American Institute of Mathematical Sciences

Advanced
September  2013, 8(3): 649-661. doi: 10.3934/nhm.2013.8.649

Coupling of microscopic and phase transition models at boundary

Mauro Garavello 1, and  Benedetto Piccoli 2,

1. 

Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 53 - Edificio U5, 20125 - Milano
2. 

Joseph and Loretta Lopez Chair Professor of Mathematics, Department of Mathematical Sciences and Program Director, Center for Computational and Integrative Biology, Rutgers University - Camden, 311 N 5th Street, Camden, NJ 08102

Received  March 2013 Revised  July 2013 Published  October 2013

This paper deals with coupling conditions between the classical microscopic Follow The Leader model and a phase transition (PT) model. We propose a solution at the interface between the two models. We describe the solution to the Riemann problem.
Keywords: coupling conditions, Follow the Leader, conservation laws, Cauchy Problem., phase transition model.
Mathematics Subject Classification: Primary: 90B20; Secondary: 35L6.
Citation: Mauro Garavello, Benedetto Piccoli. Coupling of microscopic and phase transition models at boundary. Networks & Heterogeneous Media, 2013, 8 (3) : 649-661. doi: 10.3934/nhm.2013.8.649
References:
[1]

D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws,, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 1.  doi: 10.1007/PL00001406.  Google Scholar

[2]

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.  doi: 10.1137/S0036139900380955.  Google Scholar

[3]

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

[4]

S. Blandin, J. Argote, A. M. Bayen and D. B. Work, Phase transition model of non-stationary traffic flow: Definition, properties and solution method,, Transportation Research Part B: Methodological, 52 (2013), 31.   Google Scholar

[5]

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.  doi: 10.1137/090754467.  Google Scholar

[6]

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

[7]

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.  doi: 10.1137/090752468.  Google Scholar

[8]

C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow,, Transportation Research Part B, 29 (1995), 277.  doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar

[9]

F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws,, J. Differential Equations, 71 (1988), 93.  doi: 10.1016/0022-0396(88)90040-X.  Google Scholar

[10]

M. Garavello and B. Piccoli, Coupling of lwr and phase transition models at boundary,, Journal of Hyperbolic Differential Equations, 10 (2013), 577.   Google Scholar

[11]

D. C. Gazis, R. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow,, Operations Res., 9 (1961), 545.  doi: 10.1287/opre.9.4.545.  Google Scholar

[12]

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

[13]

D. Helbing, From microscopic to macroscopic traffic models,, in, (1998), 122.  doi: 10.1007/BFb0104959.  Google Scholar

[14]

D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro- and macro-simulation of freeway traffic,, Traffic flow-modelling and simulation, 35 (2002), 517.  doi: 10.1016/S0895-7177(02)80019-X.  Google Scholar

[15]

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

[16]

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.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[17]

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

[18]

H. J. Payne, Models of freeway traffic and control, in mathematical models of public systems,, Simul. Counc. Proc., 1 (1971).   Google Scholar

[19]

I. Prigogine and R. Herman, Kinetic theory of vehicular traffic,, American Elsevier Pub. Co., (1971).   Google Scholar

[20]

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

[21]

G. B. Whitham, "Linear and Nonlinear Waves,'', Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], (1974).   Google Scholar

[22]

D. B. Work, S. Blandin, O.-P. Tossavainen, B. Piccoli and A. M. Bayen, A traffic model for velocity data assimilation,, Appl. Math. Res. Express. AMRX, 1 (2010), 1.   Google Scholar

[23]

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

show all references

References:
[1]

D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws,, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 1.  doi: 10.1007/PL00001406.  Google Scholar

[2]

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.  doi: 10.1137/S0036139900380955.  Google Scholar

[3]

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

[4]

S. Blandin, J. Argote, A. M. Bayen and D. B. Work, Phase transition model of non-stationary traffic flow: Definition, properties and solution method,, Transportation Research Part B: Methodological, 52 (2013), 31.   Google Scholar

[5]

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.  doi: 10.1137/090754467.  Google Scholar

[6]

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

[7]

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.  doi: 10.1137/090752468.  Google Scholar

[8]

C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow,, Transportation Research Part B, 29 (1995), 277.  doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar

[9]

F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws,, J. Differential Equations, 71 (1988), 93.  doi: 10.1016/0022-0396(88)90040-X.  Google Scholar

[10]

M. Garavello and B. Piccoli, Coupling of lwr and phase transition models at boundary,, Journal of Hyperbolic Differential Equations, 10 (2013), 577.   Google Scholar

[11]

D. C. Gazis, R. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow,, Operations Res., 9 (1961), 545.  doi: 10.1287/opre.9.4.545.  Google Scholar

[12]

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

[13]

D. Helbing, From microscopic to macroscopic traffic models,, in, (1998), 122.  doi: 10.1007/BFb0104959.  Google Scholar

[14]

D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro- and macro-simulation of freeway traffic,, Traffic flow-modelling and simulation, 35 (2002), 517.  doi: 10.1016/S0895-7177(02)80019-X.  Google Scholar

[15]

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

[16]

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.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[17]

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

[18]

H. J. Payne, Models of freeway traffic and control, in mathematical models of public systems,, Simul. Counc. Proc., 1 (1971).   Google Scholar

[19]

I. Prigogine and R. Herman, Kinetic theory of vehicular traffic,, American Elsevier Pub. Co., (1971).   Google Scholar

[20]

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

[21]

G. B. Whitham, "Linear and Nonlinear Waves,'', Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], (1974).   Google Scholar

[22]

D. B. Work, S. Blandin, O.-P. Tossavainen, B. Piccoli and A. M. Bayen, A traffic model for velocity data assimilation,, Appl. Math. Res. Express. AMRX, 1 (2010), 1.   Google Scholar

[23]

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

[1]

Marco Di Francesco, Graziano Stivaletta. Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 233-266. doi: 10.3934/dcds.2020010
[2]

Elena Rossi. A justification of a LWR model based on a follow the leader description. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 579-591. doi: 10.3934/dcdss.2014.7.579
[3]

Pierluigi Colli, Antonio Segatti. Uniform attractors for a phase transition model coupling momentum balance and phase dynamics. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 909-932. doi: 10.3934/dcds.2008.22.909
[4]

Jérôme Fehrenbach, Jacek Narski, Jiale Hua, Samuel Lemercier, Asja Jelić, Cécile Appert-Rolland, Stéphane Donikian, Julien Pettré, Pierre Degond. Time-delayed follow-the-leader model for pedestrians walking in line. Networks & Heterogeneous Media, 2015, 10 (3) : 579-608. doi: 10.3934/nhm.2015.10.579
[5]

João-Paulo Dias, Mário Figueira. On the Riemann problem for some discontinuous systems of conservation laws describing phase transitions. Communications on Pure & Applied Analysis, 2004, 3 (1) : 53-58. doi: 10.3934/cpaa.2004.3.53
[6]

Mauro Garavello. Boundary value problem for a phase transition model. Networks & Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89
[7]

Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225
[8]

Laurent Lévi, Julien Jimenez. Coupling of scalar conservation laws in stratified porous media. Conference Publications, 2007, 2007 (Special) : 644-654. doi: 10.3934/proc.2007.2007.644
[9]

Wen Shen. Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition. Networks & Heterogeneous Media, 2018, 13 (3) : 449-478. doi: 10.3934/nhm.2018020
[10]

Helge Holden, Nils Henrik Risebro. Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Networks & Heterogeneous Media, 2018, 13 (3) : 409-421. doi: 10.3934/nhm.2018018
[11]

Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure & Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755
[12]

Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011
[13]

Martin Gugat, Michael Herty, Siegfried Müller. Coupling conditions for the transition from supersonic to subsonic fluid states. Networks & Heterogeneous Media, 2017, 12 (3) : 371-380. doi: 10.3934/nhm.2017016
[14]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185
[15]

Helge Holden, Nils Henrik Risebro. The continuum limit of Follow-the-Leader models — a short proof. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 715-722. doi: 10.3934/dcds.2018031
[16]

Darko Mitrovic. New entropy conditions for scalar conservation laws with discontinuous flux. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1191-1210. doi: 10.3934/dcds.2011.30.1191
[17]

Jun Yang. Coexistence phenomenon of concentration and transition of an inhomogeneous phase transition model on surfaces. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 965-994. doi: 10.3934/dcds.2011.30.965
[18]

Pavel Krejčí, Jürgen Sprekels. Long time behaviour of a singular phase transition model. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1119-1135. doi: 10.3934/dcds.2006.15.1119
[19]

Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh. Emergence of aggregation in the swarm sphere model with adaptive coupling laws. Kinetic & Related Models, 2019, 12 (2) : 411-444. doi: 10.3934/krm.2019018
[20]

Boris Andreianov, Mohamed Karimou Gazibo. Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 203-222. doi: 10.3934/nhm.2016.11.203

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences