Numerical integration with polynomial exactness over a spherical cap

  • Authors:

  • Kerstin Hesse;Robert S. Womersley

  • Affiliations:

  • Handelshochschule Leipzig gGmbH, Leipzig, Germany 04109;School of Mathematics and Statistics, University of New South Wales, Sydney, Australia 2052

  • Venue:
  • Advances in Computational Mathematics
  • Year:
  • 2012

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Abstract

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere $\mathbb{S}^2$ , we discuss tensor product rules with n 2/2驴+驴O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree 驴驴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 $\mathbb{S}^2$ . For arbitrary d驴驴驴2, this strategy is extended to provide rules for numerical integration over spherical caps on $\mathbb{S}^d$ that have O(n d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree 驴驴n. We also show that positive weight rules for numerical integration over spherical caps on $\mathbb{S}^d$ that are exact for all spherical polynomials of degree 驴驴n have at least O(n d ) nodes and possess a certain regularity property.