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Comparison of an adaptive
wavelet method and nonlinearly filtered pseudo-spectral methods for two-dimensional
turbulence
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Kai SCHNEIDER, Nicholas KEVLAHAN and Marie
FARGE
K.S.: Institut
für Chemische Technik, Universität Karlsruhe, Kaiserstrasse 12,
76128 Karlsruhe, Germany;
and Centre de Physique Théorique, CNRS - Luminy, Case 907, 13288
Marseille cedex 09, France
N.K. & M.F.: LMD-CNRS, École Normale Supérieure,
24, rue Lhomond, 75231 Paris cedex 05, France
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Abstract |
An adaptive wavelet method for solving the two-dimensional
Navier-Strokes equations is compared with nonlinear Fourier filtering and
nonlinear wavelet filtering of the pseudospectral method at each time step.
The methods are each applied to a highly nonlinear flow typical of two-dimensional
turbulence, the merger of two positive vortices pushed together by a weaker
negative vortex, and the results are compared with a reference classical
pseudospectral method. Nonlinear Fourier filtering uses 1.7 times fewer
active modes than the reference simulation at the time of merger (when the
flow is most complicated) and retains the overall dynamics and structure
of the flow. However, it induces spurious oscillations in the background.
Nonlinear wavelet filtering simulation uses 9.2 times fewer modes than the
reference simulation at the time of merger, and reduces the errors in the
solution. The adaptative wavelet simulation replicate precisely the dynamics
and spatial structure of the reference simulation while retaining the high
compression rate of the nonlinear wavelet filtering simulation. In addition
we observe that the number of active wavelet modes remains quasi-constant
during the whole merging process, independent of the strength of the vorticity
gradients. On the contrary, the number of active Fourier modes is multiplied
by 5 when the vorticity gradients are strongest. The increased accuracy
of the adaptive wavelet simulation is due to the security zone added around
the active coefficients and to the compression of the nonlinear term of
the Navier-Strokes equations in the wavelet basis. These results suggest
that nonlinear Fourier filtering of a classical pseudospectral method cannot
produce significant improvement, but that the adaptive wavelet method combines
a consistently high compression rate with high accuracy.
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Comparaison entre une méthode adaptative
en ondelettes et un filtrage pseudo-spectral
non linéaire pour la turbulence bidimensionnelle
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Résumé |
Cet article compare l'intégration des équations de Navier-Stokes
bidimensionnelles dans une base d'ondelettes adaptative avec celle pseudo-spectrale
filtrée à chaque pas de temps de façon non linéaire, pour l'étude de l'interaction
fortement non linéaire de trois tourbillons. |