April  2020, 13(2): 279-307. doi: 10.3934/krm.2020010

The BGK approximation of kinetic models for traffic

1. 

RWTH Aachen University, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52062 Aachen, Germany

2. 

"La Sapienza" Università di Roma, Dipartimento di Matematica, Piazza Aldo Moro 5, 00185 Roma, Italy

3. 

University of Reading, Department of Mathematics and Statistics, Exhibition Road, London SW7 2AZ, UK

* Corresponding author: Giuseppe Visconti

Received  December 2018 Revised  July 2019 Published  January 2020

We study spatially non-homogeneous kinetic models for vehicular traffic flow. Classical formulations, as for instance the BGK equation, lead to unconditionally unstable solutions in the congested regime of traffic. We address this issue by deriving a modified formulation of the BGK-type equation. The new kinetic model allows to reproduce conditionally stable non-equilibrium phenomena in traffic flow. In particular, stop and go waves appear as bounded backward propagating signals occurring in bounded regimes of the density where the model is unstable. The BGK-type model introduced here also offers the mesoscopic description between the microscopic follow-the-leader model and the macroscopic Aw-Rascle and Zhang model.

Citation: Michael Herty, Gabriella Puppo, Sebastiano Roncoroni, Giuseppe Visconti. The BGK approximation of kinetic models for traffic. Kinetic & Related Models, 2020, 13 (2) : 279-307. doi: 10.3934/krm.2020010
References:
[1]

A. AwA. KlarT. 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.  Google Scholar

[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.  Google Scholar

[3]

M. BandoK. HasebeA. NakayamaA. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), 1035-1042.   Google Scholar

[4]

P. L. BhatnagarE. P. Gross and M. Krook, A model for collision processes in gases. Ⅰ. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.  Google Scholar

[5] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford University Press, 2000.   Google Scholar
[6]

G. Q. ChenC. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math, 47 (1992), 787-830.  doi: 10.1002/cpa.3160470602.  Google Scholar

[7]

A. Corli and L. Malaguti, Viscous profiles in models of collective movements with negative diffusivities, Z. Angew. Math. Phys., 70 (2019), Art. 47, 22 pp, arXiv1806.00652. doi: 10.1007/s00033-019-1094-2.  Google Scholar

[8]

C. F. Daganzo, Requiem for second-order fluid approximation to traffic flow, Transport. Res. B-Meth., 29 (1995), 277-286.  doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar

[9]

M. Delitala and A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), 901-932.  doi: 10.1142/S0218202507002157.  Google Scholar

[10]

M. Di FrancescoS. Fagioli and Ro sini, Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic, Math. Biosci. Eng., 14 (2017), 127-141.  doi: 10.3934/mbe.2017009.  Google Scholar

[11]

M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, Follow-the-leader approximations of macroscopic models for vehicular and pedestrian flows, in Active Particles. Vol. 1. Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2017, 333–378.  Google Scholar

[12]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to modeling vehicular traffic, SIAM J. Appl. Math., 73 (2013), 1533-1556.  doi: 10.1137/120897110.  Google Scholar

[13]

L. Fermo and A. Tosin, Fundamental diagrams for kinetic equations of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 449-462.  doi: 10.3934/dcdss.2014.7.449.  Google Scholar

[14]

F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 7625-7648.  doi: 10.1016/j.jcp.2010.06.017.  Google Scholar

[15]

D. GazisR. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow, Oper. Res., 9 (1961), 545-567.  doi: 10.1287/opre.9.4.545.  Google Scholar

[16]

B. H. Gilding and A. Tesei, The Riemann problem for a forward-backward parabolic equation, Phys. D, 239 (2010), 291-311.  doi: 10.1016/j.physd.2009.10.006.  Google Scholar

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M. HertyS. Moutari and G. Visconti, Macroscopic modeling of multilane motorways using a two-dimensional second-order model of traffic flow, SIAM J. Appl. Math., 78 (2018), 2252-2278.  doi: 10.1137/17M1151821.  Google Scholar

[18]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179.  doi: 10.3934/krm.2010.3.165.  Google Scholar

[19]

M. HertyL. Pareschi and M. Seaid, Discrete velocity models and relaxation schemes for traffic flows, SIAM J. Sci. Comput., 28 (2006), 1582-1596.  doi: 10.1137/04061982X.  Google Scholar

[20]

H. Holden and N. H. Risebro, The continuum limit of Follow-the-Leader models – a short proof, Discrete Cont. Dyn-A, 38 (2018), 715-722.  doi: 10.3934/dcds.2018031.  Google Scholar

[21]

N. H. Risebro and H. Holden, Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow, Netw. Heterog. Media, 13 (2018), 409-421.  doi: 10.3934/nhm.2018018.  Google Scholar

[22]

R. IllnerA. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1-12.  doi: 10.4310/CMS.2003.v1.n1.a1.  Google Scholar

[23]

S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math, 48 (1995), 235-276.  doi: 10.1002/cpa.3160480303.  Google Scholar

[24]

B. S. Kerner, The Physics of Traffic, Understanding Complex Systems, Springer, Berlin, 2004. doi: 10.1007/978-3-540-40986-1.  Google Scholar

[25]

T. Kim and H. M. Zhang, A stochastic wave propagation model, Transport. Res. B-Meth., 42 (2008), 619-634.  doi: 10.1016/j.trb.2007.12.002.  Google Scholar

[26]

A. Klar and R. Wegener, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport. Theor. Stat., 25 (1996), 785-798.   Google Scholar

[27]

A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic, J. Stat. Phys., 87 (1997), 91-114.  doi: 10.1007/BF02181481.  Google Scholar

[28]

P. Lafitte and C. Mascia, Numerical exploration of a forward-backward diffusion equation, Math. Models Methods Appl. Sci., 22 (2012), 1250004, 33pp. doi: 10.1142/S0218202512500042.  Google Scholar

[29]

R. J. LeVeque, Numerical Methods for Conservation Laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-5116-9.  Google Scholar

[30]

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.  Google Scholar

[31]

M. Lo Schiavo, A personalized kinetic model of traffic flow, Math. Comput. Modelling, 35 (2002), 607-622.  doi: 10.1016/S0895-7177(02)80024-3.  Google Scholar

[32]

C. MasciaA. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation, Arch. Ration. Mech. Anal., 194 (2009), 887-925.  doi: 10.1007/s00205-008-0185-6.  Google Scholar

[33]

A. R. Méndez and R. M. Velasco, Kerner's free-synchronized phase transition in a macroscopic traffic flow model with two classes of drivers, J. Phys. A: Math. Theor., 46 (2013), 462001, 9 pp. doi: 10.1088/1751-8113/46/46/462001.  Google Scholar

[34]

L. Pareschi and G. Russo, Implicit-explicit runge-kutta methods and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005), 129-155.  doi: 10.1007/s10915-004-4636-4.  Google Scholar

[35]

L. Pareschi and G. Russo, Efficient Asymptotic Preserving Deterministic Methods for the Boltzmann Equation, Lecture series, von Karman Institute, Rhode St. Genèse, Belgium, 2011, AVT-194 RTO AVT/VKI, Models and Computational Methods for Rarefied Flows. Google Scholar

[36] L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013.   Google Scholar
[37]

S. L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: a critical review of the basic model and an alternative proposal for dilute traffic analysis,, Transport. Res., 9 (1975), 225-235.  doi: 10.1016/0041-1647(75)90063-5.  Google Scholar

[38]

H. J. Payne, Models of freeway traffic and control, Math. Models Publ. Sys., Simulation Council Proc. 28, 1 (1971), 51-61.   Google Scholar

[39]

S. Pieraccini and G. Puppo, Implicit-explicit schemes for BGK kinetic equations, J. Sci. Comput., 32 (2006), 1-28.  doi: 10.1007/s10915-006-9116-6.  Google Scholar

[40]

I. Prigogine, A Boltzmann-like approach to the statistical theory of traffic flow, in Theory of Traffic Flow (ed. R. Herman), Elsevier, Amsterdam, 1961, 158–164.  Google Scholar

[41]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Co., New York, 1971. Google Scholar

[42]

G. PuppoM. SempliceA. Tosin and G. Visconti, Analysis of a multi-population kinetic model for traffic flow, Commun. Math. Sci., 15 (2017), 379-412.  doi: 10.4310/CMS.2017.v15.n2.a5.  Google Scholar

[43]

G. PuppoM. SempliceA. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Mod., 10 (2017), 823-854.  doi: 10.3934/krm.2017033.  Google Scholar

[44]

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

[45]

S. Roncoroni, Kinetic Modelling of Vehicular Traffic Flow, Technical report, Università degli Studi dell'Insubria, 2017, Master Thesis. Google Scholar

[46]

B. SeiboldM. R. FlynnA. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models, Netw. Heterog. Media, 8 (2013), 745-772.  doi: 10.3934/nhm.2013.8.745.  Google Scholar

[47]

G. B. Whitham, Some comments on wave propagation and shock wave structure with application to magnetohydrodynamics, Comm. Pure Appl. Math., 12 (1959), 113-158.  doi: 10.1002/cpa.3160120107.  Google Scholar

[48]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transport. Res. B-Meth., 36 (2002), 275-290.  doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

[49]

H. M. Zhang and T. Kim, A car-following theory for multiphase vehicular traffic flow, Transport. Res. B-Meth., 39 (2005), 385-399.  doi: 10.1016/j.trb.2004.06.005.  Google Scholar

show all references

References:
[1]

A. AwA. KlarT. 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.  Google Scholar

[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.  Google Scholar

[3]

M. BandoK. HasebeA. NakayamaA. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), 1035-1042.   Google Scholar

[4]

P. L. BhatnagarE. P. Gross and M. Krook, A model for collision processes in gases. Ⅰ. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.  Google Scholar

[5] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford University Press, 2000.   Google Scholar
[6]

G. Q. ChenC. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math, 47 (1992), 787-830.  doi: 10.1002/cpa.3160470602.  Google Scholar

[7]

A. Corli and L. Malaguti, Viscous profiles in models of collective movements with negative diffusivities, Z. Angew. Math. Phys., 70 (2019), Art. 47, 22 pp, arXiv1806.00652. doi: 10.1007/s00033-019-1094-2.  Google Scholar

[8]

C. F. Daganzo, Requiem for second-order fluid approximation to traffic flow, Transport. Res. B-Meth., 29 (1995), 277-286.  doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar

[9]

M. Delitala and A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), 901-932.  doi: 10.1142/S0218202507002157.  Google Scholar

[10]

M. Di FrancescoS. Fagioli and Ro sini, Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic, Math. Biosci. Eng., 14 (2017), 127-141.  doi: 10.3934/mbe.2017009.  Google Scholar

[11]

M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, Follow-the-leader approximations of macroscopic models for vehicular and pedestrian flows, in Active Particles. Vol. 1. Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2017, 333–378.  Google Scholar

[12]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to modeling vehicular traffic, SIAM J. Appl. Math., 73 (2013), 1533-1556.  doi: 10.1137/120897110.  Google Scholar

[13]

L. Fermo and A. Tosin, Fundamental diagrams for kinetic equations of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 449-462.  doi: 10.3934/dcdss.2014.7.449.  Google Scholar

[14]

F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 7625-7648.  doi: 10.1016/j.jcp.2010.06.017.  Google Scholar

[15]

D. GazisR. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow, Oper. Res., 9 (1961), 545-567.  doi: 10.1287/opre.9.4.545.  Google Scholar

[16]

B. H. Gilding and A. Tesei, The Riemann problem for a forward-backward parabolic equation, Phys. D, 239 (2010), 291-311.  doi: 10.1016/j.physd.2009.10.006.  Google Scholar

[17]

M. HertyS. Moutari and G. Visconti, Macroscopic modeling of multilane motorways using a two-dimensional second-order model of traffic flow, SIAM J. Appl. Math., 78 (2018), 2252-2278.  doi: 10.1137/17M1151821.  Google Scholar

[18]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179.  doi: 10.3934/krm.2010.3.165.  Google Scholar

[19]

M. HertyL. Pareschi and M. Seaid, Discrete velocity models and relaxation schemes for traffic flows, SIAM J. Sci. Comput., 28 (2006), 1582-1596.  doi: 10.1137/04061982X.  Google Scholar

[20]

H. Holden and N. H. Risebro, The continuum limit of Follow-the-Leader models – a short proof, Discrete Cont. Dyn-A, 38 (2018), 715-722.  doi: 10.3934/dcds.2018031.  Google Scholar

[21]

N. H. Risebro and H. Holden, Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow, Netw. Heterog. Media, 13 (2018), 409-421.  doi: 10.3934/nhm.2018018.  Google Scholar

[22]

R. IllnerA. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1-12.  doi: 10.4310/CMS.2003.v1.n1.a1.  Google Scholar

[23]

S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math, 48 (1995), 235-276.  doi: 10.1002/cpa.3160480303.  Google Scholar

[24]

B. S. Kerner, The Physics of Traffic, Understanding Complex Systems, Springer, Berlin, 2004. doi: 10.1007/978-3-540-40986-1.  Google Scholar

[25]

T. Kim and H. M. Zhang, A stochastic wave propagation model, Transport. Res. B-Meth., 42 (2008), 619-634.  doi: 10.1016/j.trb.2007.12.002.  Google Scholar

[26]

A. Klar and R. Wegener, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport. Theor. Stat., 25 (1996), 785-798.   Google Scholar

[27]

A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic, J. Stat. Phys., 87 (1997), 91-114.  doi: 10.1007/BF02181481.  Google Scholar

[28]

P. Lafitte and C. Mascia, Numerical exploration of a forward-backward diffusion equation, Math. Models Methods Appl. Sci., 22 (2012), 1250004, 33pp. doi: 10.1142/S0218202512500042.  Google Scholar

[29]

R. J. LeVeque, Numerical Methods for Conservation Laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-5116-9.  Google Scholar

[30]

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.  Google Scholar

[31]

M. Lo Schiavo, A personalized kinetic model of traffic flow, Math. Comput. Modelling, 35 (2002), 607-622.  doi: 10.1016/S0895-7177(02)80024-3.  Google Scholar

[32]

C. MasciaA. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation, Arch. Ration. Mech. Anal., 194 (2009), 887-925.  doi: 10.1007/s00205-008-0185-6.  Google Scholar

[33]

A. R. Méndez and R. M. Velasco, Kerner's free-synchronized phase transition in a macroscopic traffic flow model with two classes of drivers, J. Phys. A: Math. Theor., 46 (2013), 462001, 9 pp. doi: 10.1088/1751-8113/46/46/462001.  Google Scholar

[34]

L. Pareschi and G. Russo, Implicit-explicit runge-kutta methods and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005), 129-155.  doi: 10.1007/s10915-004-4636-4.  Google Scholar

[35]

L. Pareschi and G. Russo, Efficient Asymptotic Preserving Deterministic Methods for the Boltzmann Equation, Lecture series, von Karman Institute, Rhode St. Genèse, Belgium, 2011, AVT-194 RTO AVT/VKI, Models and Computational Methods for Rarefied Flows. Google Scholar

[36] L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013.   Google Scholar
[37]

S. L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: a critical review of the basic model and an alternative proposal for dilute traffic analysis,, Transport. Res., 9 (1975), 225-235.  doi: 10.1016/0041-1647(75)90063-5.  Google Scholar

[38]

H. J. Payne, Models of freeway traffic and control, Math. Models Publ. Sys., Simulation Council Proc. 28, 1 (1971), 51-61.   Google Scholar

[39]

S. Pieraccini and G. Puppo, Implicit-explicit schemes for BGK kinetic equations, J. Sci. Comput., 32 (2006), 1-28.  doi: 10.1007/s10915-006-9116-6.  Google Scholar

[40]

I. Prigogine, A Boltzmann-like approach to the statistical theory of traffic flow, in Theory of Traffic Flow (ed. R. Herman), Elsevier, Amsterdam, 1961, 158–164.  Google Scholar

[41]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Co., New York, 1971. Google Scholar

[42]

G. PuppoM. SempliceA. Tosin and G. Visconti, Analysis of a multi-population kinetic model for traffic flow, Commun. Math. Sci., 15 (2017), 379-412.  doi: 10.4310/CMS.2017.v15.n2.a5.  Google Scholar

[43]

G. PuppoM. SempliceA. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Mod., 10 (2017), 823-854.  doi: 10.3934/krm.2017033.  Google Scholar

[44]

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

[45]

S. Roncoroni, Kinetic Modelling of Vehicular Traffic Flow, Technical report, Università degli Studi dell'Insubria, 2017, Master Thesis. Google Scholar

[46]

B. SeiboldM. R. FlynnA. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models, Netw. Heterog. Media, 8 (2013), 745-772.  doi: 10.3934/nhm.2013.8.745.  Google Scholar

[47]

G. B. Whitham, Some comments on wave propagation and shock wave structure with application to magnetohydrodynamics, Comm. Pure Appl. Math., 12 (1959), 113-158.  doi: 10.1002/cpa.3160120107.  Google Scholar

[48]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transport. Res. B-Meth., 36 (2002), 275-290.  doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

[49]

H. M. Zhang and T. Kim, A car-following theory for multiphase vehicular traffic flow, Transport. Res. B-Meth., 39 (2005), 385-399.  doi: 10.1016/j.trb.2004.06.005.  Google Scholar

Figure 1.  Schematic summary of the work
Figure 2.  Equilibrium flux diagrams (top row) and equilibrium speed diagrams (bottom row) of the homogeneous kinetic model (4)-(6) with $ N = 2 $ and $ N = 3 $ speeds for the case $ \Delta a = \frac12 $ and $ \Delta b = 0 $
Figure 3.  Left: experimental flux diagram from measurements by the Minnesota Department of Transportation, reproduced by kind permission of Seibold et al. [46]. Right: comparison between experimental data and the flux diagram resulting from the model (4)-(6), with $ N = 3 $ speeds and $ P(\rho) = 1 - \rho^{\frac14} $ for the case $ \Delta a = \frac12 $ and $ \Delta b = 0 $
Figure 4.  Left: characteristic speeds at equilibrium of the homogeneous kinetic model (4). Right: concavity of the equilibrium flux $ Q_\text{eq} $ given in (7) for the homogeneous kinetic model (4)
Figure 5.  Propagation of a low density perturbation up to the final time $ T_M = 0.2 $. Top left: solution at equilibrium with $ \epsilon = 0 $. Top right: solution with $ \epsilon = 10^{-1} $. Bottom left: comparison of the two solutions at final time. Bottom right: flux-diagram during the evolution of the BGK model with $ \epsilon = 10^{-1} $ and at equilibrium (red line)
Figure 6.  Propagation of a high density perturbation up to the final time $ T_M = 0.2 $. Top left: solution at equilibrium with $ \epsilon = 0 $. Top right: solution with $ \epsilon = 10^{-1} $. Bottom left: comparison of the two solutions at final time. Bottom right: flux-diagram during the evolution of the BGK model with $ \epsilon = 10^{-1} $ and at equilibrium (red line)
Figure 7.  Propagation of a large perturbation up to the final time $ T_M = 0.2 $. Top left: solution at equilibrium with $ \epsilon = 0 $. Top right: solution with $ \epsilon = 10^{-1} $. Bottom left: comparison of the two solutions at final time. Bottom right: flux-diagram during the evolution of the BGK model with $ \epsilon = 10^{-1} $ and at equilibrium (red line)
Figure 8.  Propagation of a high density perturbation up to the time in which the density profile is bounded by the maximum value of the density. Left: solution with $ \epsilon = 10^{-1} $. Right: solution with $ \epsilon = 10^{-2} $
Figure 9.  The right bottom panels show the sign of the diffusion coefficient (13) for the BGK model (9) with the equilibrium distribution of the homogeneous kinetic model (4)-(6). The blue line corresponds to the positive sign of the coefficient and the red line to the negative sign
Figure 10.  The right bottom panels show the sign of $ \mu(\rho) $ for the ARZ model (21) with different closures and pressure functions. Blue lines correspond to regimes where $ \mu(\rho)>0 $ and red lines to regimes where $ \mu(\rho)<0 $
Figure 11.  Evolution of a high density perturbation with the ARZ model. Top-left: density profile at different instants in time. Top-right: zoom of the density profile in the region where unstable waves occur. Bottom-left: speed profile at different instants in time. Bottom-right: density profile at final time $ T_M = 3 $
Figure 12.  The right bottom panels show the sign of the diffusion coefficient (38) for the BGK-type model (30) with the closure provided by the homogeneous model in Section 2 and different pressure functions. Blue lines correspond to regimes where the coefficient is positive and red lines to regimes where it is negative
Table 1.  Classification of flow regimes for the Boltzmann-type kinetic equation (1)
$ \epsilon $ Regime Kinetic model Continuum flow model
$ 0 $ Equilibrium flow Boltzmann Mass conservation law (LWR model)
$ \to 0 $ Viscous flow Diffusion equation
$ \asymp 1 $ Transitional Extended hydrodynamic equations
$>1 $ Rarefied -
$ \to\infty $ Free molecular flow Collision-less Boltzmann -
$ \epsilon $ Regime Kinetic model Continuum flow model
$ 0 $ Equilibrium flow Boltzmann Mass conservation law (LWR model)
$ \to 0 $ Viscous flow Diffusion equation
$ \asymp 1 $ Transitional Extended hydrodynamic equations
$>1 $ Rarefied -
$ \to\infty $ Free molecular flow Collision-less Boltzmann -
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