A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit

We prove existence and uniqueness of solutions to a transport equation modelling vehicular traffic in which the velocity field depends non-locally on the downstream traffic density via a discontinuous anisotropic kernel. The result is obtained recasting the problem in the space of probability measures equipped with the $\infty$-Wasserstein distance. We also show convergence of solutions of a finite dimensional system, which provide a particle method to approximate the solutions to the original problem

The Aw-Rascle traffic model with locally constrained flow

We consider solutions of the Aw-Rascle model for traffic flow fulfilling a constraint on the flux at $x=0$. Two different kinds of solutions are proposed: at $x=0$ the first one conserves both the number of vehicles and the generalized momentum, while the second one conserves only the number of cars. We study the invariant domains for these solutions and we compare the two Riemann solvers in terms of total variation of relevant quantities. Finally we construct ad hoc finite volume numerical schemes to compute these solutions.Comment: 24 page

Macroscopic modeling and simulations of room evacuation

We analyze numerically two macroscopic models of crowd dynamics: the classical Hughes model and the second order model being an extension to pedestrian motion of the Payne-Whitham vehicular traffic model. The desired direction of motion is determined by solving an eikonal equation with density dependent running cost, which results in minimization of the travel time and avoidance of congested areas. We apply a mixed finite volume-finite element method to solve the problems and present error analysis for the eikonal solver, gradient computation and the second order model yielding a first order convergence. We show that Hughes' model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit.Comment: 22 page

Validation of traffic flow models on processed GPS data

Macroscopic traffic flow models allow describing the spatio-temporal evolution of traffic density. Their sound mathematical structure consisting of partial differential equations of hyperbolic type and the related efficient numerical schemes enable fast computations to monitor traffic evolution. The aim of the internship was to validate these models against processed data provided by the industrial partners Autoroutes Trafic and VINCI Autoroutes. Targeted applications included congestion detection, congestion starting and ending points location, congestion evolution in time and traveling time estimation. To this end, we used the first-order Lighthill-Whitham-Richards model with a parabolic-linear flux function. The first part of the internship has been devoted to parameters identification, performing different calibration methods and finally choosing a hybrid compromise in order to exploit to the best the available data. Afterwards, numerical simulations have been performed on a selected case study, and results have been compared to real data to assess the validity and relevancy of the model. Numerical simulations consisted in established finite volume discretization of the hyperbolic partial differential equation. Numerical results show that, while reproducing traffic evolution during all the morning is really challenging, short term predictions are reliable

The Cauchy problem at a node with buffer

International audienceWe consider the Lighthill-Whitham-Richards traffic flow model on a network composed by an arbitrary number of incoming and outgoing arcs connected together by a node with a buffer. Similar to [M. Herty, J.-P. Lebacque, and S. Moutari. A novel model for intersections of vehicular traffic flow. Netw. Heterog. Media, 4(4):813–826 (electronic), 2009], we define the solution to the Riemann problem at the node and we prove existence and well posedness of solutions to the Cauchy problem, by using the wave-front tracking technique and the generalized tangent vectors