Non-local conservation laws for traffic flow modeling

In this thesis, we provide mathematical traffic flow models with non-local fluxes and adapted numerical schemes to compute approximate solutions to such kind of equations. More precisely, we consider flux functions depending on an integral evaluation of the conserved variables through a convolution product. First of all, we prove the well-posedness of entropy weak solutions for a class of scalar conservation laws with non-local flux arising in traffic modeling. This model is intended to describe the reaction of drivers that adapt their velocity with respect to what happens in front of them. Here, the support of the convolution kernel is proportional to the look-ahead distance of drivers. We approximate the problem by a Lax- Friedrichs scheme and we provide some estimates for the sequence of approximate solutions. Stability with respect to the initial data is obtained through the doubling of variable technique. We study also the limit model as the kernel support tends to infinity. After that, we prove the stability of entropy weak solutions of a class of scalar conservation laws with non-local flux under higher regularity assumptions. We obtain an estimate of the dependence of the solution with respect to the kernel function, the speed and the initial datum. We also prove the existence for small times of weak solutions for non-local systems in one space dimension, given by a non-local multi-class model intended to describe the behaviour of different groups drivers or vehicles. We approximate the problem by a Godunov-type numerical scheme and we provide uniform L∞ and BV estimates for the sequence of approximate solutions, locally in time. We 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. Furthermore, we propose alternative simple schemes to numerically integrate non-local multi- class systems in one space dimension. We obtain these schemes by splitting the non-local conservation laws into two different equations, namely, the Lagrangian and the remap steps. We provide some estimates recovered by approximating the problem with the Lagrangian- Antidiffusive Remap (L-AR) schemes, and we prove the convergence to weak solutions in the scalar case. Finally, we show some numerical simulations illustrating the efficiency of the LAR schemes in comparison with classical first and second order numerical schemes. Moreover, we recover the numerical approximation of the non-local multi-class traffic flow model proposed, presenting the multi-class version of the Finite Volume WENO (FV-WENO) schemes, in order to obtain higher order of accuracy. Simulations using FV-WENO schemes for a multi-class model for autonomous and human-driven traffic flow are presented. Finally, we introduce a traffic model for a class of non-local conservation laws at road junctions. Instead of a single velocity function for the whole road, we consider two different road segments, which may differ for their speed law and number of lanes. We use an upwind type numerical scheme to construct a sequence of approximate solutions and we provide uniform L∞ and BV estimates. Using a Lax-Wendroff type argument, we prove the well-posedness of the proposed model. Some numerical simulations are compared with the corresponding (discontinuous) local model.