Paola Goatin

Christophe Chalons, Paola Goatin, Luis Villada

This paper focuses on the numerical approximation of the solutions of non-local conservation laws in one space dimension. These equations are motivated by two distinct applications, namely a traffic flow model in which the mean velocity depends on a weighted mean of the downstream traffic density, and a sedimentation model where either the solid phase velocity or the solid-fluid relative velocity depends on the concentration in a neighborhood. In both models, the velocity is a function of a convolution product between the unknown and a kernel function with compact support. It turns out that the solutions of such equations may exhibit oscillations that are very difficult to approximate using classical first-order numerical schemes. We propose to design Discontinuous Galerkin (DG) schemes and Finite Volume WENO (FV-WENO) schemes to obtain high-order approximations. As we will see, the DG schemes give the best numerical results but their CFL condition is very restrictive. On the contrary, FV-WENO schemes can be used with larger time steps. We will see that the evaluation of the convolution terms necessitates the use of quadratic polynomials reconstructions in each cell in order to obtain the high-order accuracy with the FV-WENO approach. Simulations using DG and FV-WENO schemes are presented for both applications.

Oliver Kolb, Guillaume Costeseque, Paola Goatin et al.

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 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.

Sosina Gashaw, Jérôme Härri, Paola Goatin

Lagrangian formulation of kinematic wave provides a more accurate representation than the most commonly used Eulerian formulation. Furthermore, Lagrangian representation offers a flexibility to study certain traffic phenomena (e.g. capacity drop, traffic delay, trajectory). The natural resemblance of Lagrangian representation to road traffic data collection methods (probe vehicles) also renders it suitable for applications such as traffic motoring. This paper presents a multi-class Lagrangian representation for a traffic flow consisting of cars and two-wheelers. A numerical method for solving the continuity equation is presented. We compare the results obtained with Eulerian and Lagrangian formulations.