Here we redirect attention to a
fast pseudospectral method on the sphere developed by Merilees in
1973, recently revived by Fornberg. In these works, the required
spatial derivatives are computed by the formal differentiation of
one-dimensional Fourier series approximations to both scalar and
vector functions on the surface of the sphere. Filters must be used
to alleviate prohibitive time-stepping restrictions and maintain
stability on the non-isotropic latitude-longitude grids. Merilees'
original filter was eventually found to be unusable, as it was
unstable for longer runs. In this paper we examine alternatives to
Merilees' filter. In particular, we first use a harmonic filter that
consists of a harmonic analysis followed directly by a synthesis. The
resulting stability and accuracy are identical to the traditional
spectral transform method. Fewer Legendre transforms are required
since they are limited to the filter and not used to compute spatial
derivatives. In theory, this approach can also be viewed as a fast
spectral method since fast harmonic filters exist in the
literature. Next we examine alternate fast Fourier filters with intent
to reproduce the accuracy and stability provided by the harmonic
filter. Computational examples are provided with both high order
difference and Fourier derivative calculations. In addition, results
are presented for both harmonic and Fourier filters.
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