The continuum limit of Follow-the-Leader models - a short proof

We offer a simple and self-contained proof that the Follow-the-Leader model converges to the Lighthill-Whitham-Richards model for traffic flow.


Introduction
The problem of convergence of particle models to continuum models is fundamental. We here study it in the context of traffic flow. In this case there are two fundamentally different models: The first one is based on individual vehicles whose dynamics is determined by the behavior of the vehicle immediately in front of it. This gives the Follow-the-Leader (FtL) model, which constitutes a system of ordinary differential equations describing the dynamics of individual vehicles. The other model is based on the assumption of heavy traffic where the individual vehicles are represented by a density. Assuming that the number of vehicles is conserved, we get the classical Lighthill-Whitham-Richards (LWR) model [11,12], which is nothing but a scalar hyperbolic conservation law. The question that we address in this paper is in what sense the FtL model approaches or approximates the LWR model in the case of dense traffic.
The principal assumption in FtL models is that the velocity V of any given vehicle is a function of the distance to the vehicle in front of it. We shall write this function as where ∆Z denotes the distance to nearest vehicle in front, and the length of each vehicle. For obvious reasons, ∆Z ≥ . It is commonly assumed that V is an increasing positive function defined in [1, ∞), such that lim y→∞ V (y) = v max < ∞. Consider N vehicles with length and position Z 1 (t) < · · · < Z N (t) on the real axis with dynamics given by To close this system, we must prescribe the velocity of the first vehicle at Z N . It is natural to model this by lettingŻ N = v max .
In this paper we analyze the limit of this system of ordinary differential equations when N → ∞ and → 0. We show that where intuitively Z i+1 , Z i → z, and where ρ is an entropy solution to the scalar conservation law This problem has also been addressed by several other researchers. We here mention [1,2,4,5,7,8]. The long and technically demanding paper [6] shows this convergence, while in [3,13], the convergence of the discrete system is assumed rather than proved. The approach here resembles [10] where FtL models are viewed as a numerical approximation of the LWR model, and the proof of convergence depends on classical results by Crandall-Majda and Wagner for a grid approximation.
Here we offer is a simple and straightforward proof of the continuum limit. Solutions to scalar conservation laws are in general not continuous, and (1.2) must be considered in the weak sense; furthermore weak solutions to the Cauchy problem are not unique, and in order for the Cauchy problem to have a unique solution, one must impose the Kružkov entropy condition [9]: A function ρ ∈ C([0, ∞); L 1 (R)) is called an entropy solution to the Cauchy problem for (1.2) if for all constants k ∈ R and all non-negative test functions More precisely, we show the following result. Assume that the velocity function satisfies the reasonable assumptions (2.1), and the initial data ρ(0, · ) ∈ L 1 (R) ∩ BV (R). Let ρ (t, z) be the density of vehicles as defined by the FtL model, see (2.14). Then we show that lim →0 ρ = ρ ∈ C([0, ∞); L 1 (R)), where ρ is the unique solution to (1.2) satisfying the entropy condition (1.3) such that ρ(0, z) = ρ 0 (z).
The rest of this note is organized as follows: In Section 2 we define the discrete model and prove some simple bounds on its solutions, and in Section 3 we give the elementary proof of convergence.

The model
We use units such where V i = V (y i ) and V N = 1. Later we will also need y N = ∞. Regarding the initial values, we assume that there is a function ρ 0 : Thus with the current scaling, the length of each vehicle is = 1/(N + 1). We will also need z −1/2 (0) = −∞. Here we choose the infimum of possible values for In particular, Proof. If y i (t) = 1, then V (y i (t)) = 0 and henceẏ i (t) ≥ 0. This gives the lower bound on y i . Using (2.1a) and the bound V i+1 ≤ 1, we geṫ By integrating this inequality, we see that the estimate (2.5) holds, and the limit (2.6) then follows trivially.
since V ≥ 0. For i = N we recall the conventions that y N = ∞ and V N = 1, and thus V (y N ) = 0. We have that 0 we could also carry out these estimates for ρ i , proving (2.8).
To simplify the notation, we henceforth write = j . Furthermore, since v(ρ ) = V , V → v(ρ). Adding (3.1) and (3.2), we get → 0, as to 0, and thus ρ is a weak solution. To show that ρ is an entropy solution, let η be a twice differentiable convex function. Since V ≥ 0, we get where Q = η V and Q i = Q(y i ). Introduce q(ρ) = Q(1/ρ) with q i = q(ρ i ). Define µ = µ(ρ) by µ(ρ) = ρη(1/ρ). As usual we write µ i = µ(ρ i ) and η i = η(y i ). Then µ is a convex function of ρ, and if µ is a convex function of ρ, then η is a convex function of y. We have that , and define q (t, z) similarly. As when establishing (3.1), we find for a non-negative test function ϕ with support in Therefore, If q (t, · ) is of bounded variation, then r → 0. We now assume that µ is (a smooth approximation to) the Kružkov entropy µ(ρ) = |ρ − k|. A short computation yields that which is consistent with (1.3). Then η(y) = y |1/y − k|, and |η (y)| ≤ |k|. If V satisfies (2.1b), the mapping ρ → q(ρ) is Lipschitz, since Hence q (t, · ) is of bounded variation, since ρ is in BV . A similar argument with a test function whose support include the initial data on t = 0, will show (1.3). We conclude that ρ is an entropy solution. Since the entropy solution is unique, we also conclude that the whole sequence, rather than just a subsequence, converges.