# American Institute of Mathematical Sciences

June  2019, 14(2): 265-288. doi: 10.3934/nhm.2019011

## Derivation of second order traffic flow models with time delays

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

Received  December 2017 Revised  December 2018 Published  April 2019

Fund Project: S. Göttlich acknowledges funding by the German Research Foundation under grant GO 1920/4-1.

Starting from microscopic follow-the-leader models, we develop hyperbolic delay partial differential equations to govern the density and velocity of vehicular traffic. The proposed models can be seen as an extension of the classical Aw-Rascle-Zhang model, where the reaction time of drivers appears as an additional term in the velocity equation. We propose numerical methods based on first principles and present a numerical study, where we focus on the impact of time delays in comparison to undelayed models.

Citation: Michael Burger, Simone Göttlich, Thomas Jung. Derivation of second order traffic flow models with time delays. Networks & Heterogeneous Media, 2019, 14 (2) : 265-288. doi: 10.3934/nhm.2019011
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##### References:
Comparison for contact discontinuity: numerical RSD and ARZ solutions vs. analytical ARZ
Comparison for shock solution: numerical RSD and ARZ solutions vs. analytical ARZ
Comparison for rarefaction wave: numerical RSD and ARZ solutions vs. analytical ARZ
Real parts $\lambda_2$ for $\gamma = 0.$
Time evolution of density and flux of the delayed microscopic and macroscopic models for $T = 0.5$
Zoom: Time evolution of density and flux of the delayed microscopic and macroscopic models for $T = 0.5$
Convergence of the microscopic to the macroscopic models at time $t = 10$ and fixed delay $T = 0.5$
Comparison of macroscopic models for $T = 0.5$ with Zoom
Comparison of macroscopic models for $T = 5$ with Zoom
Comparison TE model and convection-diffusion flow model for $T = 0.5$
Traffic light scenario
Density for the delayed model
Density for the undelayed model
Speed over time at the end of the road in the Traffic Light Situation
Fundamental relation between density and flux and fitted function
Comparison of the data-fitted delayed and undelayed ARZ model to real data
Comparison of errors for different $\Delta x$
 $\Delta x$ 5 1 0.5 0.1 RSD $||\cdot||_{2}$-Error 0.074 0.0759 0.0706 0.0335 RSD $||\cdot||_{\infty}$-Error 0.038 0.0337 0.027 0.0288 CG $||\cdot||_{2}$-Error 0.0727 0.0746 0.0692 0.0308 CG $||\cdot||_{\infty}$-Error 0.036 0.0305 0.0236 0.0288 TE $||\cdot||_{2}$-Error 0.0651 0.0683 0.0657 0.0486 TE $||\cdot||_{\infty}$-Error 0.0223 0.0181 0.0236 0.0288
 $\Delta x$ 5 1 0.5 0.1 RSD $||\cdot||_{2}$-Error 0.074 0.0759 0.0706 0.0335 RSD $||\cdot||_{\infty}$-Error 0.038 0.0337 0.027 0.0288 CG $||\cdot||_{2}$-Error 0.0727 0.0746 0.0692 0.0308 CG $||\cdot||_{\infty}$-Error 0.036 0.0305 0.0236 0.0288 TE $||\cdot||_{2}$-Error 0.0651 0.0683 0.0657 0.0486 TE $||\cdot||_{\infty}$-Error 0.0223 0.0181 0.0236 0.0288
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