Numerical integration with polynomial exactness over a spherical cap

作者：Hesse Kerstin^*; Womersley Robert S

来源：Advances in Computational Mathematics, 2012, 36(3): 451-483.

DOI：10.1007/s10444-011-9187-2

• Equal weight rules
• Markov inequalities
• Numerical integration
• Polynomial exactness
• Positive weights
• Spherical cap
• Tensor product rules

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere 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 %26lt;= 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 S-2. For arbitrary d = 2, this strategy is extended to provide rules for numerical integration over spherical caps on S-d that have O(n(d)) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree %26lt;= n. We also show that positive weight rules for numerical integration over spherical caps on Sd that are exact for all spherical polynomials of degree %26lt;= n have at least O(n(d)) nodes and possess a certain regularity property.

• 出版日期2012-4