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Moment methods for kinetic traffic flow and a class of macroscopic traffic models

Raul Borsche

https://orcid.org/0000-0001-5460-748X

Department of Mathematics, RPTU Kaiserslautern-Landau, Gottlieb-Daimler-Straße 48, 67663 Kaiserslautern, Germany

E-mail Address: borsche@rptu.de

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and

Axel Klar

https://orcid.org/0000-0003-1679-9438

Department of Mathematics, RPTU Kaiserslautern-Landau, Gottlieb-Daimler-Straße 48, 67663 Kaiserslautern, Germany

E-mail Address: klar@mathematik.uni-kl.de

Corresponding author.

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https://doi.org/10.1142/S0218202524400074Cited by:1 (Source: Crossref)

This article is part of the issue:

Special Issue on “Active Particles Methods 2024”
Guest Editors: Nicola Bellomo and Franco Brezzi

Abstract

Starting from a nonlocal version of a classical kinetic traffic model, we derive a class of second-order macroscopic traffic flow models using appropriate moment closure approaches. Under mild assumptions on the closure, we prove that the resulting macroscopic equations fulfill a set of conditions including hyperbolicity, physically reasonable invariant domains and physically reasonable bounds on the speed with which the waves propagate. Finally, numerical results for various situations are presented, illustrating the analytical findings and comparing kinetic and macroscopic solutions.

Communicated by N. Bellomo and F. Brezzi

Keywords:

AMSC: 90B20, 35L02, 35L04

References

1. G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler , Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci. 29 (2019) 1901–2005. Link, Web of Science, Google Scholar
2. A. M. Anile, S. Pennisi and M. Sammartino , A thermodynamical approach to Eddington factors, J. Math. Phys. 32 (1991) 544–550. Crossref, Web of Science, Google Scholar
3. A. Aw, A. Klar, T. Materne and M. Rascle , Derivation of continuum flow traffic models from microscopic follow-the-leader models, SIAM J. Appl. Math. 63 (2002) 259–278. Crossref, Web of Science, Google Scholar
4. A. Aw and M. Rascle , Resurrection of second order models of traffic flow? SIAM J. Appl. Math. 60 (2000) 916–938. Crossref, Web of Science, Google Scholar
5. N. Bellomo and C. Dogbe , On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev. 53 (2011) 409–463. Crossref, Web of Science, Google Scholar
6. F. Berthelin, P. Degond, M. Delitala and M. Rascle , A model for the formation and evolution of traffic jams, Arch. Ration. Mech. Anal. 187 (2008) 185–220. Crossref, Web of Science, Google Scholar
7. F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, J. Royer and M. Rascle , A traffic-flow model with constraints for the modeling of traffic jams, Math. Models Methods Appl. Sci. 18 (2008) 1269–1298. Link, Web of Science, Google Scholar
8. R. Borsche, M. Kimathi and A. Klar , Kinetic derivations of a Hamilton-Jacobi type traffic flow model, Commun. Math. Sci. 11 (2013) 739–756. Crossref, Web of Science, Google Scholar
9. R. Borsche and A. Klar , A nonlinear discrete-velocity relaxation model for traffic flow, SIAM J. Appl. Math. 78 (2018) 2891–2917. Crossref, Web of Science, Google Scholar
10. R. Borsche and A. Klar , A hierarchy of discrete velocity models for traffic flow derived from a nonlocal Prigogine-Herman model, SIAM J. Appl. Math. 84 (2024) 139–164. Crossref, Google Scholar
11. F. A. Chiarello and A. Tosin , Macroscopic limits of non-local kinetic descriptions of vehicular traffic, Kinet. Relat. Models 16 (2022) 540–564. Crossref, Google Scholar
12. R. M. Colombo , Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math. 63 (2002) 708–721. Crossref, Web of Science, Google Scholar
13. J. F. Coulombel, F. Golse and T. Goudon , Diffusion approximation and entropy-based moment closure for kinetic equations, Asymptot. Anal. 45 (2005) 1–39. Web of Science, Google Scholar
14. C. F. Daganzo , Requiem for second-order fluid approximations of traffic flow, Transp. Res. B, Methodol. 29 (1995) 277–286. Crossref, Web of Science, Google Scholar
15. G. Dimarco and A. Tosin , The Aw-Rascle traffic model: Enskog-type kinetic derivation and generalisations, J. Stat. Phys. 178 (2020) 178–210. Crossref, Web of Science, Google Scholar
16. L. Fermo and A. Tosin , A fully-discrete-state kinetic theory approach to modeling vehicular traffic, SIAM J. Appl. Math. 73 (2013) 1533–1556. Crossref, Web of Science, Google Scholar
17. P. Goatin , The Aw–Rascle vehicular traffic flow model with phase transitions, Math. Comput. Model. 44 (2006) 287–303. Crossref, Web of Science, Google Scholar
18. J. Greenberg , Extension and amplification of the Aw-Rascle model, SIAM J. Appl. Math. 62 (2001) 729–745. Crossref, Web of Science, Google Scholar
19. J. M. Greenberg , Congestion redux, SIAM J. Appl. Math. 64 (2004) 1175–1185. Crossref, Web of Science, Google Scholar
20. J. M. Greenberg, A. Klar and M. Rascle , Congestion on multilane highways, SIAM J. Appl. Math. 63 (2003) 818–833. Crossref, Web of Science, Google Scholar
21. D. Helbing , Gas-kinetic derivation of Navier-Stokes-like traffic equation, Phys. Rev. E 53 (1996) 2366–2381. Crossref, Web of Science, Google Scholar
22. D. Helbing and A. F. Johansson , On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models, Eur. Phys. J. B 69 (2009) 549–562. Crossref, Web of Science, Google Scholar
23. M. Herty, G. Puppo, S. Roncoroni and G. Visconti , The BGK approximation of kinetic models for traffic, Kinet. Relat. Models 13 (2020) 279–307. Crossref, Web of Science, Google Scholar
24. D. Hoff , Invariant regions for systems of conservation laws, Trans. Amer. Math. Soc. 289 (1985) 591–610. Crossref, Web of Science, Google Scholar
25. A. Klar and R. Wegener , Enskog-like kinetic models for vehicular traffic, J. Stat. Phys. 87 (1997) 91–114. Crossref, Web of Science, Google Scholar
26. A. Klar and R. Wegener , A hierachy of models for multilane vehicular traffic I: Modeling, SIAM J. Appl. Math. 59 (1998) 983–1001. Crossref, Web of Science, Google Scholar
27. A. Klar and R. Wegener , Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math. 60 (2000) 1749–1766. Crossref, Web of Science, Google Scholar
28. M. Lachowicz , On the hydrodynamic limit of the Enskog equation, Publ. RIMS, Kyoto Univ. 34 (1998) 191–210. Crossref, Web of Science, Google Scholar
29. C. D. Levermore , Moment closure hierarchies for kinetic theories, J. Stat. Phys. 83 (1996) 1021–1065. Crossref, Web of Science, Google Scholar
30. T. P. Liu , Hyperbolic conservation laws with relaxation, Commun. Math. Phys. 108 (1987) 153–175. Crossref, Web of Science, Google Scholar
31. I. Mueller and T. Ruggeri , Rational Extended Thermodynamics, 2nd edn., Springer Tracts in Natural Philosophy (Springer, 1998). Crossref, Google Scholar
32. P. Nelson , A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions, Transp. Theory Stat. Phys. 24 (1995) 383–408. Crossref, Google Scholar
33. L. Pareschi and G. Russo , Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput. 25 (2005) 129–155. Crossref, Web of Science, Google Scholar
34. S. Paveri-Fontana , On Boltzmann like treatments for traffic flow, Transp. Res. 9 (1975) 225–235. Crossref, Google Scholar
35. H. J. Payne , Models of freeway traffic and control, in Mathematical Models of Public Systems, Vol. 1 (Simulation Council, 1971), pp. 51–61. Google Scholar
36. I. Prigogine and R. Herman , Kinetic Theory of Vehicular Traffic (American Elsevier, 1971). Google Scholar
37. M. Rascle , An improved macroscopic model of traffic flow: Derivation and links with the Lighthill-Whitham model, Math. Comput. Model. 35 (2002) 581–590. Crossref, Web of Science, Google Scholar
38. G. Whitham , Linear and Nonlinear Waves (Wiley, 1974). Google Scholar
39. H. M. Zhang , A non-equilibrium traffic model devoid of gas-like behavior, Transp. Res. B 36 (2002) 275–290. Crossref, Google Scholar