# American Institute of Mathematical Sciences

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

## Coupling of microscopic and phase transition models at boundary

 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.
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.
 [1] Marco Di Francesco, Graziano Stivaletta. Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux. Discrete and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and 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 and Applied Analysis, 2004, 3 (1) : 53-58. doi: 10.3934/cpaa.2004.3.53 [6] 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 [7] Mauro Garavello. Boundary value problem for a phase transition model. Networks and Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89 [8] Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225 [9] Wen Shen. Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition. Networks and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2018, 38 (2) : 715-722. doi: 10.3934/dcds.2018031 [16] Darko Mitrovic. New entropy conditions for scalar conservation laws with discontinuous flux. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1191-1210. doi: 10.3934/dcds.2011.30.1191 [17] Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh. Emergence of aggregation in the swarm sphere model with adaptive coupling laws. Kinetic and Related Models, 2019, 12 (2) : 411-444. doi: 10.3934/krm.2019018 [18] Jun Yang. Coexistence phenomenon of concentration and transition of an inhomogeneous phase transition model on surfaces. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 965-994. doi: 10.3934/dcds.2011.30.965 [19] Pavel Krejčí, Jürgen Sprekels. Long time behaviour of a singular phase transition model. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1119-1135. doi: 10.3934/dcds.2006.15.1119 [20] Zhen Cheng, Wenjun Wang. The Cauchy problem of a two-phase flow model for a mixture of non-interacting compressible fluids. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4155-4176. doi: 10.3934/cpaa.2021151

2021 Impact Factor: 1.41