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

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**Résumé:**

**Abstract: **

In this paper, we present a rational approximations based on Fourier series representation. Considering periodic piecewise analytic functions, the well known Gibbs phenomenon deteriorates 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. It is shown 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 Padé approximants increase. Numerical results are demonstrated for several examples and the collocation method is applied as a post-processing 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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**Date:** 2003-05-01