Jin, Detailed traffic animation for urban road networks, Graphical Models, 74 (2012), 265-282. doi: 10.1016/j.gmod.2012.04.002.

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