Spherical Harmonic Decomposition on a Cubic Grid

TL;DR

This paper introduces a method to decompose functions on a cubic grid into spherical harmonic components at a fixed radius, facilitating boundary condition treatment in numerical simulations without requiring specialized coordinates.

Contribution

It presents a novel approach combining interpolation and integration steps to extract spherical harmonic amplitudes directly from grid data without specialized coordinate systems.

Findings

Enables boundary condition application at larger radii in grid-based simulations.

Eliminates the need for coordinate adaptation to the sphere.

Provides a computationally efficient way to analyze spherical harmonic components.

Abstract

A method is described by which a function defined on a cubic grid (as from a finite difference solution of a partial differential equation) can be resolved into spherical harmonic components at some fixed radius. This has applications to the treatment of boundary conditions imposed at radii larger than the size of the grid, following Abrahams, Rezzola, Rupright et al.(gr-qc/9709082}. In the method described here, the interpolation of the grid data to the integration 2-sphere is combined in the same step as the integrations to extract the spherical harmonic amplitudes, which become sums over grid points. Coordinates adapted to the integration sphere are not needed.