Further study on the high-order double-Fourier-series spectral filtering on a sphere

• Authors: Hyeong-Bin Cheong;In-Hyuk Kwon;Tae-Young Goo
• Venue: Journal of Computational Physics
• Year: 2004

A high-order harmonic spectral filter (HSF) is further studied using double Fourier series (DFS), which performs filtering in terms of successive inversion of tridiagonal matrices with complex-valued elements. The high-order harmonics filter equation is split into multiple Helmholtz equations. It is found that the filter provides the same order of accuracy as the spectral filter in [J. Comput. Phys. 177 (2002) 313] that consists of the pentadiagonal matrices with real-valued elements. The advantage of the filter over the previous one lies on the simplicity and easiness of numerical implementation or computer coding, just requiring the same complexity as Poisson's equation solver. However, the operation count associated with the filter increases by a factor of about 2. To circumvent the inefficiency while preserving the simplicity, an easy way to construct pentadiagonal matrices associated with the biharmonic equation is presented in which the tridiagonal matrices related with Poisson's equation are manipulated. Computational efficiency of the spectral filter is discussed in terms of the relative computing time to the spectral transform. It is revealed that the computing cost (requiring O(N2) operations with N being the truncation) for the spectral filtering, even with the complex-valued matrices, is not significant in the DFS spectral model that is characterized by O(N2 log2 N) operations. Filtering with different DFS expansions is discussed with a focus on the accuracy and pole condition. It is shown that the DFS violating the pole conditions produces a discontinuity at poles in case of wave truncation, and its influence spreads over the globe. The spectral filter is applied to two kinds of uniform-grid data, the regular and the shifted grids, and the results are compared with each other. The operator splitting (or spherical harmonics factorization) makes it feasible to apply the finite difference method to the high-order harmonics filter with ease because only the five-point stencil computations are required. The application could also be extended to other numerical methods only if the Helmholtz equation solver is available.