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A Review of David Gottlieb's Work on the Resolution of the Gibbs Phenomenon

A Review of David Gottlieb's Work on the Resolution of the Gibbs Phenomenon
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Year:    2011

Communications in Computational Physics, Vol. 9 (2011), Iss. 3 : pp. 497–519

Abstract

Given a piecewise smooth function, it is possible to construct a global expansion in some complete orthogonal basis, such as the Fourier basis. However, the local discontinuities of the function will destroy the convergence of global approximations, even in regions for which the underlying function is analytic. The global expansions are contaminated by the presence of a local discontinuity, and the result is that the partial sums are oscillatory and feature non-uniform convergence. This characteristic behavior is called the Gibbs phenomenon. However, David Gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable postprocessing. In this paper we review the history of the Gibbs phenomenon and the story of its resolution.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.301109.170510s

Communications in Computational Physics, Vol. 9 (2011), Iss. 3 : pp. 497–519

Published online:    2011-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    23

Keywords:   

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