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A nonlocal Lagrangian traffic flow model and the zero-filter limit

Giuseppe M. Coclite, Kenneth H. Karlsen, Nils Henrik Risebro

Abstract

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the $L^1$ sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the ``zero-filter'' (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the ``filter size'' and (ii) the dissipation of an arbitrary convex entropy.

A nonlocal Lagrangian traffic flow model and the zero-filter limit

Abstract

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the ``zero-filter'' (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the ``filter size'' and (ii) the dissipation of an arbitrary convex entropy.
Paper Structure (9 sections, 14 theorems, 140 equations, 4 figures)

This paper contains 9 sections, 14 theorems, 140 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Analysis of a fully discrete scheme
  3. The nonlocal Lagrangian PDE for $y=1/u$
  4. Eulerian formulation
  5. Zero-filter limit of the nonlocal model
  6. Numerical examples
  7. Comparing different models
  8. The zero-filter ($\alpha\to 0$) limit
  9. Convergence of $y_\alpha$ and the effect of different filters.

Key Result

Lemma 2.1

If ${\Delta t}$ and ${\Delta x}$ are chosen such that the CFL -condition holds, then the scheme eq:wscheme is monotone in the sense that where $\widetilde{w}^{n+1}$ is a corresponding solution of eq:wscheme.

Figures (4)

  • Figure 1: Numerical solutions of \ref{['eq:numLWR']} -- \ref{['eq:numbar']} computed by the explicit Euler scheme. Left: $\ell=0.06$ , right: $\ell=0.005$.
  • Figure 2: Solutions of \ref{['eq:wschemep']}, \ref{['eq:numbar']},with initial data given in \ref{['eq:rho01']}, \ref{['eq:initdata']}. For all computations $t=1.2$ and $\ell=1/2000$. For comparisons we also show a numerical solution of \ref{['eq:rholaw']}. Upper left: $\alpha=1/2$, upper right: $\alpha=1/8$, lower left: $\alpha=1/32$, lower right: $\alpha=1/128$.
  • Figure 3: Solutions of \ref{['eq:wschemep']}, \ref{['eq:numbar']},with initial data given in \ref{['eq:rho01']}, \ref{['eq:initdata']}. For all computations $t=1.2$ and $\ell=1/5000$. In the left column, $\Phi=\Phi_{\mathrm{tri}}$, in the right column, $\Phi=\Phi_{\mathrm{box}}$.
  • Figure 4: Solutions of \ref{['eq:wschemep']}, \ref{['eq:numbar']},with initial data given in \ref{['eq:rho01']}, \ref{['eq:initdata']}, using the discontinuous filter $\Phi_{\mathrm{box}}$ with $\alpha=1/256$ and $\ell=1/10000$. The figure to the right is just an enlargement of a region of the left figure.

Theorems & Definitions (32)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 22 more