Advertisement

Foundations of Computational Mathematics

, Volume 13, Issue 2, pp 139–159 | Cite as

ENO Reconstruction and ENO Interpolation Are Stable

  • Ulrik S. Fjordholm
  • Siddhartha Mishra
  • Eitan Tadmor
Article

Abstract

We prove that the ENO reconstruction and ENO interpolation procedures are stable in the sense that the jump of the reconstructed ENO point values at each cell interface has the same sign as the jump of the underlying cell averages across that interface. Moreover, we prove that the size of these jumps after reconstruction relative to the jump of the underlying cell averages is bounded. Similar sign properties and the boundedness of the jumps hold for the ENO interpolation procedure. These estimates, which are shown to hold for ENO reconstruction and interpolation of arbitrary order of accuracy and on nonuniform meshes, indicate a remarkable rigidity of the piecewise polynomial ENO procedure.

Keywords

Newton interpolation Adaptivity ENO reconstruction Sign property 

Mathematics Subject Classification

65D05 65M12 

Notes

Acknowledgements

S.M. thanks Prof. Mike Floater of CMA, Oslo for useful discussions. The research of E.T. was supported by grants from the National Science Foundation, DMS#10-08397, and the Office of Naval Research, ONR#N000140910385 and #000141210318.

References

  1. 1.
    S. Amat, F. Arandiga, A. Cohen, R. Donat, Tensor product multiresolution with error control, Signal Process., 82, 587–608 (2002). MATHCrossRefGoogle Scholar
  2. 2.
    F. Arandiga, A. Cohen, R. Donat, N. Dyn, Interpolation and approximation of piecewise smooth functions, SIAM J. Numer. Anal. 43(1), 41–57 (2005). MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    F. Arandiga, A. Cohen, R. Donat, N. Dyn, B. Matei, Approximation of piecewise smooth functions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques, Appl. Comput. Harmon. Anal., 24, 225–250 (2008). MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    R. Baraniuk, R. Claypoole, G.M. Davis, W. Sweldens, Nonlinear wavelet transforms for image coding via lifting, IEEE Trans. Image Process., 12, 1449–1459 (2003). MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Berzins, Adaptive polynomial interpolation on evenly spaced meshes, SIAM Rev. 49(4), 604–627 (2007). MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    T. Chan, H.M. Zhou, ENO-wavelet transforms for piecewise smooth functions, SIAM J. Numer. Anal., 40(4), 1369–1404 (2002). MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    A. Cohen, N. Dyn, M. Matel, Quasilinear subdivison schemes with applications to ENO interpolation, Appl. Comput. Harmon. Anal., 15, 89–116 (2003). MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    U.S. Fjordholm, S. Mishra, E. Tadmor, Entropy stable ENO scheme, in Hyperbolic Problem: Theory, Numerics and Applications. Proc. of HYP2010—the 13th International Conference on Hyperbolic Problems Held in Beijing (2012, to appear). Google Scholar
  9. 9.
    U.S. Fjordholm, S. Mishra, E. Tadmor, Arbitrarily high-order essentially non-oscillatory entropy stable schemes for systems of conservation laws, SIAM J. Numer. Anal. (in press). Google Scholar
  10. 10.
    A. Harten, ENO schemes with subcell resolution, J. Comput. Phys. 83, 148–184 (1989). MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    A. Harten, Recent developments in shock-capturing schemes, in Proc. International Congress of Mathematicians, vols. I, II, Kyoto, 1990 (Math. Soc. Japan, Tokyo, 1991), pp. 1549–1559. Google Scholar
  12. 12.
    A. Harten, Multi-resolution analysis for ENO schemes, in Algorithmic Trends in Computational Fluid Dynamics (1991) (Springer, New York, 1993), pp. 287–302. CrossRefGoogle Scholar
  13. 13.
    A. Harten, Adaptive multiresolution schemes for shock computations, J. Comput. Phys. 115(2), 319–338 (1994). MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    A. Harten, Multiresolution representation of cell-averaged data: a promotional review, in Signal and Image Representation in Combined Spaces. Wavelet Anal. Appl., vol. 7 (Academic Press, San Diego, 1998), pp. 361–391. CrossRefGoogle Scholar
  15. 15.
    A. Harten, B. Engquist, S. Osher, S.R. Chakravarty, Some results on high-order accurate essentially non-oscillatory schemes, Appl. Numer. Math. 2, 347–377 (1986). MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    A. Harten, B. Engquist, S. Osher, S.R. Chakravarty, Uniformly high order accurate essentially non-oscillatory schemes, J. Comput. Phys. 71(2), 231–303 (1987). MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    G. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126, 202–226 (1996). MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    D. Levy, G. Puppo, G. Russo, Central WENO schemes for hyperbolic systems of conservation laws, Math. Model. Numer. Anal. 33, 547–571 (1999). MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    B. Matei, Méthodes multirésolution non-linéaires- applications au traitement d’image. Ph.D. thesis, University Paris VI (2002). Google Scholar
  20. 20.
    J. Qiu, C.-W. Shu, On the construction, comparison, and local characteristic decompositions for high order central WENO schemes, J. Comput. Phys. 183, 187–209 (2002). MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    C.W. Shu, Numerical experiments on the accuracy of ENO and modified ENO schemes, J. Sci. Comput. 5(2), 127–149 (1990). MATHCrossRefGoogle Scholar
  22. 22.
    C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. ICASE Technical report, NASA, 1997. Google Scholar
  23. 23.
    C.W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory schemes—II, J. Comput. Phys. 83, 32–78 (1989). MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    P. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21, 995–1011 (1984). MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© SFoCM 2012

Authors and Affiliations

  • Ulrik S. Fjordholm
    • 1
  • Siddhartha Mishra
    • 1
  • Eitan Tadmor
    • 2
  1. 1.Seminar for Applied MathematicsETH ZürichZürichSwitzerland
  2. 2.Department of Mathematics, Center of Scientific Computation and Mathematical Modeling (CSCAMM), Institute for Physical Sciences and Technology (IPST)University of MarylandCollege ParkUSA

Personalised recommendations