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Adaptive Filtered Schemes for First Order Hamilton-Jacobi Equations

  • Maurizio Falcone
  • Giulio PaolucciEmail author
  • Silvia Tozza
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

In this paper we consider a class of “filtered” schemes for some first order time dependent Hamilton-Jacobi equations. A typical feature of a filtered scheme is that at the node xj the scheme is obtained as a mixture of a high-order scheme and a monotone scheme according to a filter function F. The mixture is usually governed by F and by a fixed parameter ε = ε(Δt, Δx) > 0 which goes to 0 as (Δt, Δx) is going to 0 and does not depend on n. Here we improve the standard filtered scheme introducing an adaptive and automatic choice of the parameter ε = εn(Δt, Δx) at every iteration. To this end, we use a smoothness indicator in order to select the regions where we can compute the regularity threshold εn. The numerical tests presented confirms the effectiveness of the adaptive scheme.

References

  1. 1.
    O. Bokanowski, M. Falcone, S. Sahu, An efficient filtered scheme for some first order Hamilton-Jacobi-Bellman equations. SIAM J. Sci. Comput. 38(1), A171–A195 (2016)CrossRefGoogle Scholar
  2. 2.
    M.G. Crandall, P.L. Lions, Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43(167), 1–19 (1984)MathSciNetCrossRefGoogle Scholar
  3. 3.
    M. Falcone, R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations (SIAM, Philadelphia, 2014)zbMATHGoogle Scholar
  4. 4.
    M. Falcone, G. Paolucci, S. Tozza, Convergence of adaptive filtered schemes for first order evolutive Hamilton-Jacobi equations (submitted)Google Scholar
  5. 5.
    G. Jiang, D.-P. Peng, Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21(6), 2126–2143 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Kurganov, S. Noelle, G. Petrova, Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23(3), 707–740 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    P.L. Lions, P. Souganidis, Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton–Jacobi equations. Numer. Math. 69, 441–470 (1995)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A.M. Oberman, T. Salvador, Filtered schemes for Hamilton-Jacobi equations: a simple construction of convergent accurate difference schemes. J. Comput. Phys. 284, 367–388 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Maurizio Falcone
    • 1
  • Giulio Paolucci
    • 1
    Email author
  • Silvia Tozza
    • 2
  1. 1.Department of MathematicsSapienza University of RomeRomeItaly
  2. 2.Istituto Nazionale di Alta Matematica/Department of MathematicsSapienza University of RomeRomeItaly

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