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

Spectral methods using spherical harmonic basis functions have proven to be very eeective 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 artiicial oscillations have been the topic of several recent articles 8],,17],,19]. Navarra et.al. 19] alleviates the problem by employing various lters in one and two dimensions. Lindberg and Broccoli 17] implement a nonuniform spherical smoothing spline and zonal ltering, while Gelb 8] applies the Gegenbauer method 13] in the latitudinal direction for xed 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 ooers a two-dimensional approach to the problem by simultaneously applying the Gegenbauer method in both directions. Assuming only the knowledge of the rst (N + 1) 2 spherical harmonic coeecients, 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.