Efficient Evaluation of Ellipsoidal Harmonics for Potential Modeling

TL;DR

This paper introduces an efficient numerical method for computing ellipsoidal harmonic normalization constants by applying tanh-sinh quadrature to singular integrals, improving potential modeling accuracy.

Contribution

It demonstrates the effectiveness of tanh-sinh quadrature for ellipsoidal harmonic normalization, supported by theoretical proofs and comparative analysis.

Findings

Tanh-sinh quadrature accurately approximates normalization constants.

The integrands are proven to be suitable for tanh-sinh application.

Results outperform other change-of-variable quadratures.

Abstract

Ellipsoidal harmonics are a useful generalization of spherical harmonics but present additional numerical challenges. One such challenge is in computing ellipsoidal normalization constants which require approximating a singular integral. In this paper, we present results for approximating normalization constants using a well-known decomposition and applying tanh-sinh quadrature to the resulting integrals. Tanh-sinh has been shown to be an effective quadrature scheme for a certain subset of singular integrands. To support our numerical results, we prove that the decomposed integrands lie in the space of functions where tanh-sinh is optimal and compare our results to a variety of similar change-of-variable quadratures.