The Choice of Spectral Functions on a Sphere for Boundary and Eigenvalue Problems: A Comparison of Chebyshev, Fourier and Associated Legendre Expansions

Abstract

Modified Fourier series, as judged by criteria of accuracy, numerical efficiency and ease of programming, are the best choice of latitudinal expansion functions for general problems on the sphere. The pseudospectral and spectral methods, however, can be easily and successfully applied with all three types of orthogonal series. For special situations, such as when the latitude variable is stretched, Chebyshev polynomials are the only practical choice, but for orthodox problems on the globe, they are less efficient than the other two sets of functions. Although spherical harmonics have been universally employed in the past, Fourier series give comparable accuracy and are significantly easier to program and manipulate. Thus, in the absence of a special reason to the contrary, the simplest and most effective way to handle the north?south dependence of the solution to a boundary or eigenvalue problem on the sphere is to use a Fourier series in colatitude.