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 and 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.

[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.

[3]

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

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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. 

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[38]

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

[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.

[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.

[41]

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

[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.

[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.

[44]

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

[45]

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

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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. 

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[38]

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

[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.

[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.

[41]

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

[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.

[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.

[44]

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

[45]

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

[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.

[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.

[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.

[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.

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 -
[1]

Vincent Giovangigli, Wen-An Yong. Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion. Kinetic and Related Models, 2015, 8 (1) : 79-116. doi: 10.3934/krm.2015.8.79

[2]

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[3]

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