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# Non-local multi-class traffic flow models

• * Corresponding author: Paola Goatin
• We prove the existence for small times of weak solutions for a class of non-local systems in one space dimension, arising in traffic modeling. We approximate the problem by a Godunov type numerical scheme and we provide uniform ${{\mathbf{L}}^\infty }$ and BV estimates for the sequence of approximate solutions, locally in time. We finally present some numerical simulations illustrating the behavior of different classes of vehicles and we analyze two cost functionals measuring the dependence of congestion on traffic composition.

Mathematics Subject Classification: Primary: 35L65, 90B20; Secondary: 65M08.

 Citation: • • Figure 1.  Numerical simulation illustrating that the simplex $\mathcal{S}$ is not an invariant domain for (1). We take $M = 2$ and we consider the initial conditions $\rho_1(0, x) = 0.9 \chi_{[-0.5, -0.3]}$ and $\rho_2(0, x) = 0.1 \chi_{]-\infty, 0]}+\chi_{]0, +\infty[}$ depicted in (a), the constant kernels $\omega_1(x) = \omega_2(x) = 1/\eta$, $\eta = 0.5$, and the speed functions given by $v^{max}_1 = 0.2$, $v^{max}_2 = 1$, $\psi(\xi) = \max\{1-\xi, 0\}$ for $\xi\geq 0$. The space and time discretization steps are $\Delta x = 0.001$ and $\Delta t = 0.4 \Delta x$. Plots (b) and (c) show the density profiles of $\rho_1$, $\rho_2$ and their sum $r$ at times $t = 1.8, ~2.8$. The function $\max_{x\in\mathbb{R}} r(t, x)$ is plotted in (d), showing that $r$ can take values greater than 1, even if $r(0, x) = \rho_1(0, x)+\rho_2(0, x)\leq1$

Figure 2.  Density profiles of cars and trucks at increasing times corresponding to the non-local model (28)

Figure 3.  Density profiles corresponding to the non-local problem (29) with $\beta = 0.9$ at different times

Figure 4.  Functional $J$ (left) and $\Psi$ (right)

Figure 5.  $(t, x)$-plots of the total traffic density $r(t, x) = \rho_1(t, x)+\rho_2(t, x)$ in (29) corresponding to different values of $\beta$: (a) no autonomous vehicles are present; (b) point of minimum for $\Psi$; (c) point of minimum for $J$; (d) point of maximum for $J$

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