Error and symmetry analysis of Misner's algorithm for spherical harmonic decomposition on a cubic grid

TL;DR

This paper analyzes the error behavior and symmetry-based optimizations of Misner's algorithm for spherical harmonic decomposition on cubic grids, providing guidelines for parameter selection and improved implementation strategies.

Contribution

It offers a detailed analysis of truncation errors in Misner's algorithm and introduces symmetry-based enhancements for more efficient computation.

Findings

Truncation errors depend on algorithm parameters and data properties.

Symmetry considerations enable more efficient implementation.

Guidelines for parameter selection improve accuracy in practical applications.

Abstract

Computing spherical harmonic decompositions is a ubiquitous technique that arises in a wide variety of disciplines and a large number of scientific codes. Because spherical harmonics are defined by integrals over spheres, however, one must perform some sort of interpolation in order to compute them when data is stored on a cubic lattice. Misner (2004, Class. Quant. Grav., 21, S243) presented a novel algorithm for computing the spherical harmonic components of data represented on a cubic grid, which has been found in real applications to be both efficient and robust to the presence of mesh refinement boundaries. At the same time, however, practical applications of the algorithm require knowledge of how the truncation errors of the algorithm depend on the various parameters in the algorithm. Based on analytic arguments and experience using the algorithm in real numerical simulations, I explore these dependencies and provide a rule of thumb for choosing the parameters based on the truncation errors of the underlying data. I also demonstrate that symmetries in the spherical harmonics themselves allow for an even more efficient implementation of the algorithm than was suggested by Misner in his original paper.