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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[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.

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