Numerische Mathematik

, Volume 115, Issue 1, pp 1–44 | Cite as

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

  • Olivier Bokanowski
  • Nadia Megdich
  • Hasnaa Zidani


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.

Mathematics Subject Classification (2000)

65M12 65M15 35F25 35F99 49L25 


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  1. 1.
    Agbrall R., Augoula S.: High order numerical discretization for Hamilton-Jacobi equations on triangular meshes. J. Sci. Comput. 15(2), 197–229 (2000)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. In: Systems and Control: Foundations and Applications. Birkhäuser, Boston (1997)Google Scholar
  3. 3.
    Barles G.: Solutions de viscosité des équations de Hamilton-Jacobi, volume 17 of Mathématiques & Applications (Berlin). Springer-Verlag, Paris (1994)Google Scholar
  4. 4.
    Barles G., Souganidis P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4, 271–283 (1991)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bokanowski, O., Forcadel, N., Zidani, H.: L 1-error estimates for numerical approximations of Hamilton-Jacobi-Bellman equations in dimension 1. Math. Comput. (to appear) (2009)Google Scholar
  6. 6.
    Bokanowski O., Martin S., Munos R., Zidani H.: An anti-diffusive scheme for viability problems. Appl. Numer. Math. 56(9), 1135–1254 (2006)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bokanowski O., Megdich N., Zidani H.: An adaptative antidissipative method for optimal control problems. Arima 5, 256–271 (2006)Google Scholar
  8. 8.
    Bokanowski O., Zidani H.: Anti-diffusive schemes for linear advection and application to Hamilton-Jacobi-Bellman equations. J. Sci. Comput. 30(1), 1–33 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cockburn B., Gremaud P.-A.: A priori error estimates for numerical methods for scalar conservation laws. II. Flux-splitting monotone schemes on irregular Cartesian grids. Math. Comput. 66(218), 547–572 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cockburn, B., Gremaud, P.-A., Yang, J.X.: A priori error estimates for numerical methods for scalar conservation laws. III. Multidimensional flux-splitting monotone schemes on non-Cartesian grids. SIAM J. Numer. Anal. 35(5), 1775–1803 (electronic) (1998)Google Scholar
  11. 11.
    Crandall M.G., Lions P.-L.: Two approximations of solutions of Hamilton Jacobi equations. Math. Comput. 43, 1–19 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Desprès, B.: Lax theorem and finite volume schemes. Math. Comput. 73(247), 1203–1234 (electronic) (2004)Google Scholar
  13. 13.
    Desprès B., Lagoutière F.: Contact discontinuity capturing schemes for linear advection and compressible gas dynamics. J. Sci. Comput. 16, 479–524 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Falcone, M.: A numerical approach to the infinite horizon problem. Appl. Math. Optim. 15(13), 213–214 (1987), and 23 (1991)Google Scholar
  15. 15.
    Falcone M., Ferretti R.: Semi-lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods. J. Comput. Phys. 175, 559–575 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Godlewski, E., Raviart, P.-A.: Hyperbolic Systems of Conservation Laws. SMAI. Ellipses (1991)Google Scholar
  17. 17.
    Lagoutiere F.: A non-dissipative entropic scheme for convex scalar equations via discontinuous cell reconstruction. C. R. Acad. Sci. 338(7), 549–554 (2004)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Lions P.L., Souganidis P.E.: Convergence of muscle and filtered schemes for scalar conservation laws and Hamilton Jacobi equations. Numer. Math. 69, 441–470 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Osher S., Shu C.-W.: High essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907–922 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rudin W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)zbMATHGoogle Scholar
  21. 21.
    Saint-Pierre P.: Approximation of viability kernel. Appl. Math. Optim. 29, 187–209 (1994)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Olivier Bokanowski
    • 1
    • 2
  • Nadia Megdich
    • 3
    • 4
  • Hasnaa Zidani
    • 4
  1. 1.Laboratoire J.-L. LionsUniversité Pierre et Marie CurieParis Cedex 05France
  2. 2.UFR de Mathématiques, Site ChevaleretUniversité Paris-DiderotParis CedexFrance
  3. 3.ISECS, Institut supérieur d’électronique et communication de SfaxSfaxTunisia
  4. 4.Project Commands ENSTA, Inria Saclay, CMAP, UMAParis Cedex 15France

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