Article Contents

2021, Volume 16, Issue 1: 31-47. Doi: 10.3934/nhm.2020032

This issue Previous Article A 2-dimensional shape optimization problem for tree branches Next Article Pointwise long time behavior for the mixed damped nonlinear wave equation in

$ \mathbb{R}^n_+ $

Properties of the LWR model with time delay

1.
University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany
2.
Fraunhofer Institute ITWM, 67663 Kaiserslautern, Germany

^* Corresponding author: Elisa Iacomini
^* Corresponding author: Elisa Iacomini

Received: March 2020

Revised: October 2020

Early access: December 2020

Published: March 2021

Abstract / Introduction Full Text(HTML) Figure(9) Related Papers Cited by

Abstract

In this article, we investigate theoretical and numerical properties of the first-order Lighthill-Whitham-Richards (LWR) traffic flow model with time delay. Since standard results from the literature are not directly applicable to the delayed model, we mainly focus on the numerical analysis of the proposed finite difference discretization. The simulation results also show that the delay model is able to capture Stop & Go waves.

Keywords:

Macroscopic traffic flow models,
hyperbolic delay partial differential equation,
Lax-Friedrichs discretization,
numerical simulations,
stop & go waves.

Mathematics Subject Classification: 35L65, 90B20, 65M06.

Citation:

FullText(HTML)

Figure 1. Comparison between density profiles computed with (right) and without(left) CFL condition in case of a rarefaction wave

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Figure 2. Comparison between density profiles computed with (right) and without (left) CFL condition in a shock framework

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Figure 3. Test 0: Comparing the density evolution computed by the delayed model (left) and the LWR model (right)

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Figure 4. Test 0: Comparison between density profiles corresponding to different grid steps size

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Figure 5. Test 0: Density evolution and profile, at time $ T = \frac{1}{3}T_f $, in case of a too high delay

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Figure 6. Test 1: Reproducing the simulation presented in [7], with $ \rho^0(x,1) $ on the left and $ \rho^0(x,2) $ on the right

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Figure 7. Test 2: Density values in the $ (x,t) $-plane (left) and density profile at time $ T = T_f $ (right)

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Figure 8. Test 2: Density values in the $ (x,t) $-plane with low delay term, $ T_\Delta = 4 \Delta t $, (left) and density profile at $ T = T_f $ (right)

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Figure 9. Triggering of Stop & Go waves: Density values in the $ (x,t) $-plane

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