In simulating interstellar dust clouds astrophysicists need high accuracy
integration formulae for functions on the sphere. To construct better
formulae than previously used, almost equidistantly spaced nodes on the
sphere and weights belonging to these nodes are required.
This problem is closely related to a facility dispersion problem on the sphere
and to the theories of spherical designs and multivariate Gauss quadrature
formulae.
We propose a two-stage algorithm to compute optimal facility locations
on the unit sphere with high accuracy and an appropriate algorithm to
calculate the corresponding weights of the cubature formulae.
These algorithms can be extended to other facility dispersion problems.
Numerical examples show that the constructed formulae
yield impressive small integration errors of approximately $10^{-12}$.