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

Helge Holden; Nils Henrik Risebro

doi:10.3934/nhm.2018018

Article Contents

2018, Volume 13, Issue 3: 409-421. Doi: 10.3934/nhm.2018018

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

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, P.O. Box 1053, Blindern, NO-0316 Oslo, Norway

^* Corresponding author: Helge Holden
^* Corresponding author: Helge Holden

Received: September 2017

Revised: January 2018

Published: August 2018

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 35L02; Secondary: 35Q35, 82B21.

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Figure 1. 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 $n = 0, \ldots, 40$

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Figure 2. The approximate density $\rho_\ell$ for $t = 0$ and $t = 2$ in Eulerian coordinates

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