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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

B. L. Keyfitz and H. C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal., 72 (1979/80), 219-241. doi: 10.1007/BF00281590.

[19]

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

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

B. L. Keyfitz and H. C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal., 72 (1979/80), 219-241. doi: 10.1007/BF00281590.

[19]

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

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

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