Transform Method for the Calculation of Vector-Coupled Sums: Application to the Spectral Form of the Vorticity Equation

Steven A. Orszag National Center for Atmospheric Research, Boulder, Colo

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Abstract

A transform method is developed for the fast calculation of vector-coupled sums appearing in the spectrally truncated vorticity equation. The method involves a kind of finite convolution theorem for expansions in surface harmonics. The method succeeds because it turns out to be much faster to transform the spectral representation to physical space, multiply the physical-space functions, and then inverse transform back to the spectral representation, than to evaluate the vector-coupled sums directly in spectral form. Direct evaluation of the sums requires order N5 operations when the spectral representation is truncated at surface harmonies of degree N, while the transform method requires about 10N3+40N2 log2N real operations per time step. Direct evaluation of the sums also requires storage of order N5, while the transform method requires storage of roughly 3N3 real words. An explanation is also given how to specialize the transform method to the “hemispheric” models of Baer and Platzman with a savings of nearly a factor of 8 in speed. For the hemispheric model with N = 19, the transform method requires at least three times fewer calculations than are required by direct evaluation. The transform method is comparable in speed with direct evaluation of the vector-coupled sums when N = 10 (N = 15 for the hemispheric models), with the transform method improving its advantage over direct evaluation as N2. These new methods suggest that spectral representations with a large number of retained modes may be in a not too unfavorable competitive position with finite-difference jury methods (which require at least order N2 log2N operations per time step) for the solution of the equations of incompressible spherical hydrodynamics.

Abstract

A transform method is developed for the fast calculation of vector-coupled sums appearing in the spectrally truncated vorticity equation. The method involves a kind of finite convolution theorem for expansions in surface harmonics. The method succeeds because it turns out to be much faster to transform the spectral representation to physical space, multiply the physical-space functions, and then inverse transform back to the spectral representation, than to evaluate the vector-coupled sums directly in spectral form. Direct evaluation of the sums requires order N5 operations when the spectral representation is truncated at surface harmonies of degree N, while the transform method requires about 10N3+40N2 log2N real operations per time step. Direct evaluation of the sums also requires storage of order N5, while the transform method requires storage of roughly 3N3 real words. An explanation is also given how to specialize the transform method to the “hemispheric” models of Baer and Platzman with a savings of nearly a factor of 8 in speed. For the hemispheric model with N = 19, the transform method requires at least three times fewer calculations than are required by direct evaluation. The transform method is comparable in speed with direct evaluation of the vector-coupled sums when N = 10 (N = 15 for the hemispheric models), with the transform method improving its advantage over direct evaluation as N2. These new methods suggest that spectral representations with a large number of retained modes may be in a not too unfavorable competitive position with finite-difference jury methods (which require at least order N2 log2N operations per time step) for the solution of the equations of incompressible spherical hydrodynamics.

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