Model of vehicle interactions with autonomous cars and its properties

Figure 1.  Left: flux–density diagram (4). Middle: variance of microscopic speeds at equilibrium (9) as function of the density $ \rho $. Right: diffusion coefficient (13) in the diffusive limit of the BGK model (11) as function of the density $ \rho $. The plots are obtained using the Maxwellian of the spatially homogeneous model (3)–(6) with $ \Delta v = \frac{ v_{\max}}{3} $. Here, $ \rho_{\max} = 1 $ and $ v_{\max} = 1 $

Figure 2.  Schematic representation of the Ansatz (29) for the kinetic distribution with $ \rho = 0.8 \rho_{\max} $ and $ \delta_1 = 1 $ (dotted line), $ \delta_2 = 4 $ (dashed line), $ \delta_3 = 7 $ (solid line). The values $ \overline{f}_i $, $ i = 1,2,3 $, are found in order to satisfy mass conservation

Figure 3.  Fundamental diagrams of the single–distribution model (21) for autonomous and human–driven vehicles for three choices of the parameter $ \bar{\rho} $. The different diagrams refer to different penetration rates, $ p\leq 0.4 $. Here $ v_{\max} = 1 = \rho_{\max} = 1 $, $ \Delta v = \frac13 $ and $ \hat{u}(\rho) = \frac{1}{\rho}\int_{\mathcal{V}} v f(t,v) \mathrm{d} v $

Figure 4.  Variance of microscopic speeds at equilibrium of the single–distribution model (21) for autonomous and human–driven vehicles. Three choices of the parameter $ \bar{\rho} $ and different penetration rates $ p $ are considered. Here $ v_{\max} = 1 = \rho_{\max} = 1 $, $ \Delta v = \frac13 $ and $ \hat{u}(\rho) = \frac{1}{\rho}\int_{\mathcal{V}} v f(t,v) \mathrm{d} v $

Figure 5.  Top row: sign of the diffusion coefficient $ \mu(\rho) $ (13) of the single–distribution model (21) for autonomous and human–driven vehicles. Three choices of the parameter $ \bar{\rho} $ and different penetration rates $ p $ are considered. Here $ v_{\max} = 1 = \rho_{\max} = 1 $, $ \Delta v = \frac13 $ and $ \hat{u}(\rho) = \frac{1}{\rho}\int_{\mathcal{V}} v f(t,v) \mathrm{d} v $. Bottom row: left boundary $ \alpha $ and right boundary $ \beta $ of the interval of instability, and its amplitude $ \Gamma $, as function of the penetration rate $ p $