# American Institute of Mathematical Sciences

doi: 10.3934/nhm.2020032

## 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

Received  March 2020 Revised  October 2020 Published  December 2020

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.

Citation: Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, doi: 10.3934/nhm.2020032
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Comparison between density profiles computed with (right) and without(left) CFL condition in case of a rarefaction wave
Comparison between density profiles computed with (right) and without (left) CFL condition in a shock framework
Test 0: Comparing the density evolution computed by the delayed model (left) and the LWR model (right)
Test 0: Comparison between density profiles corresponding to different grid steps size
Test 0: Density evolution and profile, at time $T = \frac{1}{3}T_f$, in case of a too high delay
Test 1: Reproducing the simulation presented in [7], with $\rho^0(x,1)$ on the left and $\rho^0(x,2)$ on the right
Test 2: Density values in the $(x,t)$-plane (left) and density profile at time $T = T_f$ (right)
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)
Triggering of Stop & Go waves: Density values in the $(x,t)$-plane
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