# American Institute of Mathematical Sciences

December  2017, 12(4): 591-617. doi: 10.3934/nhm.2017024

## Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams

 1 Université Côte d'Azur, Inria, CNRS, LJAD, Parc Valrose, 06108 Nice, France 2 Institut de Mécanique des Fluides de Toulouse, CNRS UMR 5502, France 3 Université d'Orléans, MAPMO, UMR CNRS 7349, France

Received  December 2016 Revised  August 2017 Published  October 2017

We discuss numerical strategies to deal with PDE systems describing traffic flows, taking into account a density threshold, which restricts the vehicle density in the situation of congestion. These models are obtained through asymptotic arguments. Hence, we are interested in the simulation of approached models that contain stiff terms and large speeds of propagation. We design schemes intended to apply with relaxed stability conditions.

Citation: Florent Berthelin, Thierry Goudon, Bastien Polizzi, Magali Ribot. Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams. Networks & Heterogeneous Media, 2017, 12 (4) : 591-617. doi: 10.3934/nhm.2017024
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##### References:
Numerical results in the case of transport (15) - (16). Density (left) and velocity (right) with the Glimm scheme (top) and the explicit-implicit scheme (bottom). The results are given for the three different pressures under consideration: pressure (VO1) in black, pressure (VO2) in blue and pressure (VO3) in green. The exact solution is plotted in red
Numerical results in the case of decongestion (17) - (18). Comparison of the two schemes. Density (left) and velocity (right) at final time $T=0.2$. Top: pressure (VO1) with Glimm scheme (black). On the top, the simulations are performed implicit-explicit scheme (blue) and pressure (VO2) with implicit-explicit scheme (green). Bottom: pressure (VO3) with Glimm scheme (black) and with the explicit-implicit scheme (blue). The exact solution is plotted in red. Parameters : $\gamma=2, \varepsilon =10^{-3}$ for pressures (VO1) and (VO2) and $\gamma=4$ for pressure (VO3)
Numerical results in the case of decongestion (17) - (18). Pressure (VO3) for different values of $\gamma$. Density (left) and velocity (right) at final time $T=0.2$: $\gamma=4$ (black), $\gamma=20$ (blue) and $\gamma=100$ (green). The simulations are performed with the implicit-explicit scheme and the exact solution is plotted in red
Numerical results in the case of decongestion (17) - (18). Pressure (VO2) for different values of $\varepsilon$ and $\gamma$. Density (left) and velocity (right) at final time $T=0.2$: (VO2) for $\gamma=2, \varepsilon =10^{-3}$ (black), $\gamma=2, \varepsilon =10^{-5}$ (blue), $\gamma=3, \varepsilon =10^{-5}$ (green) and $\gamma=3, \varepsilon =10^{-7}$ (orange). The simulations are performed with the implicit-explicit scheme and the exact solution is plotted in red
Numerical results in the case of congestion -Comparison of the two schemes. Density (left) and velocity (right) at final time $T=0.01$. Glimm scheme (top) and implicit-explicit scheme (bottom), with pressure (VO1) (black), (VO2) (blue) and (VO3) (green). The exact solution is plotted in red. Parameters : $\gamma=2, \varepsilon =10^{-3}$ for pressures (VO1) and (VO2) and $\gamma=4$ for pressure (VO3)
Numerical results in the case of congestion -Comparison of the two schemes -Different parameters. For (VO1) and (VO2), we use $\gamma=2$ and $\varepsilon =10^{-5}$; for (VO3), we take $\gamma=50$. Pressure (VO1) in black, pressure (VO2) in blue and pressure (VO3) in green. The exact solution is plotted in red
Numerical results in the case of congestion -Pressure (VO3) for different values of $\gamma$. Density (left) and velocity (right) at final time $T=0.01$. Pressure (VO3) for $\gamma=20$ (black), $\gamma=50$ (blue) and $\gamma=100$ (green). The simulations are performed with the Glimm scheme (top) and the mplicit-explicit scheme (bottom). The exact solution is plotted in red
Numerical results in the case of congestion -Pressure (VO2) for different values of $\gamma$ and $\varepsilon$. Density (left) and velocity (right) at final time $T=0.01$. We compare the pressure (VO2) for $\gamma=2, \varepsilon =10^{-5}$ (black), $\gamma=3, \varepsilon =10^{-5}$ (blue) and $\gamma=3, \varepsilon =10^{-7}$ (green). The simulations are performed with the Glimm scheme (top) and the implicit-explicit scheme (bottom). The exact solution is plotted in red
Numerical results in the case of a shock between two blocks -Pressure (VO1) for $\gamma=2$ and different values of $\varepsilon$: $\varepsilon =10^{-3}$ (black), $\varepsilon =10^{-4}$ (blue), $\varepsilon =10^{-5}$ (green) and $\varepsilon =10^{-6}$ (orange). Figure 9a (on the left) represents the densities whereas figure 9b (on the right) represents the velocities. The exact solution is plotted in red
Numerical results in the case of a shock between two blocks -Pressure VO2 for $\gamma=2$ and different values of $\varepsilon$: $\varepsilon =10^{-3}$ (black), $\varepsilon =10^{-4}$ (blue), $\varepsilon =10^{-5}$ (green) and $\varepsilon =10^{-6}$ (orange). On top, simulations are performed with the Glimm scheme and on the bottom, with the implicit-explicit scheme. We display the densities on the left and the velocities on the right. The exact solution is plotted in red
Numerical results in the case of a shock between two blocks -Pressure VO3 for different values of $\gamma$: $\gamma=16$ (black), $\gamma=32$ (blue), $\gamma=64$ (green) and $\gamma=128$ (orange). On top, simulations are performed with the Glimm scheme and on the bottom, with the implicit-explicit scheme. We display the densities on the left and the velocities on the right. The exact solution is plotted in red
Comparaison of the numerical scheme and velocity offset for the initial data (AI). We display the densities on the left and the velocities on the right, at $t=0.2$ (top), $t=0.4$ (middle) and $t=0.6$ (bottom). Glimm scheme with pressures (VO1) (in black), (VO2) (in green) and (VO3) (in purple); implicit-explicit scheme with pressures (VO1) (in blue), (VO2) (in orange) and (VO3) (in grey). The exact solution is plotted in red
Comparaison of the numerical scheme and velocity offset for the initial data (AⅢ). We display the densities on the left and the velocities on the right, at $t=0.27$ (top), $t=0.53$ (middle) and $t=0.8$ (bottom). Glimm scheme with pressures (VO1) (in black), (VO2) (in green) and (VO3) (in purple); implicit-explicit scheme with pressures (VO1) (in blue), (VO2) (in orange) and (VO3) (in grey). The exact solution is plotted in red
Time steps -comparison between Glimm scheme and the explicit-implicit scheme for the congestion case. Pressure (VO3) for different values of γ and pressure (VO2) for γ = 2 and different values of ". The time step is the smallest time step used during the simulation and the factor is the ratio of the time step for the explicit-implicit scheme over the time step for the Glimm scheme
 Pressure & param Time step Glimm scheme Time step explicit-implicit scheme Factor Pressure (VO2), ε = 10−4 ∆t = 2·10−6 ∆t = 2·10−6 1 Pressure (VO2), ε = 10−5 ∆t = 7·10−7 ∆t = 10−6 1.39 Pressure (VO2), ε = 10−6 ∆t = 2·10−7 ∆t = 7:7·10−7 3.22 Pressure (VO2), ε = 10−7 ∆t = 7:5·10−8 ∆t = 6:2·10−7 8.18 Pressure (VO3), γ = 50 ∆t = 9·10−6 ∆t = 10−5 1.12 Pressure (VO3), γ = 100 ∆t = 4:8·10−6 ∆t = 6:4·10−6 1.36 Pressure (VO3), γ = 200 ∆t = 2:4·10−6 ∆t = 5:6·10−6 2.33 Pressure (VO3), γ = 500 ∆t = 9:5·10−7 ∆t = 2:7·10−5 27.95
 Pressure & param Time step Glimm scheme Time step explicit-implicit scheme Factor Pressure (VO2), ε = 10−4 ∆t = 2·10−6 ∆t = 2·10−6 1 Pressure (VO2), ε = 10−5 ∆t = 7·10−7 ∆t = 10−6 1.39 Pressure (VO2), ε = 10−6 ∆t = 2·10−7 ∆t = 7:7·10−7 3.22 Pressure (VO2), ε = 10−7 ∆t = 7:5·10−8 ∆t = 6:2·10−7 8.18 Pressure (VO3), γ = 50 ∆t = 9·10−6 ∆t = 10−5 1.12 Pressure (VO3), γ = 100 ∆t = 4:8·10−6 ∆t = 6:4·10−6 1.36 Pressure (VO3), γ = 200 ∆t = 2:4·10−6 ∆t = 5:6·10−6 2.33 Pressure (VO3), γ = 500 ∆t = 9:5·10−7 ∆t = 2:7·10−5 27.95
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