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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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References

  1. Agbrall R., Augoula S.: High order numerical discretization for Hamilton-Jacobi equations on triangular meshes. J. Sci. Comput. 15(2), 197–229 (2000)

    Article  MathSciNet  Google Scholar 

  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)

  3. Barles G.: Solutions de viscosité des équations de Hamilton-Jacobi, volume 17 of Mathématiques & Applications (Berlin). Springer-Verlag, Paris (1994)

    Google Scholar 

  4. Barles G., Souganidis P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4, 271–283 (1991)

    MATH  MathSciNet  Google Scholar 

  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)

  6. Bokanowski O., Martin S., Munos R., Zidani H.: An anti-diffusive scheme for viability problems. Appl. Numer. Math. 56(9), 1135–1254 (2006)

    Article  MathSciNet  Google Scholar 

  7. Bokanowski O., Megdich N., Zidani H.: An adaptative antidissipative method for optimal control problems. Arima 5, 256–271 (2006)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Crandall M.G., Lions P.-L.: Two approximations of solutions of Hamilton Jacobi equations. Math. Comput. 43, 1–19 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  12. Desprès, B.: Lax theorem and finite volume schemes. Math. Comput. 73(247), 1203–1234 (electronic) (2004)

  13. Desprès B., Lagoutière F.: Contact discontinuity capturing schemes for linear advection and compressible gas dynamics. J. Sci. Comput. 16, 479–524 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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. Falcone M., Ferretti R.: Semi-lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods. J. Comput. Phys. 175, 559–575 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Godlewski, E., Raviart, P.-A.: Hyperbolic Systems of Conservation Laws. SMAI. Ellipses (1991)

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  19. Osher S., Shu C.-W.: High essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907–922 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rudin W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)

    MATH  Google Scholar 

  21. Saint-Pierre P.: Approximation of viability kernel. Appl. Math. Optim. 29, 187–209 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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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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