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Nonlinear Schrödinger Equations on Periodic Metric Graphs
The continuum limit of Follow-the-Leader models — a short proof
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 |
We offer a simple and self-contained proof that the Follow-the-Leader model converges to the Lighthill-Whitham-Richards model for traffic flow.
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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