CRPC-TR97681							February 1997

Title: Recovering Grid-Point Values Without Gibbs Oscillations in Two 
	Dimensional Domains on the Sphere

Authors: Anne Gelb, Antonio Navarra

Abstract:
Spectral methods using spherical harmonic basis functions have proven to 
be very effective in geophysical and astrophysical simulations. It is an 
unfortunate fact, however, that spurious oscillations, known as the Gibbs 
phenomenon, contaminate these spectral solutions, particularly in regions 
where discontinuities or steep gradients occur. They are also apparent in 
the polar regions even when considering analytical periodic functions.

These undesirable artificial oscillations have been the topic of several 
recent articles [8],[17],[19]. Navarra et.al. [19] alleviates the problem 
by employing various filters in one and two dimensions. Lindberg and 
Broccoli [17] implement a nonuniform spherical smoothing spline and zonal 
filtering, while Gelb [8] applies the Gegenbauer method [13] in the 
latitudinal direction for fixed longitudinal coordinates.

Since the physical problems solved on spheres often involve 
discontinuities or steep gradients in the longitudinal direction, and 
since spherical harmonic spectral methods always introduce oscillations 
in the polar regions, it is clear that an ideal numerical method should 
incorporate the removal of the Gibbs phenomenon in both directions, as 
suggested in both [17] and [19]. This paper offers a two-dimensional 
approach to the problem by simultaneously applying the Gegenbauer method 
in both directions. Assuming only the knowledge of the first (N+1)^2 
spherical harmonic coefficients, we prove an exponentially convergent 
approximation to a piecewise smooth function in regions composed of 
arbitrary rectangles for which the function is continuous, thereby 
entirely removing the Gibbs phenomenon.

Key Words: Gibbs phenomenon, Gegenbauer polynomials, spherical harmonics

AMS(MOS) Subject Classifications: 42A10, 42A20, 33C55, 65M70, 85-08