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February  2018, 38(2): 715-722. doi: 10.3934/dcds.2018031

The continuum limit of Follow-the-Leader models — a short proof

Helge Holden 1,, and  Nils Henrik Risebro 2,

1. 

Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
2. 

Department of Mathematics, University of Oslo, Blindern, NO-0316 Oslo, Norway

* Corresponding author: Helge Holden

We dedicate this paper to the memory of Hans Petter Langtangen (1962–2016)

Received  July 2017 Published  February 2018

Fund Project: Research was supported by the grant Waves and Nonlinear Phenomena (WaNP) from the Research Council of Norway. The research was done while the authors were at Institut MittagLeffler, Stockholm.

We offer a simple and self-contained proof that the Follow-the-Leader model converges to the Lighthill-Whitham-Richards model for traffic flow.

Keywords: Follow-the-Leader model, Lighthill-Whitham-Richards model, traffic flow, continuum limit, conservation laws.
Mathematics Subject Classification: Primary: 35L02; Secondary: 35Q35, 82B21.
Citation: Helge Holden, Nils Henrik Risebro. The continuum limit of Follow-the-Leader models — a short proof. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 715-722. doi: 10.3934/dcds.2018031
References:
[1]

B. Argall, E. Cheleshkin, J. M. Greenberg, C. Hinde and P.-J. Lin, A rigorous treatment of a follow-the-leader traffic model with traffic lights present, SIAM J. Appl. Math., 63 (2002), 149-168.  doi: 10.1137/S0036139901391215.  Google Scholar

[2]

A. Aw, A. Klar, T. 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.  Google Scholar

[3]

R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, Rend. Sem. Math. Univ. Padova, 131 (2014), 217-235.  doi: 10.4171/RSMUP/131-13.  Google Scholar

[4]

E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, Networks and Heterogeneous Media, 11 (2016), 395-413.  doi: 10.3934/nhm.2016002.  Google Scholar

[5]

M. Di Francesco, S. Fagioli and M. D. Rosini, Deterministic particle approximation of scalar conservation laws, Boll. Unione Mat. Ital., 10 (2017), 487-501.  doi: 10.1007/s40574-017-0132-2.  Google Scholar

[6]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871.  doi: 10.1007/s00205-015-0843-4.  Google Scholar

[7]

P. Goatin and F. Rossi, A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit, Commun. Math. Sci., 15 (2017), 261-287.  doi: 10.4310/CMS.2017.v15.n1.a12.  Google Scholar

[8]

K. Han, T. Yaob and T. L. Friesz, Lagrangian-based hydrodynamic model: Freeway traffic estimation, Preprint, arXiv: 1211.4619v1, 2012. Google Scholar

[9]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag, New York, 2015, Second edition. doi: 10.1007/978-3-662-47507-2.  Google Scholar

[10]

H. Holden and N. H. Risebro. Follow-the-leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Preprint, arXiv: 1702.01718, 2017. Google Scholar

[11]

M. J. Lighthill and G. B. Whitham, Kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. (London), Series A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[12]

P. I. Richards, Shockwaves on the highway, Operations Research, 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

[13]

E. Rossi, A justification of a LWR model based on a follow the leader description, Discrete Cont. Dyn. Syst. Series S, 7 (2014), 579-591.  doi: 10.3934/dcdss.2014.7.579.  Google Scholar

show all references

We dedicate this paper to the memory of Hans Petter Langtangen (1962–2016)

References:
[1]

B. Argall, E. Cheleshkin, J. M. Greenberg, C. Hinde and P.-J. Lin, A rigorous treatment of a follow-the-leader traffic model with traffic lights present, SIAM J. Appl. Math., 63 (2002), 149-168.  doi: 10.1137/S0036139901391215.  Google Scholar

[2]

A. Aw, A. Klar, T. 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.  Google Scholar

[3]

R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, Rend. Sem. Math. Univ. Padova, 131 (2014), 217-235.  doi: 10.4171/RSMUP/131-13.  Google Scholar

[4]

E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, Networks and Heterogeneous Media, 11 (2016), 395-413.  doi: 10.3934/nhm.2016002.  Google Scholar

[5]

M. Di Francesco, S. Fagioli and M. D. Rosini, Deterministic particle approximation of scalar conservation laws, Boll. Unione Mat. Ital., 10 (2017), 487-501.  doi: 10.1007/s40574-017-0132-2.  Google Scholar

[6]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871.  doi: 10.1007/s00205-015-0843-4.  Google Scholar

[7]

P. Goatin and F. Rossi, A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit, Commun. Math. Sci., 15 (2017), 261-287.  doi: 10.4310/CMS.2017.v15.n1.a12.  Google Scholar

[8]

K. Han, T. Yaob and T. L. Friesz, Lagrangian-based hydrodynamic model: Freeway traffic estimation, Preprint, arXiv: 1211.4619v1, 2012. Google Scholar

[9]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag, New York, 2015, Second edition. doi: 10.1007/978-3-662-47507-2.  Google Scholar

[10]

H. Holden and N. H. Risebro. Follow-the-leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Preprint, arXiv: 1702.01718, 2017. Google Scholar

[11]

M. J. Lighthill and G. B. Whitham, Kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. (London), Series A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[12]

P. I. Richards, Shockwaves on the highway, Operations Research, 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

[13]

E. Rossi, A justification of a LWR model based on a follow the leader description, Discrete Cont. Dyn. Syst. Series S, 7 (2014), 579-591.  doi: 10.3934/dcdss.2014.7.579.  Google Scholar

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