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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 and 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), 1-42. doi: 10.1007/PL00001406.

[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), 259-278. doi: 10.1137/S0036139900380955.

[3]

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

[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), 31-55.

[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), 107-127. doi: 10.1137/090754467.

[6]

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

[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), 2652-2666. doi: 10.1137/090752468.

[8]

C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.

[9]

F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122. doi: 10.1016/0022-0396(88)90040-X.

[10]

M. Garavello and B. Piccoli, Coupling of lwr and phase transition models at boundary, Journal of Hyperbolic Differential Equations, 10 (2013), 577-636.

[11]

D. C. Gazis, R. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Res., 9 (1961), 545-567. doi: 10.1287/opre.9.4.545.

[12]

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.

[13]

D. Helbing, From microscopic to macroscopic traffic models, in "A perspective look at nonlinear media,'' Lecture Notes in Phys., Springer, (1998), 122-139. doi: 10.1007/BFb0104959.

[14]

D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro- and macro-simulation of freeway traffic, Traffic flow-modelling and simulation, Math. Comput. Modelling, 35 (2002), 517-547. doi: 10.1016/S0895-7177(02)80019-X.

[15]

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.

[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), 317-345. doi: 10.1098/rspa.1955.0089.

[17]

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

[18]

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

[19]

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

[20]

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

[21]

G. B. Whitham, "Linear and Nonlinear Waves,'' Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[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), 1-35.

[23]

H. M. 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]

D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 1-42. doi: 10.1007/PL00001406.

[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), 259-278. doi: 10.1137/S0036139900380955.

[3]

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

[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), 31-55.

[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), 107-127. doi: 10.1137/090754467.

[6]

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

[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), 2652-2666. doi: 10.1137/090752468.

[8]

C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.

[9]

F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122. doi: 10.1016/0022-0396(88)90040-X.

[10]

M. Garavello and B. Piccoli, Coupling of lwr and phase transition models at boundary, Journal of Hyperbolic Differential Equations, 10 (2013), 577-636.

[11]

D. C. Gazis, R. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Res., 9 (1961), 545-567. doi: 10.1287/opre.9.4.545.

[12]

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.

[13]

D. Helbing, From microscopic to macroscopic traffic models, in "A perspective look at nonlinear media,'' Lecture Notes in Phys., Springer, (1998), 122-139. doi: 10.1007/BFb0104959.

[14]

D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro- and macro-simulation of freeway traffic, Traffic flow-modelling and simulation, Math. Comput. Modelling, 35 (2002), 517-547. doi: 10.1016/S0895-7177(02)80019-X.

[15]

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.

[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), 317-345. doi: 10.1098/rspa.1955.0089.

[17]

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

[18]

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

[19]

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

[20]

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

[21]

G. B. Whitham, "Linear and Nonlinear Waves,'' Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[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), 1-35.

[23]

H. M. 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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