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

  • Authors:

  • Olivier Bokanowski;Nadia Megdich;Hasnaa Zidani

  • Affiliations:

  • Université Pierre et Marie Curie, Laboratoire J.-L. Lions, 75252, Paris Cedex 05, France and Université Paris-Diderot, UFR de Mathématiques, Site Chevaleret, 75205, Paris Cede ...;ISECS, Institut supérieur d’électronique et communication de Sfax, Route de Menzel Chaker, 3000, Sfax, Tunisia and Project Commands ENSTA, Inria Saclay, CMAP, UMA, 32 Boulevar ...;Project Commands ENSTA, Inria Saclay, CMAP, UMA, 32 Boulevard Victor, 75739, Paris Cedex 15, France

  • Venue:
  • Numerische Mathematik
  • Year:
  • 2010

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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 + maxα(f(x, α)vx) = 0, v(0, x) = v0(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 L1-norm, towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on several numerical examples.