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In this paper, we consider first order Hamilton-Jacobi (HJ) equations posedon a ``junction'', that is to say the union of a finite number of half-lineswith a unique common point. For this continuous HJ problem, we propose a finitedifference scheme and prove two main results. As a first result, we show boundson the discrete gradient and time derivative of the numerical solution. Oursecond result is the convergence (for a subsequence) of the numerical solutiontowards a viscosity solution of the continuous HJ problem, as the mesh sizegoes to zero. When the solution of the continuous HJ problem is unique, werecover the full convergence of the numerical solution. We apply this scheme tocompute the densities of cars for a traffic model. We recover the well-knownGodunov scheme outside the junction point and we give a numerical illustration.

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