# American Institute of Mathematical Sciences

June  2014, 7(3): 579-591. doi: 10.3934/dcdss.2014.7.579

## A justification of a LWR model based on a follow the leader description

 1 Department of Mathematics and Applications, University of Milano-Bicocca, Via Cozzi, 53, 20125 Milano, Italy

Received  May 2013 Revised  July 2013 Published  January 2014

We investigate the correlations between a macroscopic Lighthill--Whitham and Richards model and a microscopic follow-the-leader model for traffic flow. We prove that the microscopic model tends to the macroscopic one in a sort of kinetic limit, i.e. as the number of individuals tends to infinity, keeping the total mass fixed. Based on this convergence result, we approximately compute the solutions to a conservation law by means of the integration of an ordinary differential system. From the numerical point of view, the limiting procedure is then extended to the case of several populations, referring to the macroscopic model in [2] and to the natural multi--population analogue of the microscopic one.
Citation: Elena Rossi. A justification of a LWR model based on a follow the leader description. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 579-591. doi: 10.3934/dcdss.2014.7.579
##### 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] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266. [3] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000. [4] R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759-772. doi: 10.1017/S0308210500002663. [5] R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666. doi: 10.1137/090752468. [6] R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, to appear in Rend. Semin. Mat. Univ. Padova. [7] G. Costeseque, Analyse et Modelisation du Trafic Routier: Passage du Microscopique au Macroscopique, Master's Thesis, Ecole des Ponts ParisTech, 2011. [8] S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. [9] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [10] M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [11] K. Radhakrishnan and A. C. Hindmarsh, Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations, LLNL report UCRL-ID-113855, National Aeronautics and Space Administration, Lewis Research Center, 1993. doi: 10.2172/15013302. [12] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [13] E. Rossi, On the Micro-Macro Limit in Traffic Flow, Master's thesis, Università Cattolica del Sacro Cuore, Brescia, 2012.

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] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612. doi: 10.1017/S0956792503005266. [3] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000. [4] R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759-772. doi: 10.1017/S0308210500002663. [5] R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666. doi: 10.1137/090752468. [6] R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, to appear in Rend. Semin. Mat. Univ. Padova. [7] G. Costeseque, Analyse et Modelisation du Trafic Routier: Passage du Microscopique au Macroscopique, Master's Thesis, Ecole des Ponts ParisTech, 2011. [8] S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. [9] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [10] M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [11] K. Radhakrishnan and A. C. Hindmarsh, Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations, LLNL report UCRL-ID-113855, National Aeronautics and Space Administration, Lewis Research Center, 1993. doi: 10.2172/15013302. [12] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [13] E. Rossi, On the Micro-Macro Limit in Traffic Flow, Master's thesis, Università Cattolica del Sacro Cuore, Brescia, 2012.

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