Spherical harmonic projectors

Authors:
Paul N. Swarztrauber and William F. Spotz

Journal:
Math. Comp. **73** (2004), 753-760

MSC (2000):
Primary 65M70; Secondary 42C10, 74S25

DOI:
https://doi.org/10.1090/S0025-5718-03-01597-7

Published electronically:
October 2, 2003

MathSciNet review:
2031404

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Abstract | References | Similar Articles | Additional Information

Abstract: The harmonic projection (HP), which is implicit in the numerous harmonic transforms between physical and spectral spaces, is responsible for the reliability of the spectral method for modeling geophysical phenomena. As currently configured, the HP consists of a forward transform from physical to spectral space (harmonic analysis) immediately followed by a harmonic synthesis back to physical space. Unlike its Fourier counterpart in Cartesian coordinates, the HP does not identically reconstruct the original function on the surface of the sphere but rather replaces it with a weighted least-squares approximation. The importance of the HP is that it uniformly resolves waves on the surface of the sphere and therefore eliminates high frequencies that are artificially induced by the clustering of grid points in the neighborhood of the poles. The HP also maintains spectral accuracy when combined with the double Fourier method. Originally the HP required storage where is the number of latitudinal points. However, this was recently reduced to using an algorithm that also provided a savings of up to 50 percent in compute time. The HP was also generalized to an arbitrary latitudinal distribution of points. However, the HP as a composite of analysis and synthesis can be subject to considerable error depending on the point distribution. Here we define a variant of the traditional HP that is well conditioned, with condition number 1, for any point distribution. In addition, storage requirements are further reduced because the projections corresponding to all longitudinal wave numbers are expressed in terms of a single orthogonal matrix.

**1.**J. C. Adams and P. N. Swarztrauber, SPHEREPACK 3.0: A model development facility,*Mon. Wea. Rev.*,**127**(1999) pp. 1872-1878.**2.**SPHEREPACK 3.0, A model development facility, http://www.scd.ucar.edu/css/software/ spherepack/.**3.**P. E. Merilees, Numerical experiments with the pseudospectral method in spherical coordinates,*Atmosphere*,**12**(1974) pp. 77-96.**4.**W. F. Spotz, M. A. Taylor, and P. N. Swarztrauber, Fast shallow-water equation solvers in latitude-longitude coordinates,*J. Comp. Phys.*,**145**(1998) pp. 432-444.**5.**C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Society for Industrial and Applied Mathematics, Philadelphia, 1995. MR**96d:65067****6.**R. Jakob-Chien and B. K. Alpert, A fast spherical filter with uniform resolution,*J. Comp. Phys.*,**136**(1997) pp. 580-584.**7.**Netlib Repository of mathematical software, papers, and databases, http://www.netlib.org/.**8.**W. F. Spotz and P. N. Swarztrauber, A performance comparison of associated Legendre projections,*J. Comp. Phys.*,**168**(2001) pp. 339-355.**9.**G. W. Stewart, Matrix Algorithms Volume I: Basic Decompositions, Society for Industrial and Applied Mathematics, Philadelphia, 1998.**10.**P. N. Swarztrauber, On the spectral approximation of discrete scalar and vector functions on the sphere,*SIAM J. Numer. Anal.*,**16**(1979), pp. 934-949. MR**81c:65011****11.**P. N. Swarztrauber, Spectral transform methods for solving the shallow water equations on the sphere,*Mon. Wea. Rev.*,**124**(1996), pp. 730-744.**12.**P. N. Swarztrauber, D. L. Williamson, and J. B. Drake, The Cartesian method for solving partial differential equations in spherical geometry,*Dyn. Atmos. Oceans*,**27**(1997), pp. 679-706.**13.**P. N. Swarztrauber and W. F. Spotz, Generalized discrete spherical harmonic transforms,*J. Comp. Phys.*,**159**(2000), pp. 213-230. MR**2001i:33011****14.**N. Yarvin and V. Rokhlin, A generalized one-dimensional fast multipole method with application to filtering of spherical harmonics.*J. Comp. Phys.*,**147**(1998), pp. 594-609. MR**99h:78007**

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Additional Information

**Paul N. Swarztrauber**

Affiliation:
National Center for Atmospheric Research, P.O. Box 3000, Boulder, Colorado 80307-3000

Email:
pauls@ucar.edu

**William F. Spotz**

Affiliation:
Sandia Corporation, P.O. Box 5800, Albuquerque, New Mexico 87123-1110

Email:
wfspotz@sandia.gov

DOI:
https://doi.org/10.1090/S0025-5718-03-01597-7

Keywords:
Spectral method,
spherical harmonics,
projections

Received by editor(s):
July 19, 2002

Received by editor(s) in revised form:
November 11, 2002

Published electronically:
October 2, 2003

Additional Notes:
The first author was supported in part by the DOE and UCAR Climate Change Prediction Program under Cooperative Agreement No. DE-FC03-97ER62402. UCAR is sponsored by the NSF

Article copyright:
© Copyright 2003
American Mathematical Society