Research

We consider first order Hamilton-Jacobi (HJ) equations posed on a "junction", that is to say the union of a finite number of half-lines with a unique common point. For this continuous HJ problem, we propose a finite difference scheme and prove two main results. As a first result, we show bounds on the discrete gradient and time derivative of the numerical solution. Our second result is the convergence (for a subsequence) of the numerical solution towards a viscosity solution of the continuous HJ problem, as the mesh size goes to zero. When the solution of the continuous HJ problem is unique, we recover the full convergence of the numerical solution. We apply this scheme to compute the densities of cars for a traffic model. We recover the well-known Godunov scheme outside the junction point and we give a numerical illustration.

This paper revisits the variational theory (VT) of traffic flow, now under the presence of continuum lateral inflows and outflows to the freeway say Eulerian source terms. It is found that a VT solution can be easily exhibited only in Eulerian coordinates when source terms are exogenous meaning that they only depend on time and space, but not when they are a function of traffic conditions, as per a merge model. In discrete time, however, these dependencies become exogenous, which allowed us to propose improved numerical solution methods. In Lagrangian and vehicle number-space coordinates, VT solutions may not exist even if source terms are exogenous.

The aim of this paper is to propose a new event-based mesoscopic approach to model multi-class traffic flow on multi-lane road sections. The mesoscopic model was first proposed by Leclercq and Bécarie (2012) and turns out to be equivalent to the resolution of the seminal LWR model in lagrangian-space coordinates n − x. It is fully consistent at a macroscopic scale with the LWR model while keeping track of individual vehicles. Our model is built on Hamilton-Jacobi equations which have been proven to provide an efficient framework in traffic flow modeling for exact numerical methods at a low computational cost. The paper revisits the multi-class problem with a continuous moving bottleneck approach (instead of a sequential one), introducing a capacity drop parameter for multi-lane sections. It also overhauls the Daganzo diverge model with a relaxed FIFO assumption.

We set a rigorous link between microscopic and macroscopic traffic models by homogenization of a system of Ordinary Differential Equations (ODEs) that describe the dynamics of vehicles. At the microscopic level, each vehicle behaves according to a given speed-spacing function depending on its leading vehicle. After a proper asymptotic rescaling that can be seen as unzooming, we show that solutions of these systems of ODEs converge to the solution of a macroscopic traffic flow model, based on Hamilton-Jacobi equation on one dimensional domain.

We derive first order continuum traffic flow models from a microscopic delayed follow-the-leader model. This micro model is based on an optimal velocity function and a reaction time parameter. The corresponding macroscopic formulations in Eulerian or Lagrangian coordinates result in first order convection-diffusion equations. More precisely, the convection is described by the optimal velocity while the diffusion term depends on the reaction time. A linear stability analysis for homogeneous solutions of both continuous and discrete models are provided. The behavior of the novel model is illustrated thanks to numerical simulations. Transitions to collision-free self-sustained stop-and-go dynamics are obtained if the reaction time is sufficiently large. The results show that the dynamics of the microscopic model can be well captured by the macroscopic equations. For non–zero reaction times we observe a scattered fundamental diagram. The scattering width is compared to real data.

We deal with a macroscopic model of multi-anticipative car-following behaviour i.e. driving behaviour taking into account several vehicles ahead. Some empirical studies have suggested that drivers not only react to the closest leader vehicle but also anticipate on traffic conditions further ahead. Using a recent mathematical result of homogenization for a general class of car-following models (and also available for multi-anticipative models), we deeply investigate the effects of multi-anticipation at the microscopic level on the macroscopic traffic flow. To investigate multi-anticipation behaviour may be fundamental to understand better cooperative traffic flow dynamics.

This paper is concerned with the numerical homogenization of Hamilton-Jacobi equations posed on a periodic network embedded in R^2. For this continuous HJ problem, we propose a finite difference scheme which was previously designed for HJ equations on a single junction. We provide some qualitative properties of the effective Hamiltonian. We also apply a derived scheme to compute the densities of vehicles for a traffic flow model on the network and we recover some macroscopic features obtained in traffic flow theory. We finally provide numerical computations of the effective Hamiltonian in the case of two consecutive traffic lights on R.

This paper deals with numerical methods providing semi-analytic solutions to a wide class of macroscopic traffic flow models for piecewise affine initial and boundary conditions. In a very recent paper, a variational principle has been proved for models of the Generic Second Order Modeling (GSOM) family, yielding an adequate framework for effective numerical methods. Any model of the GSOM family can be recast into its Lagrangian form as a Hamilton-Jacobi equation (HJ) for which the solution is interpreted as the position of vehicles. This solution can be computed thanks to Lax-Hopf like formulas and a generalization of the inf-morphism property. The efficiency of this computational method is illustrated through a numerical example and finally a discussion about future developments is provided.

We consider a macroscopic traffic model able to reproduce the boundedness of the vehicles acceleration. It is based on a modified Lighthill-Whitham-Richards (LWR) model recast as a Hamilton-Jacobi Partial Differential Equation for which there exists explicit solution methods. We developed an optimization-based framework which is able to be extended to large scale networks in order to estimate queue lengths on arterial road networks, taking into account data from classical density or flow sensors and mobile sensors. We validated our results on real data extracted from the NGSIM Lankershim Boulevard dataset, comparing the estimation result from LWR model with and without the bounded acceleration constraint. Comparing to previous method without considering bounded acceleration, our method improves the queue lengths estimation precision under most cases.

This paper details a new macroscopic traffic flow model accounting for the boundedness of traffic acceleration, which is required for physical realism. Our approach relies on the coupling between a scalar conservation law, which refers to the seminal LWR model, and a system of Ordinary Differential Equations describing the trajectory of accelerating vehicles, which we treat as moving constraints. We propose a Wave Front Tracking Algorithm to construct approximate solutions. We use this algorithm to prove the existence of solutions to the associated Cauchy Problem, and provide some numerical examples illustrating the solution behaviour.

This article deals with macroscopic traffic flow models on a road network. More precisely, we consider coupling conditions at junctions for the Aw-Rascle-Zhang (ARZ) second order model consisting of a hyperbolic system of two conservation laws. These coupling conditions conserve both the number of vehicles and the mixing of Lagrangian attributes of traffic through the junction. The proposed Riemann solver is based on assignment coefficients, multi-objective optimization of fluxes and priority parameters. We prove that this Riemann solver is well posed in the case of special junctions, including 1-to-2 diverge and 2-to-1 merge.

This paper is concerned with a generic class of second order models (GSOM) on networks. Such higher order models can be more easily solved in a Lagrangian framework whose coordinates move with the traffic. The difficulty is to deal with Eulerian-fixed discontinuities such as junctions. The aim of this work is twofold: first, to propose adapted junction models for second order macroscopic traffic flow models and second, to solve the resulting model in a moving framework. Our numerical methodology is based on a finite difference scheme that bridges the gap with a microscopic description of the traffic.