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September 2018, 13(3): 409-421. doi: 10.3934/nhm.2018018

## Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow

 1 Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 2 Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, NO-0316 Oslo, Norway

* Corresponding author: Helge Holden

Received  September 2017 Revised  January 2018 Published  July 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 show how to view the standard Follow-the-Leader (FtL) model as a numerical method to compute numerically the solution of the Lighthill-Whitham-Richards (LWR) model for traffic flow. As a result we offer a simple proof that FtL models converge to the LWR model for traffic flow when traffic becomes dense. The proof is based on techniques used in the analysis of numerical schemes for conservation laws, and the equivalence of weak entropy solutions of conservation laws in the Lagrangian and Eulerian formulation.

Citation: Helge Holden, Nils Henrik Risebro. Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Networks & Heterogeneous Media, 2018, 13 (3) : 409-421. doi: 10.3934/nhm.2018018
##### 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] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1-21. doi: 10.1090/S0025-5718-1980-0551288-3. [5] 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. [6] 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. [7] M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, A deterministic particle approximation for non-linear conservation laws, In N. Bellomo, P. Degond, E. Tadmor (eds. ) Active Particles, Birkhäuser, 1 (2017), 333-378. [8] 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. [9] P. Goatin and F. Rossi, A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit, Comm. Math. Sci., 15 (2017), 261-287. doi: 10.4310/CMS.2017.v15.n1.a12. [10] K. Han, T. Yaob and T. L. Friesz, Lagrangian-based hydrodynamic model: Freeway traffic estimation, Preprint, arXiv: 1211.4619v1, 2012. [11] 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. [12] H. Holden and N. H. Risebro, The continuum limit of Follow-the-Leader models — a short proof, Discrete Cont. Dyn. Syst. A, 38 (2018), 715-722. doi: 10.3934/dcds.2018031. [13] 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. [14] P. I. Richards, Shockwaves on the highway, Operations Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [15] 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. [16] D. H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Diff. Eqn., 68 (1987), 118-136. doi: 10.1016/0022-0396(87)90188-4.

show all references

##### 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] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1-21. doi: 10.1090/S0025-5718-1980-0551288-3. [5] 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. [6] 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. [7] M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, A deterministic particle approximation for non-linear conservation laws, In N. Bellomo, P. Degond, E. Tadmor (eds. ) Active Particles, Birkhäuser, 1 (2017), 333-378. [8] 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. [9] P. Goatin and F. Rossi, A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit, Comm. Math. Sci., 15 (2017), 261-287. doi: 10.4310/CMS.2017.v15.n1.a12. [10] K. Han, T. Yaob and T. L. Friesz, Lagrangian-based hydrodynamic model: Freeway traffic estimation, Preprint, arXiv: 1211.4619v1, 2012. [11] 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. [12] H. Holden and N. H. Risebro, The continuum limit of Follow-the-Leader models — a short proof, Discrete Cont. Dyn. Syst. A, 38 (2018), 715-722. doi: 10.3934/dcds.2018031. [13] 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. [14] P. I. Richards, Shockwaves on the highway, Operations Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [15] 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. [16] D. H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Diff. Eqn., 68 (1987), 118-136. doi: 10.1016/0022-0396(87)90188-4.
Left: the Lagrangian grid $\{ (t^n, x_{i-1/2}) \}_{i = 1}^N$. Right: the Eulerian grid $\{ (t^n, z^n_{i-1/2}) \}_{i = 1}^N$. In both cases $n = 0, \ldots, 40$
The approximate density $\rho_\ell$ for $t = 0$ and $t = 2$ in Eulerian coordinates
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