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June  2019, 14(2): 371-387. doi: 10.3934/nhm.2019015

Non-local multi-class traffic flow models

 Inria Sophia Antipolis - Méditerranée, Université Côte d'Azur, Inria, CNRS, LJAD, 2004 route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France

* Corresponding author: Paola Goatin

Received  August 2018 Revised  October 2018 Published  April 2019

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.

Citation: Felisia Angela Chiarello, Paola Goatin. Non-local multi-class traffic flow models. Networks & Heterogeneous Media, 2019, 14 (2) : 371-387. doi: 10.3934/nhm.2019015
References:
 [1] A. Aggarwal, R. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983.  doi: 10.1137/140975255.  Google Scholar [2] P. Amorim, R. Colombo and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM M2AN, 49 (2015), 19-37.  doi: 10.1051/m2an/2014023.  Google Scholar [3] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916–938 (electronic), URL http://dx.doi.org/10.1137/S0036139997332099. doi: 10.1137/S0036139997332099.  Google Scholar [4] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587–612, URL https://doi.org/10.1017/S0956792503005266. doi: 10.1017/S0956792503005266.  Google Scholar [5] S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), 217–241, URL http://dx.doi.org/10.1007/s00211-015-0717-6. doi: 10.1007/s00211-015-0717-6.  Google Scholar [6] R. M. Colombo and A. Groli, Minimising stop and go waves to optimise traffic flow, Appl. Math. Lett., 17 (2004), 697-701.  doi: 10.1016/S0893-9659(04)90107-3.  Google Scholar [7] F. A. Chiarello and P. Goatin, Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel, ESAIM: M2AN, 52 (2018), 163–180, URL https://doi.org/10.1051/m2an/2017066. doi: 10.1051/m2an/2017066.  Google Scholar [8] J. Friedrich, O. Kolb and S. Göttlich, A Godunov type scheme for a class of scalar conservation laws with non-local flux, Netw. Heterog. Media, 13 (2018), 531–547, URL http://dx.doi.org/10.3934/nhm.2018024. Google Scholar [9] P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107–121, URL http://dx.doi.org/10.3934/nhm.2016.11.107. doi: 10.3934/nhm.2016.11.107.  Google Scholar [10] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar [11] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar [12] H. Payne, Models of Freeway Traffic and Control, Simulation Councils, Incorporated, 1971. Google Scholar [13] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar [14] A. Sopasakis and M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 66 (2006), 921–944 (electronic). doi: 10.1137/040617790.  Google Scholar [15] G. Whitham, Linear and Nonlinear Waves, Pure and applied mathematics, Wiley, 1974.  Google Scholar [16] H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275–290, URL http://www.sciencedirect.com/science/article/pii/S0191261500000503. doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

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References:
 [1] A. Aggarwal, R. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983.  doi: 10.1137/140975255.  Google Scholar [2] P. Amorim, R. Colombo and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM M2AN, 49 (2015), 19-37.  doi: 10.1051/m2an/2014023.  Google Scholar [3] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916–938 (electronic), URL http://dx.doi.org/10.1137/S0036139997332099. doi: 10.1137/S0036139997332099.  Google Scholar [4] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587–612, URL https://doi.org/10.1017/S0956792503005266. doi: 10.1017/S0956792503005266.  Google Scholar [5] S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), 217–241, URL http://dx.doi.org/10.1007/s00211-015-0717-6. doi: 10.1007/s00211-015-0717-6.  Google Scholar [6] R. M. Colombo and A. Groli, Minimising stop and go waves to optimise traffic flow, Appl. Math. Lett., 17 (2004), 697-701.  doi: 10.1016/S0893-9659(04)90107-3.  Google Scholar [7] F. A. Chiarello and P. Goatin, Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel, ESAIM: M2AN, 52 (2018), 163–180, URL https://doi.org/10.1051/m2an/2017066. doi: 10.1051/m2an/2017066.  Google Scholar [8] J. Friedrich, O. Kolb and S. Göttlich, A Godunov type scheme for a class of scalar conservation laws with non-local flux, Netw. Heterog. Media, 13 (2018), 531–547, URL http://dx.doi.org/10.3934/nhm.2018024. Google Scholar [9] P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107–121, URL http://dx.doi.org/10.3934/nhm.2016.11.107. doi: 10.3934/nhm.2016.11.107.  Google Scholar [10] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar [11] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar [12] H. Payne, Models of Freeway Traffic and Control, Simulation Councils, Incorporated, 1971. Google Scholar [13] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar [14] A. Sopasakis and M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 66 (2006), 921–944 (electronic). doi: 10.1137/040617790.  Google Scholar [15] G. Whitham, Linear and Nonlinear Waves, Pure and applied mathematics, Wiley, 1974.  Google Scholar [16] H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275–290, URL http://www.sciencedirect.com/science/article/pii/S0191261500000503. doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar
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 $r(0, x) = \rho_1(0, x)+\rho_2(0, x)\leq1$
Density profiles of cars and trucks at increasing times corresponding to the non-local model (28)
Density profiles corresponding to the non-local problem (29) with $\beta = 0.9$ at different times
Functional $J$ (left) and $\Psi$ (right)
$(t, x)$-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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