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June  2014, 7(3): 543-556. doi: 10.3934/dcdss.2014.7.543

Free-congested and micro-macro descriptions of traffic flow

Francesca Marcellini 1,

1. 

Università di Milano-Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano, Italy

Received  May 2013 Revised  July 2013 Published  January 2014

We present two frameworks for the description of traffic, both consisting in the coupling of systems of different types. First, we consider the Free--Congested model [7,11], where a scalar conservation law is coupled with a $2\times2$ system. Then, we present the coupling of a micro- and a macroscopic models, the former consisting in a system of ordinary differential equations and the latter in the usual LWR conservation law, see [10]. A comparison between the two different frameworks is also provided.
Keywords: macroscopic traffic models, microscopic traffic models., Traffic models, hyperbolic systems of conservation laws, ordinary differential equations.
Mathematics Subject Classification: 35L65, 90B2.
Citation: Francesca Marcellini. Free-congested and micro-macro descriptions of traffic flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 543-556. doi: 10.3934/dcdss.2014.7.543
References:
[1]

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

[2]

A. Aw and M. Rascle, Resurrection of "second order'' models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic). doi: 10.1137/S0036139997332099.  Google Scholar

[3]

P. Bagnerini, R. M. Colombo and A. Corli, On the role of source terms in continuum traffic flow models, Math. Comput. Modelling, 44 (2006), 917-930. doi: 10.1016/j.mcm.2006.02.019.  Google Scholar

[4]

P. Bagnerini and M. Rascle, A multiclass homogenized hyperbolic model of traffic flow, SIAM J. Math. Anal., 35 (2003), 949-973 (electronic). doi: 10.1137/S0036141002411490.  Google Scholar

[5]

S. Benzoni Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Europ. J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266.  Google Scholar

[6]

S. Benzoni-Gavage, R. M. Colombo and P. Gwiazda, Measure valued solutions to conservation laws motivated by traffic modelling, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 1791-1803. doi: 10.1098/rspa.2005.1649.  Google Scholar

[7]

S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127. doi: 10.1137/090754467.  Google Scholar

[8]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721. doi: 10.1137/S0036139901393184.  Google Scholar

[9]

R. M. Colombo, P. Goatin and F. S. Priuli, Global well posedness of traffic flow models with phase transitions, Nonlinear Anal., 66 (2007), 2413-2426. doi: 10.1016/j.na.2006.03.029.  Google Scholar

[10]

R. M. Colombo and F. Marcellini, A mixed ode-pde model for vehicular traffic, {preprint}, (2013). Google Scholar

[11]

R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666. doi: 10.1137/090752468.  Google Scholar

[12]

R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759-772. doi: 10.1017/S0308210500002663.  Google Scholar

[13]

L. C. Edie, Car-following and steady-state theory for noncongested traffic, Operations Res., 9 (1961), 66-76. doi: 10.1287/opre.9.1.66.  Google Scholar

[14]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303. doi: 10.1016/j.mcm.2006.01.016.  Google Scholar

[15]

M. Godvik and H. Hanche-Olsen, Existence of solutions for the Aw-Rascle traffic flow model with vacuum, J. Hyperbolic Differ. Equ., 5 (2008), 45-63. doi: 10.1142/S0219891608001428.  Google Scholar

[16]

D. Helbing and M. Treiber, Critical discussion of synchronized flow, Cooper@tive Tr@nsport@tion Dyn@mics, 1 (2002). Google Scholar

[17]

B. S. Kerner, Phase transitions in traffic flow, in Traffic and Granular Flow '99, (eds., D. Helbing, H. Hermann, M. Schreckenberg and D. Wolf), Springer Verlag, 2000, 253-283. doi: 10.1007/978-3-642-59751-0_25.  Google Scholar

[18]

B. L. Keyfitz and H. C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory,, Arch. Rational Mech. Anal., 72 (): 219.  doi: 10.1007/BF00281590.  Google Scholar

[19]

K. M. Kockelman, Modeling traffics flow-density relation: Accommodation of multiple flow regimes and traveler types, Transportation, 28 (2001), 363-374. Google Scholar

[20]

C. Lattanzio and B. Piccoli, Coupling of microscopic and macroscopic traffic models at boundaries, Math. Models Methods Appl. Sci., 20 (2010), 2349-2370. doi: 10.1142/S0218202510004945.  Google Scholar

[21]

J. P. Lebacque, S. Mammar and H. Haj-Salem, Generic second order traffic flow modelling, in Transportation and Traffic Theory: Proceedings of the 17th International Symposium on Transportation and Traffic Theory, 2007. Google Scholar

[22]

R. J. LeVeque, Numerical Methods for Conservation Laws, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-8629-1.  Google Scholar

[23]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[24]

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

[25]

B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795. doi: 10.1090/S0002-9947-1983-0716850-2.  Google Scholar

[26]

H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

show all references

References:
[1]

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

[2]

A. Aw and M. Rascle, Resurrection of "second order'' models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic). doi: 10.1137/S0036139997332099.  Google Scholar

[3]

P. Bagnerini, R. M. Colombo and A. Corli, On the role of source terms in continuum traffic flow models, Math. Comput. Modelling, 44 (2006), 917-930. doi: 10.1016/j.mcm.2006.02.019.  Google Scholar

[4]

P. Bagnerini and M. Rascle, A multiclass homogenized hyperbolic model of traffic flow, SIAM J. Math. Anal., 35 (2003), 949-973 (electronic). doi: 10.1137/S0036141002411490.  Google Scholar

[5]

S. Benzoni Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Europ. J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266.  Google Scholar

[6]

S. Benzoni-Gavage, R. M. Colombo and P. Gwiazda, Measure valued solutions to conservation laws motivated by traffic modelling, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 1791-1803. doi: 10.1098/rspa.2005.1649.  Google Scholar

[7]

S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127. doi: 10.1137/090754467.  Google Scholar

[8]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721. doi: 10.1137/S0036139901393184.  Google Scholar

[9]

R. M. Colombo, P. Goatin and F. S. Priuli, Global well posedness of traffic flow models with phase transitions, Nonlinear Anal., 66 (2007), 2413-2426. doi: 10.1016/j.na.2006.03.029.  Google Scholar

[10]

R. M. Colombo and F. Marcellini, A mixed ode-pde model for vehicular traffic, {preprint}, (2013). Google Scholar

[11]

R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666. doi: 10.1137/090752468.  Google Scholar

[12]

R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759-772. doi: 10.1017/S0308210500002663.  Google Scholar

[13]

L. C. Edie, Car-following and steady-state theory for noncongested traffic, Operations Res., 9 (1961), 66-76. doi: 10.1287/opre.9.1.66.  Google Scholar

[14]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303. doi: 10.1016/j.mcm.2006.01.016.  Google Scholar

[15]

M. Godvik and H. Hanche-Olsen, Existence of solutions for the Aw-Rascle traffic flow model with vacuum, J. Hyperbolic Differ. Equ., 5 (2008), 45-63. doi: 10.1142/S0219891608001428.  Google Scholar

[16]

D. Helbing and M. Treiber, Critical discussion of synchronized flow, Cooper@tive Tr@nsport@tion Dyn@mics, 1 (2002). Google Scholar

[17]

B. S. Kerner, Phase transitions in traffic flow, in Traffic and Granular Flow '99, (eds., D. Helbing, H. Hermann, M. Schreckenberg and D. Wolf), Springer Verlag, 2000, 253-283. doi: 10.1007/978-3-642-59751-0_25.  Google Scholar

[18]

B. L. Keyfitz and H. C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory,, Arch. Rational Mech. Anal., 72 (): 219.  doi: 10.1007/BF00281590.  Google Scholar

[19]

K. M. Kockelman, Modeling traffics flow-density relation: Accommodation of multiple flow regimes and traveler types, Transportation, 28 (2001), 363-374. Google Scholar

[20]

C. Lattanzio and B. Piccoli, Coupling of microscopic and macroscopic traffic models at boundaries, Math. Models Methods Appl. Sci., 20 (2010), 2349-2370. doi: 10.1142/S0218202510004945.  Google Scholar

[21]

J. P. Lebacque, S. Mammar and H. Haj-Salem, Generic second order traffic flow modelling, in Transportation and Traffic Theory: Proceedings of the 17th International Symposium on Transportation and Traffic Theory, 2007. Google Scholar

[22]

R. J. LeVeque, Numerical Methods for Conservation Laws, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-8629-1.  Google Scholar

[23]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[24]

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

[25]

B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795. doi: 10.1090/S0002-9947-1983-0716850-2.  Google Scholar

[26]

H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

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