# American Institute of Mathematical Sciences

April  2020, 13(2): 279-307. doi: 10.3934/krm.2020010

## The BGK approximation of kinetic models for traffic

 1 RWTH Aachen University, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52062 Aachen, Germany 2 "La Sapienza" Università di Roma, Dipartimento di Matematica, Piazza Aldo Moro 5, 00185 Roma, Italy 3 University of Reading, Department of Mathematics and Statistics, Exhibition Road, London SW7 2AZ, UK

* Corresponding author: Giuseppe Visconti

Received  December 2018 Revised  July 2019 Published  January 2020

We study spatially non-homogeneous kinetic models for vehicular traffic flow. Classical formulations, as for instance the BGK equation, lead to unconditionally unstable solutions in the congested regime of traffic. We address this issue by deriving a modified formulation of the BGK-type equation. The new kinetic model allows to reproduce conditionally stable non-equilibrium phenomena in traffic flow. In particular, stop and go waves appear as bounded backward propagating signals occurring in bounded regimes of the density where the model is unstable. The BGK-type model introduced here also offers the mesoscopic description between the microscopic follow-the-leader model and the macroscopic Aw-Rascle and Zhang model.

Citation: Michael Herty, Gabriella Puppo, Sebastiano Roncoroni, Giuseppe Visconti. The BGK approximation of kinetic models for traffic. Kinetic & Related Models, 2020, 13 (2) : 279-307. doi: 10.3934/krm.2020010
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Schematic summary of the work
Equilibrium flux diagrams (top row) and equilibrium speed diagrams (bottom row) of the homogeneous kinetic model (4)-(6) with $N = 2$ and $N = 3$ speeds for the case $\Delta a = \frac12$ and $\Delta b = 0$
Left: experimental flux diagram from measurements by the Minnesota Department of Transportation, reproduced by kind permission of Seibold et al. [46]. Right: comparison between experimental data and the flux diagram resulting from the model (4)-(6), with $N = 3$ speeds and $P(\rho) = 1 - \rho^{\frac14}$ for the case $\Delta a = \frac12$ and $\Delta b = 0$
Left: characteristic speeds at equilibrium of the homogeneous kinetic model (4). Right: concavity of the equilibrium flux $Q_\text{eq}$ given in (7) for the homogeneous kinetic model (4)
Propagation of a low density perturbation up to the final time $T_M = 0.2$. Top left: solution at equilibrium with $\epsilon = 0$. Top right: solution with $\epsilon = 10^{-1}$. Bottom left: comparison of the two solutions at final time. Bottom right: flux-diagram during the evolution of the BGK model with $\epsilon = 10^{-1}$ and at equilibrium (red line)
Propagation of a high density perturbation up to the final time $T_M = 0.2$. Top left: solution at equilibrium with $\epsilon = 0$. Top right: solution with $\epsilon = 10^{-1}$. Bottom left: comparison of the two solutions at final time. Bottom right: flux-diagram during the evolution of the BGK model with $\epsilon = 10^{-1}$ and at equilibrium (red line)
Propagation of a large perturbation up to the final time $T_M = 0.2$. Top left: solution at equilibrium with $\epsilon = 0$. Top right: solution with $\epsilon = 10^{-1}$. Bottom left: comparison of the two solutions at final time. Bottom right: flux-diagram during the evolution of the BGK model with $\epsilon = 10^{-1}$ and at equilibrium (red line)
Propagation of a high density perturbation up to the time in which the density profile is bounded by the maximum value of the density. Left: solution with $\epsilon = 10^{-1}$. Right: solution with $\epsilon = 10^{-2}$
The right bottom panels show the sign of the diffusion coefficient (13) for the BGK model (9) with the equilibrium distribution of the homogeneous kinetic model (4)-(6). The blue line corresponds to the positive sign of the coefficient and the red line to the negative sign
The right bottom panels show the sign of $\mu(\rho)$ for the ARZ model (21) with different closures and pressure functions. Blue lines correspond to regimes where $\mu(\rho)>0$ and red lines to regimes where $\mu(\rho)<0$
Evolution of a high density perturbation with the ARZ model. Top-left: density profile at different instants in time. Top-right: zoom of the density profile in the region where unstable waves occur. Bottom-left: speed profile at different instants in time. Bottom-right: density profile at final time $T_M = 3$
The right bottom panels show the sign of the diffusion coefficient (38) for the BGK-type model (30) with the closure provided by the homogeneous model in Section 2 and different pressure functions. Blue lines correspond to regimes where the coefficient is positive and red lines to regimes where it is negative
Classification of flow regimes for the Boltzmann-type kinetic equation (1)
 $\epsilon$ Regime Kinetic model Continuum flow model $0$ Equilibrium flow Boltzmann Mass conservation law (LWR model) $\to 0$ Viscous flow Diffusion equation $\asymp 1$ Transitional Extended hydrodynamic equations $>1$ Rarefied - $\to\infty$ Free molecular flow Collision-less Boltzmann -
 $\epsilon$ Regime Kinetic model Continuum flow model $0$ Equilibrium flow Boltzmann Mass conservation law (LWR model) $\to 0$ Viscous flow Diffusion equation $\asymp 1$ Transitional Extended hydrodynamic equations $>1$ Rarefied - $\to\infty$ Free molecular flow Collision-less Boltzmann -
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