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Numerische Mathematik

, Volume 115, Issue 4, pp 609–645 | Cite as

Finite volume schemes for locally constrained conservation laws

  • Boris Andreianov
  • Paola Goatin
  • Nicolas Seguin
Article

Abstract

This paper is devoted to the numerical analysis of the road traffic model proposed by Colombo and Goatin (J. Differ. Equ. 234(2):654–675, 2007). The model involves a standard conservation law supplemented by a local unilateral constraint on the flux at the point x = 0 (modelling a road light, a toll gate, etc.). We first show that the problem can be interpreted in terms of the theory of conservation laws with discontinuous flux function, as developed by Adimurthi et al. (J. Hyperbolic Differ. Equ. 2(4):783–837, 2005) and Bürger et al. (SIAM J. Numer. Anal. 47(3):1684–1712, 2009). We reformulate accordingly the notion of entropy solution introduced by Colombo and Goatin (J. Differ. Equ. 234(2):654–675, 2007), and extend the well-posedness results to the L framework. Then, starting from a general monotone finite volume scheme for the non-constrained conservation law, we produce a simple scheme for the constrained problem and show its convergence. The proof uses a new notion of entropy process solution. Numerical examples modelling a “green wave” are presented.

Mathematics Subject Classification (2000)

35L65 65M12 76M12 90B20 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Boris Andreianov
    • 1
  • Paola Goatin
    • 2
  • Nicolas Seguin
    • 3
  1. 1.Laboratoire de MathématiquesUniversité de Franche-ComtéBesançon CedexFrance
  2. 2.ISITV, Université du Sud Toulon-VarLa Valette du Var CedexFrance
  3. 3.Laboratoire J.-L. Lions, UPMC Univ Paris 06Paris Cedex 05France

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