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

Guillaume Costeseque (1, 2) , Jean-Patrick Lebacque (2) , Régis Monneau (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 IFSTTAR/COSYS/GRETTIA - Génie des Réseaux de Transport Terrestres et Informatique Avancée

In 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.

Mots clés: Hamilton-Jacobi equations; junctions; viscosity solutions; numerical scheme; traffic problems

Domaines: Analyse numérique [math.NA]

Guillaume Costeseque, Jean-Patrick Lebacque, Régis Monneau. A convergent scheme for Hamilton-Jacobi equations on a junction: application to traffic. Numerische Mathematik, 2015, 129 (3), pp.405-447. ⟨hal-00829044v2⟩