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Fourier-Padé approximations and filtering for spectral simulations of an incompressible Boussinesq convection problem

Authors: M. S. Min, S. M. Kaber and W. S. Don
Journal: Math. Comp. 76 (2007), 1275-1290
MSC (2000): Primary 41A20, 41A21, 41A25, 65T10, 65T20
Published electronically: February 16, 2007
MathSciNet review: 2299774
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Abstract: In this paper, we present rational approximations based on Fourier series representation. For periodic piecewise analytic functions, the well-known Gibbs phenomenon hampers the convergence of the standard Fourier method. Here, for a given set of the Fourier coefficients from a periodic piecewise analytic function, we define Fourier-Padé-Galerkin and Fourier-Padé collocation methods by expressing the coefficients for the rational approximations using the Fourier data. We show that those methods converge exponentially in the smooth region and successfully reduce the Gibbs oscillations as the degrees of the denominators and the numerators of the Padé approximants increase.

Numerical results are demonstrated in several examples. The collocation method is applied as a postprocessing step to the standard pseudospectral simulations for the one-dimensional inviscid Burgers' equation and the two-dimensional incompressible inviscid Boussinesq convection flow.

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Additional Information

M. S. Min
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island

S. M. Kaber
Affiliation: Laboratoire Jacques-Louis Lions, Université Paris VI, France

W. S. Don
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island

Keywords: Rational approximation, Gibbs phenomenon, Fourier--Pad\'e--Galerkin method, Fourier--Pad\'e collocation, postprocessing
Received by editor(s): June 3, 2003
Received by editor(s) in revised form: July 7, 2004
Published electronically: February 16, 2007
Additional Notes: This research was supported by Grant AFOSR F49620-02-1-0113.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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