Abstract
We couple, at a fixed interface, the microscopic Follow the Leader model and the macroscopic Lighthill–Whitham–Richards model, both used for describing vehicular traffic. The coupling is obtained by suitable boundary conditions. We prove existence of solutions for the corresponding Cauchy problem, by using the wave-front tracking method and classical results on ordinary differential equations.
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Aw, A., Klar, A., Materne, T., Rascle, M.: Derivation of continuum traffic flow models from microscopic follow-the-leader models. SIAM J. Appl. Math. 63(1), 259–278 (2002)
Bourrel, E., Lesort, J.: Mixing microscopic and macroscopic representations of traffic flow: hybrid model based on Lighthill–Whitham–Richards theory. Transp. Res. Rec. 193–200, 2003 (1852)
Coclite, G.M., Garavello, M., Piccoli, B.: Traffic flow on a road network. SIAM J. Math. Anal. 36(6), 1862–1886 (2005). (electronic)
Colombo, R.M., Marcellini, F.: A mixed ODE-PDE model for vehicular traffic. Math. Methods Appl. Sci. 38(7), 1292–1302 (2015)
Colombo, R.M., Rossi, E.: On the micro-macro limit in traffic flow. Rend. Semin. Mat. Univ. Padova 131, 217–235 (2014)
Cristiani, E., Sahu, S.: On the micro-to-macro limit for first-order traffic flow models on networks. Netw. Heterog. Media 11(3), 395–413 (2016)
Di Francesco, M., Fagioli, S., Rosini, M.D.: Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic. Math. Biosci. Eng. 14(1), 127–141 (2017)
Di Francesco, M., Rosini, M.D.: Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit. Arch. Ration. Mech. Anal. 217(3), 831–871 (2015)
Garavello, M., Han, K., Piccoli, B.: Models for vehicular traffic on networks, volume 9 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield (2016)
Garavello, M., Piccoli, B.: Conservation laws on complex networks. Ann. H. Poincaré 26(5), 1925–1951 (2009)
Garavello, M., Piccoli, B.: Coupling of Lighthill–Whitham–Richards and phase transition models. J. Hyperbolic Differ. Equ. 10(3), 577–636 (2013)
Garavello, M., Piccoli, B.: Coupling of microscopic and phase transition models at boundary. Netw. Heterog. Media 8(3), 649–661 (2013)
Gazis, D.C., Herman, R., Rothery, R.W.: Nonlinear follow-the-leader models of traffic flow. Oper. Res. 9, 545–567 (1961)
Greenberg, J.M.: Extensions and amplifications of a traffic model of Aw and Rascle. SIAM J. Appl. Math. 62(3), 729–745 (2001/2002)
Lattanzio, C., Piccoli, B.: Coupling of microscopic and macroscopic traffic models at boundaries. Math. Models Methods Appl. Sci. 20(12), 2349–2370 (2010)
Lebacque, J.: The godunov scheme and what it means for first order macroscopic traffic flow models. In: Lesort JB (ed.) Proceedings of the 13th ISTTT, pp. 647–677 (1996)
Lighthill, M.J., Whitham, G.B.: On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A 229, 317–345 (1955)
Marcellini, F.: Free-congested and micro-macro descriptions of traffic flow. Discrete Contin. Dyn. Syst. Ser. S 7(3), 543–556 (2014)
Prigogine, I., Herman, R.: Kinetic Theory of Vehicular Traffic. American Elsevier Pub. Co, New York (1971)
Richards, P.I.: Shock waves on the highway. Oper. Res. 4, 42–51 (1956)
Rossi, E.: A justification of a LWR model based on a follow the leader description. Discrete Contin. Dyn. Syst. Ser. S 7(3), 579–591 (2014)
Work, D.B., Blandin, S., Tossavainen, O.-P., Piccoli, B., Bayen, A.M.: A traffic model for velocity data assimilation. Appl. Math. Res. Express 1, 1–35 (2010)
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This article is part of the topical collection “Hyperbolic PDEs, Fluids, Transport and Applications: Dedicated to Alberto Bressan for his 60th birthday” guest edited by Fabio Ancona, Stefano Bianchini, Pierangelo Marcati, Andrea Marson.
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Garavello, M., Piccoli, B. Boundary coupling of microscopic and first order macroscopic traffic models. Nonlinear Differ. Equ. Appl. 24, 43 (2017). https://doi.org/10.1007/s00030-017-0467-5
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DOI: https://doi.org/10.1007/s00030-017-0467-5