Numerical integration with polynomial exactness over a spherical cap

摘要

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere mathbbS2mathbb{S}^2, we discuss tensor product rules with n 2/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n 3) nodes for numerical integration over spherical caps on mathbbS2mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on mathbbSdmathbb{S}^d that have O(n d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on mathbbSdmathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n d ) nodes and possess a certain regularity property.