Uncertainty quantification in hierarchical vehicular flow models

Figure 1.  Outline of the model hierarchy. The left two columns indicate the kinetic and fluid description of traffic flow as presented in [24]. The third column refers to the diffusion coefficient $ \mu(\rho) $ to classify traffic instabilities. The green hierarchy is deterministic while the blue includes a parametric uncertainty $ \xi. $ The indicated links are established in this paper

Figure 2.  Probability (53) for a Maxwellian $ M_f $ with different number of discrete velocities: $ n_v = 3 $, $ n_v = 10 $. On the x-axis $ \rho_0 $ is shown, see (61)

Figure 3.  Probability (53) for different velocities samples: $ n_v = 3 $ (blue line), $ n_v = 10 $(red line) for different values of $ \rho_0 = 0.4 $(left) $ \rho_0 = 0.6 $(right) when the standard deviation $ \sigma $ ranges from zero to $ 0.2 $

Figure 4.  Probability (53) for different hesitation functions: $ h(\rho) = \rho $(blue line) and $ h(\rho) = \rho^3 $(red line)

Figure 5.  Mean and variance of the density at $ t = T_f $ for different

Figure 6.  Probability (53) at time $ t = 0 $ (black line), $ t = \frac{{T_f}}{2} $ (green line) $ t = T $ (blue line) for $ K = 4 $ (top-left), $ K = 8 $ (top-right), $ K = 32 $ (bottom-left), $ K = 64 $ (bottom-right)

Figure 7.  Probability of negative diffusion coefficient in a rarefaction case at different time: $ t = 0 $, $ t = \frac{T_f}{2} $, $ t = T_f $, $ K = 64 $, and comparison with the confident region of the density at $ t = T_f $

Figure 8.  Density profile and probability of negative diffusion coefficient in a shock case at $ t = T_f $, $ K = 64 $