We study the possibility of replacing the popular spectral transform method with the double Fourier method. The method resides in spectral space, and transforms to grid space only for nonlinear terms. This approach minimizes the number of transposes required and thus minimizes the required inter-processor communication. Compared to the standard spectral transform method, the double Fourier method requires an equally spaced latitude grid instead of a Gauss distribution and utilizes FFTs in the latitude direction rather than the more expensive associated Legendre transforms. A desirable aspect of the double Fourier approach is that its accuracy and stability are identical to the spectral transform method on an equispaced grid within machine precision. When an Eulerian time integration method is used, numerical stability is maintained by projecting data onto the spherical harmonics. Thus, the Eulerian approach offers little saving over the standard spectral transform method. (Currently, the most efficient spectral transform method requires 9 Legendre transforms per time step; the filter approach reduces this to 6.) However, we believe that when combined with a semi-Lagrangian advection scheme, it may be possible for the method to maintain stability without using a projection (thus, requiring none or very few Legendre transforms). This is because the nonlinearity in the SWEs arise mostly from the advection terms. which do not appear explicitly in the Lagrangian formulation of the equations. Thus, the semi-Lagrangian double Fourier method, with a computational complexity of O(N^2 log N), may be significantly more efficient than the spectral transform method. |
