Abstract
Pseudospectral Methods based on global polynomial approximation yield exponential accuracy when the underlying function is analytic. The presence of discontinuities destroys the extreme accuracy of the methods and the well-known Gibbs phenomenon appears. Several types of postprocessing methods have been developed to lessen the effects of the Gibbs phenomenon or even to restore spectral accuracy. The most powerful of the methods require that the locations of the discontinuities be precisely known. In this work we discuss postprocessing algorithms that are applicable when it is impractical, or difficult, or undesirable to pinpoint all discontinuity locations.
Similar content being viewed by others
References
Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York (2000)
Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)
Chan, T., Osher, S., Shen, J.: The digital TV filter and nonlinear denoising. IEEE Trans. Image Process. 10(2), 231–241 (2001)
Gelb, A.: A hybrid approach to spectral reconstruction of piecewise smooth functions. J. Sci. Comput. 15, 293–322 (2001)
Gelb, A., Cates, D.: Detection of edges in spectral data III: Refinement of the concentration method. J. Sci. Comput. 36(1), 1–43 (2008)
Gelb, A., Tadmor, E.: Detection of edges in spectral data. Appl. Comput. Harmon. Anal. 7, 101–135 (1999)
Gelb, A., Tadmor, E.: Detection of edges in spectral data II: Nonlinear enhancement. SIAM J. Numer. Anal. 38(4), 1389–1408 (2000)
Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)
Hesthaven, J.S., Kirby, R.M.: Filtering in Legendre spectral methods. Math. Comput. 77, 1425–1452 (2008)
Jung, J.-H., Shizgal, B.: Inverse polynomial reconstruction of two dimensional Fourier images. J. Sci. Comput. 25, 367–399 (2005)
Kosloff, R., Tal-Ezer, H.: A modified Chebyshev pseudospectral method with an O(1/n) time step restriction. J. Comput. Phys. 104, 457–469 (1993)
Ma, H.: Chebyshev-Legendre super spectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 35, 893–908 (1998)
Osher, S., Shen, J.: Digitized PDE method for data restoration. In: Anastassiou, G. (ed.) Analytic-Computational Methods in Applied Mathematics, Chap. 16, pp. 751–771. Chapman and Hall/CRC (2000)
Sarra, S.A.: Digital Total Variation filtering as postprocessing for Chebyshev pseudospectral methods for conservation laws. Numer. Algorithms 41, 17–33 (2006)
Sarra, S.A.: Digital Total Variation filtering as postprocessing for Radial Basis Function Approximation Methods. Comput. Math. Appl. 52, 1119–1130 (2006)
Sarra, S.A.: The Matlab postprocessing toolkit. Submitted to ACM Trans. Math. Softw. (2009)
Sod, G.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978)
Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)
Vandeven, H.: Family of spectral filters for discontinuous problems. SIAM J. Sci. Comput. 6, 159–192 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by NSF grant DMS-0609747.
Rights and permissions
About this article
Cite this article
Sarra, S.A. Edge Detection Free Postprocessing for Pseudospectral Approximations. J Sci Comput 41, 49–61 (2009). https://doi.org/10.1007/s10915-009-9287-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-009-9287-z