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

TL;DR

This paper develops a finite difference scheme for Hamilton-Jacobi equations on junctions, proving convergence to viscosity solutions, and applies it to traffic flow modeling at road intersections.

Contribution

It introduces a convergent numerical scheme for Hamilton-Jacobi equations on junctions, with proven bounds and convergence results, and demonstrates its application to traffic density computations.

Findings

Established bounds on discrete gradient and time derivative.

Proved convergence of the scheme to viscosity solutions.

Applied the scheme to traffic flow, recovering known methods and providing numerical illustrations.

Abstract

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.