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Spectral Reconstruction of Piecewise Smooth Functions from Their Discrete Data

  • Anne Gelb (a1) and Eitan Tadmor (a2)

Abstract

This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coefficients and physical space interpolants have been discussed extensively in the literature, and it is clear that an a priori knowledge of the jump discontinuity location is essential for any reconstruction technique to yield spectrally accurate results with high resolution near the discontinuities. Hence detection of the jump discontinuities is critical for all methods. Here we formulate a new localized reconstruction method adapted from the method developed in Gottlieb and Tadmor (1985) and recently revisited in Tadmor and Tanner (in press). Our procedure incorporates the detection of edges into the reconstruction technique. The method is robust and highly accurate, yielding spectral accuracy up to a small neighborhood of the jump discontinuities. Results are shown in one and two dimensions.

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[1] N.S. Banerjee and J. Geer, Exponential approximations using Fourier series partial sums, ICASE Report No. 97-56, NASA Langley Research Center (1997).
[2] N. Bary, Treatise of Trigonometric Series. The Macmillan Company, New York (1964).
[3] H.S. Carslaw, Introduction to the Theory of Fourier's Series and Integrals. Dover (1950).
[4] Eckhoff, K.S., Accurate reconstructions of functions of finite regularity from truncated series expansions. Math. Comp. 64 (1995) 671-690.
[5] Eckhoff, K.S., On a high order numerical method for functions with singularities. Math. Comp. 67 (1998) 1063-1087.
[6] Gelb, A. and Tadmor, E., Detection of edges in spectral data. Appl. Comput. Harmon. Anal. 7 (1999) 101-135.
[7] Gelb, A. and Tadmor, E., Detection of edges in spectral data. II. Nonlinear Enhancement. SIAM J. Numer. Anal. 38 (2001) 1389-1408.
[8] Golubov, B.I., Determination of the jump of a function of bounded p-variation by its Fourier series. Math. Notes 12 (1972) 444-449.
[9] D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon and its resolution. SIAM Rev. (1997).
[10] D. Gottlieb and E. Tadmor, Recovering pointwise values of discontinuous data within spectral accuracy, in Progress and Supercomputing in Computational Fluid Dynamics, Proceedings of a 1984 U.S.-Israel Workshop, Progress in Scientific Computing, Vol. 6, E.M. Murman and S.S. Abarbanel Eds., Birkhauser, Boston (1985) 357-375.
[11] Kvernadze, G., Determination of the jump of a bounded function by its Fourier series. J. Approx. Theory 92 (1998) 167-190.
[12] E. Tadmor and J. Tanner, Adaptive mollifiers for high resolution recovery of piecewise smooth data from its spectral information, Foundations of Comput. Math. Online publication DOI: 10.1007/s002080010019 (2001), in press.
[13] A. Zygmund, Trigonometric Series. Cambridge University Press (1959).

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Spectral Reconstruction of Piecewise Smooth Functions from Their Discrete Data

  • Anne Gelb (a1) and Eitan Tadmor (a2)

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