A derivation of the Aw–Rascle traffic models from Fokker–Planck type kinetic models
Authors:
R. Illner, C. Kirchner and R. Pinnau
Journal:
Quart. Appl. Math. 67 (2009), 39-45
MSC (2000):
Primary 35Qxx, 82C31
DOI:
https://doi.org/10.1090/S0033-569X-09-01075-7
Published electronically:
January 22, 2009
MathSciNet review:
2495070
Full-text PDF Free Access
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Abstract: We show how the Aw–Rascle model, a hyperbolic system of PDEs modeling traffic flow, can be derived from a simplified Fokker–Planck type kinetic equation.
References
- A. Aw and M. Rascle, Resurrection of “second order” models of traffic flow, SIAM J. Appl. Math. 60 (2000), no. 3, 916–938. MR 1750085, DOI https://doi.org/10.1137/S0036139997332099
- Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. MR 1307620
- M. Herty, R. Illner, A. Klar, and V. Panferov, Qualitative properties of solutions to systems of Fokker-Planck equations for multilane traffic flow, Transport Theory Statist. Phys. 35 (2006), no. 1-2, 31–54. MR 2284533, DOI https://doi.org/10.1080/00411450600878573
- R. Illner, A. Klar and T. Materne. On Vlasov-Fokker-Planck Type Kinetic Models for Multilane Traffic Flow. Preprint, 2003.
- Reinhard Illner, Axel Klar, and Thorsten Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci. 1 (2003), no. 1, 1–12. MR 1979839
- Axel Klar and Raimund Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math. 60 (2000), no. 5, 1749–1766. MR 1761769, DOI https://doi.org/10.1137/S0036139999356181
- C. David Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys. 83 (1996), no. 5-6, 1021–1065. MR 1392419, DOI https://doi.org/10.1007/BF02179552
References
- A. Aw and M. Rascle. Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math., Vol. 60(3), pp. 916–938, 2000. MR 1750085 (2001a:35111)
- C. Cercignani, R. Illner, and M. Pulvirenti. The Mathematical Theory of Dilute Gases, Springer-Verlag, 1994. MR 1307620 (96g:82046)
- M. Herty, R. Illner, A. Klar, and V. Panferov. Qualitative Properties of Solutions to Systems of Fokker-Planck equations for Multilane Traffic Flow. Transport Theory and Statistical Physics, Vol. 35, pp. 31–54, 2006. MR 2284533
- R. Illner, A. Klar and T. Materne. On Vlasov-Fokker-Planck Type Kinetic Models for Multilane Traffic Flow. Preprint, 2003.
- R. Illner, A. Klar and T. Materne. Vlasov-Fokker-Planck Models for Multilane Traffic Flow. Commun. Math. Sci., Vol. 1(1), pp. 1–12, 2003. MR 1979839
- A. Klar and R. Wegener. Kinetic Derivation of Macroscopic Anticipation Models for Vehicular Traffic. SIAM J. Appl. Math., Vol. 60, pp. 1749–1766, 2000. MR 1761769 (2001f:90009)
- C. D. Levermore. Moment Closure Hierarchies for Kinetic Theories. J. Stat. Phys., Vol. 83, pp. 1021–1065, 1996. MR 1392419 (97e:82041)
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Additional Information
R. Illner
Affiliation:
Department of Mathematics and Statistics, University of Victoria, PO Box 3045 STN CSC, Victoria, B.C., Canada V8W 3P4
Email:
rillner@math.uvic.ca
C. Kirchner
Affiliation:
Fachbereich Mathematik, TU Kaiserslautern, D-67653 Kaiserslautern, Germany
Email:
kirchner@mathematik.uni-kl.de
R. Pinnau
Affiliation:
Fachbereich Mathematik, TU Kaiserslautern, D-67653 Kaiserslautern, Germany
Email:
pinnau@mathematik.uni-kl.de
Keywords:
Traffic flow,
kinetic equation,
Fokker–Planck,
Aw–Rascle model,
multilane traffic
Received by editor(s):
June 1, 2007
Published electronically:
January 22, 2009
Article copyright:
© Copyright 2009
Brown University
The copyright for this article reverts to public domain 28 years after publication.
