A hybrid approach to spectral reconstruction of piecewise smooth functions

Anne Gelb

Research output: Contribution to journalArticle

26 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)293-322
Number of pages30
JournalJournal of Scientific Computing
Volume15
Issue number3
DOIs
StatePublished - Sep 2000

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Piecewise Smooth Functions
Hybrid Approach
Discontinuity
Rounding error
Filtering
Gibbs Phenomenon
Exponential Convergence
Costs
Coefficient
Partial Sums
Spectral Methods
Smooth function
Susceptibility
Convergence Rate
Polynomials
Analytic function
Simplicity
High Resolution
Recovery
Oscillation

ASJC Scopus subject areas

  • Computer Networks and Communications
  • Hardware and Architecture
  • Software

Cite this

A hybrid approach to spectral reconstruction of piecewise smooth functions. / Gelb, Anne.

In: Journal of Scientific Computing, Vol. 15, No. 3, 09.2000, p. 293-322.

Research output: Contribution to journalArticle

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