Comparison of an Adaptive Wavelet Method and Nonlinearly Filtered Pseudospectral Methods for Two-Dimensional Turbulence

An adaptive wavelet method for solving the two-dimensional Navier–Stokes 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 adaptive wavelet simulation replicates 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–Stokes 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. Received 22 April 1997 and accepted 11 August 1997