Abstract
We consider an extension of the macroscopic traffic model of Lighthill–Whitham and Richards to a multispecies traffic model. As has already been observed by Benzoni-Gavage and Colombo (Eur J Appl Math 14:587–612, 2003, [1]), the system of PDEs lacks strict hyperbolicity. We study the Riemann problem for the two species extension with focus on the values around the umbilic point, where the eigenvalues coalesce. For this purpose, we examine the behavior of the solutions around the critical point like it is done by Meltzer (Multispecies traffic models based on the Lighthill–Whitham and the Aw-Rascle model, 2016, [9]). Due to the difficulty of the umbilic point, we are not able to prove well-posedness analytically. But we provide an understanding of the systems properties via numerical experiments which leads to the conjecture that the Riemann problem has a unique solution not only near the umbilic point but also away from it in the set where it is defined.
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References
S. Benzoni-Gavage, R.M. Colombo, An \(n\)-populations model for traffic flow. Eur. J. Appl. Math. 14(5), 587–612 (2003). https://doi.org/10.1017/S0956792503005266. ISSN 0956-7925
A. Bressan, Hyperbolic systems of conservation laws, in Oxford Lecture Series in Mathematics and its Applications, vol. 20. (Oxford University Press, Oxford, 2000), (The one-dimensional Cauchy problem). ISBN 0-19-850700-3
C.M. Dafermos, Hyperbolic conservation laws in continuum physics, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 4th edn., vol. 325. (Springer, Berlin, 2016), https://doi.org/10.1007/978-3-642-04048-1, https://doi.org/10.1007/978-3-662-49451-6. ISBN 978-3-642-04047-4
D. Hoff, Invariant regions for systems of conservation laws. Trans. Amer. Math. Soc. 289(2), 591–610 (1985). https://doi.org/10.2307/2000254. ISSN 0002-9947
B.L. Keyfitz, H.C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch. Rational Mech. Anal. 72(3), 219–241 (1979/80). https://doi.org/10.1007/BF00281590. ISSN 0003-9527
P.D. Lax, Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537–566 (1957). ISSN 0010-3640
M.J. Lighthill, G.B. Whitham, On kinematic waves II. a theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A 229, 317–345 (1955). ISSN 0962-8444
S. Liu, F. Chen, Existence of global \(L^p\) solutions to a symmetric system of Keyfitz–Kranzer type. Appl. Math. Lett. 52, 96–101 (2016). https://doi.org/10.1016/j.aml.2015.08.011. ISSN 0893-9659
M.-C. Meltzer, Multispecies traffic models based on the Lighthill–Whitham and the Aw-Rascle model, Thesis (2016), https://www.mathematik.uni-wuerzburg.de/~klingen/Workgroup_files/Thesis_Marie-Christine_Meltzer.pdf
P.I. Richards, Shock waves on the highway. Oper. Res. 4, 42–51 (1956). ISSN 0030-364X
D. Serre, Systems of Conservation Laws 1 (Cambridge University Press, Cambridge, 1999). https://doi.org/10.1017/CBO9780511612374. Hyperbolicity, entropies, shock waves, Translated from the 1996 French original by I. N. Sneddon. ISBN 0-521-58233-4
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Colombo, R.M., Klingenberg, C., Meltzer, MC. (2018). A Multispecies Traffic Model Based on the Lighthill-Whitham and Richards Model. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems I. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 236. Springer, Cham. https://doi.org/10.1007/978-3-319-91545-6_30
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DOI: https://doi.org/10.1007/978-3-319-91545-6_30
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