American Institute of Mathematical Sciences

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

Free-congested and micro-macro descriptions of traffic flow

 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.
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.
 [1] Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks and Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255 [2] Gabriella Puppo, Matteo Semplice, Andrea Tosin, Giuseppe Visconti. Kinetic models for traffic flow resulting in a reduced space of microscopic velocities. Kinetic and Related Models, 2017, 10 (3) : 823-854. doi: 10.3934/krm.2017033 [3] Matteo Piu, Gabriella Puppo. Stability analysis of microscopic models for traffic flow with lane changing. Networks and Heterogeneous Media, 2022  doi: 10.3934/nhm.2022006 [4] Michael Herty, Reinhard Illner. Analytical and numerical investigations of refined macroscopic traffic flow models. Kinetic and Related Models, 2010, 3 (2) : 311-333. doi: 10.3934/krm.2010.3.311 [5] Paola Goatin, Elena Rossi. Comparative study of macroscopic traffic flow models at road junctions. Networks and Heterogeneous Media, 2020, 15 (2) : 261-279. doi: 10.3934/nhm.2020012 [6] Michael Herty, Gabriella Puppo, Sebastiano Roncoroni, Giuseppe Visconti. The BGK approximation of kinetic models for traffic. Kinetic and Related Models, 2020, 13 (2) : 279-307. doi: 10.3934/krm.2020010 [7] Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161 [8] Tong Li. Qualitative analysis of some PDE models of traffic flow. Networks and Heterogeneous Media, 2013, 8 (3) : 773-781. doi: 10.3934/nhm.2013.8.773 [9] Paola Goatin. Traffic flow models with phase transitions on road networks. Networks and Heterogeneous Media, 2009, 4 (2) : 287-301. doi: 10.3934/nhm.2009.4.287 [10] Mauro Garavello, Benedetto Piccoli. On fluido-dynamic models for urban traffic. Networks and Heterogeneous Media, 2009, 4 (1) : 107-126. doi: 10.3934/nhm.2009.4.107 [11] Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic and Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165 [12] Michael Burger, Simone Göttlich, Thomas Jung. Derivation of second order traffic flow models with time delays. Networks and Heterogeneous Media, 2019, 14 (2) : 265-288. doi: 10.3934/nhm.2019011 [13] Felisia Angela Chiarello, Paola Goatin. Non-local multi-class traffic flow models. Networks and Heterogeneous Media, 2019, 14 (2) : 371-387. doi: 10.3934/nhm.2019015 [14] Bertrand Haut, Georges Bastin. A second order model of road junctions in fluid models of traffic networks. Networks and Heterogeneous Media, 2007, 2 (2) : 227-253. doi: 10.3934/nhm.2007.2.227 [15] Michael Herty, Lorenzo Pareschi, Mohammed Seaïd. Enskog-like discrete velocity models for vehicular traffic flow. Networks and Heterogeneous Media, 2007, 2 (3) : 481-496. doi: 10.3934/nhm.2007.2.481 [16] Simone Göttlich, Oliver Kolb, Sebastian Kühn. Optimization for a special class of traffic flow models: Combinatorial and continuous approaches. Networks and Heterogeneous Media, 2014, 9 (2) : 315-334. doi: 10.3934/nhm.2014.9.315 [17] Sharif E. Guseynov, Shirmail G. Bagirov. Distributed mathematical models of undetermined "without preference" motion of traffic flow. Conference Publications, 2011, 2011 (Special) : 589-600. doi: 10.3934/proc.2011.2011.589 [18] Andrea Tosin, Mattia Zanella. Uncertainty damping in kinetic traffic models by driver-assist controls. Mathematical Control and Related Fields, 2021, 11 (3) : 681-713. doi: 10.3934/mcrf.2021018 [19] Guillaume Costeseque, Jean-Patrick Lebacque. Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 411-433. doi: 10.3934/dcdss.2014.7.411 [20] Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks and Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028

2021 Impact Factor: 1.865

Metrics

• HTML views (0)
• Cited by (7)

Other articlesby authors

• on AIMS
• on Google Scholar