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

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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.

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Acknowledgments

The authors are grateful to C. Imbert for indications about the literature, M. Hustić for his suggestions to simplify certain parts of the proofs and Ł. Paszkowski for valuable comments on the presentation. The authors also thank the two anonymous referees for their comments and corrections that improved the manuscript. This work was partially supported by the ANR (Agence Nationale de la Recherche) through HJnet project ANR-12-BS01-0008-01.

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  1. Université Paris-Est, Ecole des Ponts ParisTech, CERMICS, 6 & 8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 , Marne la Vallée Cedex 2, France

    Guillaume Costeseque & Régis Monneau

  2. Université Paris-Est, IFSTTAR, GRETTIA, 14-20 Boulevard Newton, Cité Descartes, Champs sur Marne, 77447 , Marne la Vallée Cedex 2, France

    Guillaume Costeseque & Jean-Patrick Lebacque

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  1. Guillaume Costeseque
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  2. Jean-Patrick Lebacque
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Correspondence to Guillaume Costeseque.

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Costeseque, G., Lebacque, JP. & Monneau, R. A convergent scheme for Hamilton–Jacobi equations on a junction: application to traffic. Numer. Math. 129, 405–447 (2015). https://doi.org/10.1007/s00211-014-0643-z

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  • DOI: https://doi.org/10.1007/s00211-014-0643-z

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