A method for refining high order numerical integration schemes is described. Particular focus is on integration schemes over the unit sphere with octahedral symmetry. The method is powerful enough that new integration schemes can be found from rough intuitive guesses. New schemes up to order 59 are presented. 0 1996 by John Wiley & Sons, Inc. igh order Gauss quadratures on a sphere are H of generic interest in numerical analysis.
They may have applications ranging from modeling of large scale phenomena on the globe, such as in weather forecasting, to the calculation of quantum mechanical integrals.'-3 In a Gauss quadrature on a sphere one would expect to be able to integrate three spherical harmonics per integration point having one weight and two positional parameters. For applications, it is more convenient to use a set of integration points with symmetry so that a much reduced parameter set needs to be dealt with. Because of the peculiarities of the triangulation of a sphere, the expectation just mentioned is usually not exactly fulfilled. Lower order schemes sometimes overperform or underperform quite significantly. An example of overperformance is the 72-point scheme with icosahedral symmetry of order 15.4 It integrates 256 functions, significantly more than the 216 expected. Integration schemes with octahedral symmetry, as pio-*E-mail delley