A discontinuous Galerkin solver for front propagation

International audienceWe propose a new discontinuous Galerkin (DG) method based on [Cheng and Shu, JCP, 2007] to solve a class of Hamilton-Jacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with non isotropic dynamics. Several numerical experiments show the relevance of our method, in particular for front propagation

A New Discontinuous Galerkin Finite Element Method for Directly Solving the Hamilton-Jacobi Equations

In this paper, we improve upon the discontinuous Galerkin (DG) method for Hamilton-Jacobi (HJ) equation with convex Hamiltonians in (Y. Cheng and C.-W. Shu, J. Comput. Phys. 223:398-415,2007) and develop a new DG method for directly solving the general HJ equations. The new method avoids the reconstruction of the solution across elements by utilizing the Roe speed at the cell interface. Besides, we propose an entropy fix by adding penalty terms proportional to the jump of the normal derivative of the numerical solution. The particular form of the entropy fix was inspired by the Harten and Hyman's entropy fix (A. Harten and J. M. Hyman. J. Comput. Phys. 50(2):235-269, 1983) for Roe scheme for the conservation laws. The resulting scheme is compact, simple to implement even on unstructured meshes, and is demonstrated to work for nonconvex Hamiltonians. Benchmark numerical experiments in one dimension and two dimensions are provided to validate the performance of the method

A convergent scheme for Hamilton-Jacobi equations on a junction: application to traffic

30 pagesInternational audienceIn this paper, 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

Contribution à l'étude du trafic routier sur réseaux à l'aide des équations d'Hamilton-Jacobi

This work focuses on modeling and simulation of traffic flows on a network. Modeling road traffic on a homogeneous section takes its roots in the middle of XXth century and it has generated a substantial literature since then. However, taking into account discontinuities of the network such as junctions, has attracted the attention of the scientific circle more recently. However, these discontinuities are the major sources of traffic congestion, recurring or not, that basically degrades the level of service of road infrastructure. This work therefore aims to provide a unique perspective on this issue, while focusing on scale problems and more precisely on microscopic-macroscopic passage in existing models. The first part of this thesis is devoted to the relationship between microscopic car-following models and macroscopic continuous flow models. The asymptotic passage is based on a homogenization technique for Hamilton-Jacobi equations. In a second part, we focus on the modeling and simulation of vehicular traffic flow through a junction. The considered macroscopic model is built on Hamilton-Jacobi equations as well. Finally, the third part focuses on finding analytical or semi-analytical solutions, through representation formulas aiming to solve Hamilton-Jacobi equations under adequate assumptions. In this thesis, we are also interested in a generic class of second order macroscopic traffic flow models, the so-called GSOM modelsCe travail porte sur la modélisation et la simulation du trafic routier sur un réseau. Modéliser le trafic sur une section homogène (c'est-à-dire sans entrée, ni sortie) trouve ses racines au milieu du XXème siècle et a généré une importante littérature depuis. Cependant, la prise en compte des discontinuités des réseaux comme les jonctions, n'a attiré l'attention du cercle scientifique que bien plus récemment. Pourtant, ces discontinuités sont les sources majeures des congestions, récurrentes ou non, qui dégradent la qualité de service des infrastructures. Ce travail se propose donc d'apporter un éclairage particulier sur cette question, tout en s'intéressant aux problèmes d'échelle et plus particulièrement au passage microscopique-macroscopique dans les modèles existants. La première partie de cette thèse est consacrée au lien existant entre les modèles de poursuite microscopiques et les modèles d'écoulement macroscopiques. Le passage asymptotique est assuré par une technique d'homogénéisation pour les équations d'Hamilton-Jacobi. Dans une deuxième partie, nous nous intéressons à la modélisation et à la simulation des flux de véhicules au travers d'une jonction. Le modèle macroscopique considéré est bâti autour des équations d'Hamilton-Jacobi. La troisième partie enfin, se concentre sur la recherche de solutions analytiques ou semi-analytiques, grâce à l'utilisation de formules de représentation permettant de résoudre les équations d'Hamilton-Jacobi sous de bonnes hypothèses. Nous nous intéressons également dans cette thèse, à la classe générique des modèles macroscopiques de trafic de second ordre, dits modèles GSO