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

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.

[1]  M. Rosini,et al.  Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit , 2014, 1404.7062.

[2]  Helge Holden,et al.  The continuum limit of Follow-the-Leader models - a short proof , 2017 .

[3]  M. Crandall,et al.  Monotone difference approximations for scalar conservation laws , 1979 .

[4]  S. Fagioli,et al.  Deterministic particle approximation of scalar conservation laws , 2016 .

[5]  T. Friesz,et al.  Lagrangian-based Hydrodynamic Model: Freeway Traffic Estimation , 2012, 1211.4619.

[6]  Axel Klar,et al.  Derivation of Continuum Traffic Flow Models from Microscopic Follow-the-Leader Models , 2002, SIAM J. Appl. Math..

[7]  E. Rossi.  A justification of a LWR model based on a follow the leader description , 2014 .

[8]  Brenna Argall,et al.  A Rigorous Treatment of a Follow-the-Leader Traffic Model with Traffic Lights Present , 2003, SIAM J. Appl. Math..

[9]  H. Holden,et al.  Front Tracking for Hyperbolic Conservation Laws , 2002 .

[10]  David H. Wagner,et al.  Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions , 1987 .

[11]  Francesco Rossi,et al.  A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit , 2015, 1510.04461.

[12]  E. Rossi,et al.  On the micro-macro limit in traffic flow , 2014 .

[13]  Emiliano Cristiani,et al.  On the micro-to-macro limit for first-order traffic flow models on networks , 2016, Networks Heterog. Media.

[14]  Marco Di Francesco,et al.  A Deterministic Particle Approximation for Non-linear Conservation Laws , 2016 .

[15]  M J Lighthill,et al.  ON KINEMATIC WAVES.. , 1955 .