Coupling of microscopic and phase transition models at boundary

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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

Abstract
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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

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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

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