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Convergence of a non-monotone scheme for Hamilton–Jacobi–Bellman equations with discontinous initial data

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Abstract

We prove the convergence of a non-monotonous scheme for a one-dimensional first order Hamilton–Jacobi–Bellman equation of the form v t + max α (f(x, α)v x ) = 0, v(0, x) = v 0(x). The scheme is related to the HJB-UltraBee scheme suggested in Bokanowski and Zidani (J Sci Comput 30(1):1–33, 2007). We show for general discontinuous initial data a first-order convergence of the scheme, in L 1-norm, towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on several numerical examples.

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Authors and Affiliations

  1. Laboratoire J.-L. Lions, Université Pierre et Marie Curie, 75252, Paris Cedex 05, France

    Olivier Bokanowski

  2. UFR de Mathématiques, Site Chevaleret, Université Paris-Diderot, 75205, Paris Cedex, France

    Olivier Bokanowski

  3. ISECS, Institut supérieur d’électronique et communication de Sfax, Route de Menzel Chaker, 3000, Sfax, Tunisia

    Nadia Megdich

  4. Project Commands ENSTA, Inria Saclay, CMAP, UMA, 32 Boulevard Victor, 75739, Paris Cedex 15, France

    Nadia Megdich & Hasnaa Zidani

Authors
  1. Olivier Bokanowski
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  2. Nadia Megdich
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  3. Hasnaa Zidani
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Correspondence to Olivier Bokanowski.

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Bokanowski, O., Megdich, N. & Zidani, H. Convergence of a non-monotone scheme for Hamilton–Jacobi–Bellman equations with discontinous initial data. Numer. Math. 115, 1–44 (2010). https://doi.org/10.1007/s00211-009-0271-1

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  • DOI: https://doi.org/10.1007/s00211-009-0271-1

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