Advertisement

Journal of Scientific Computing

, Volume 15, Issue 3, pp 293–322 | Cite as

A Hybrid Approach to Spectral Reconstruction of Piecewise Smooth Functions

  • Anne Gelb
Article

Abstract

Consider a piecewise smooth function for which the (pseudo-)spectral coefficients are given. It is well known that while spectral partial sums yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks in the application of spectral methods. Various types of reconstruction methods developed for the recovery of piecewise smooth functions have met with varying degrees of success. The Gegenbauer reconstruction method, originally proposed by Gottlieb et al. has the particularly impressive ability to reconstruct piecewise analytic functions with exponential convergence up to the points of discontinuity. However, it has been sharply criticized for its high cost and susceptibility to round-off error. In this paper, a new approach to Gegenbauer reconstruction is considered, resulting in a reconstruction method that is less computationally intensive and costly, yet still enjoys superior convergence. The idea is to create a procedure that combines the well known exponential filtering method in smooth regions away from the discontinuities with the Gegenbauer reconstruction method in regions close to the discontinuities. This hybrid approach benefits from both the simplicity of exponential filtering and the high resolution properties of the Gegenbauer reconstruction method. Additionally, a new way of computing the Gegenbauer coefficients from Jacobian polynomial expansions is introduced that is both more cost effective and less prone to round-off errors.

Fourier expansion Gibbs phenomenon piecewise smoothness reconstruction 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    Banerjee, N. S., and Geer, J. (1995). Exponentially accurate approximations to piecewise smooth periodic functions, ICASE Report No. 95-17, NASA Langley Research Center.Google Scholar
  2. 2.
    Bateman, H. (1953). Higher Transcendental Functions, Vol. 2, McGraw-Hill.Google Scholar
  3. 3.
    Bauer, R. (1995). Band Filters for Determining Shock Locations, Ph.D. thesis, Applied Mathematics, Brown University.Google Scholar
  4. 4.
    Cai, W., Gottlieb, D., and Shu, C.-W. (1992). On one-sided filters for spectral Fourier approximations of discontinuous functions. SIAM J. Numer. Anal. 29(4), 905–916.Google Scholar
  5. 5.
    Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1987). Spectral Methods in Fluid Dynamics.Google Scholar
  6. 6.
    Eckhoff, K. S. (1995). Accurate reconstructions of functions of finite regularity from truncated series expansions. Math. Comp. 64, 671–690.Google Scholar
  7. 7.
    Eckhoff, K. S. (1998). On a high order numerical method for functions with singularities. Math. Comp. 67, 1063–1087.Google Scholar
  8. 8.
    Gelb, A., and Tadmor, E. (1999). Detection of edges in spectral data. Appl. Comp. Harmonic Anal. 7, 101–135.Google Scholar
  9. 9.
    Gelb, A., and Tadmor, E. (2000). Detection of edges in spectral data II. Nonlinear enhan-cement. SIAM J. Numer. Anal. (in press).Google Scholar
  10. 10.
    Gelb, A., and Tadmor, E. (2000). Enhanced spectral viscosity approximations for conser-vation laws. Appl. Numer. Math. 33, 3–21.Google Scholar
  11. 11.
    Gottlieb, D., Gustafsson, B., and Forssé n, P. (2000). On the direct Fourier method for computer tomography. IEEE Trans. Med. Imag. 19, 223–232.Google Scholar
  12. 12.
    Gottlieb, D., and Orszag, S. (1977). Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS, Philadelphia.Google Scholar
  13. 13.
    Gottlieb, D., and Shu, C.-W. (1997). On the Gibbs phenomenon and its resolution. SIAM Rev. 39, 644–668.Google Scholar
  14. 14.
    Gottlieb, D., and Shu, C.-W. (1995). On the Gibbs phenomenon IV. Recovering exponen-tial accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic func-tion. Math. Comp. 64, 1081–1095.Google Scholar
  15. 15.
    Gottlieb, D., Shu, C.-W., Solomonoff, A., and Vandeven, H. (1992). On The Gibbs phenomenon I. Recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function. J. Comput. Appl. Math. 43, 81–92.Google Scholar
  16. 16.
    Gottlieb, D., and Tadmor, E. (1985). Recovering pointwise vales of discontinuous data within spectral accuracy. In Murman, E. M., and Abarbanel, S. S. (eds.), Progress and Supercomputing in Computational Fluid Dynamics, Proceedings of a 1984 U.S.-Israel Workshop, Progress in Scientific Computing, Vol. 6, Birkhauser, Boston, pp. 357–375.Google Scholar
  17. 17.
    Kvernadze, G. (1998). Determination of the jump of a bounded function by its Fourier series. J. Approx. Theory 92, 167–190.Google Scholar
  18. 18.
    Maday, Y., Ould Kaber S. M., and Tadmor, E. (1993). Legendre pseudospectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 30, 321–342.Google Scholar
  19. 19.
    Mhaskar, H. N., and Prestin, J. (2000). On the detection of singularities of a periodic function, preprint.Google Scholar
  20. 20.
    Shu, C.-W., and Wong, P. (1995). A note on the accuracy of spectral method applied to nonlinear conservation laws. J. Sci. Comput. 10, 357–369.Google Scholar
  21. 21.
    Tadmor, E. (1989). Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal. 26, 30–44.Google Scholar
  22. 22.
    Vandeven, H. (1991). Family of spectral filters for discontinuous problems. SIAM J. Sci. Comput. 48, 159–192.Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Anne Gelb
    • 1
  1. 1.Department of MathematicsArizona State UniversityTempe

Personalised recommendations