A Cubed Sphere Fast Multipole Method

TL;DR

This paper introduces a novel Fast Multipole Method tailored for spherical geometries, utilizing barycentric Lagrange interpolation on a cubed sphere grid for efficient particle interaction computations.

Contribution

It presents a kernel-independent FMM variant using a quadtree on the cubed sphere, applicable to various equations on the sphere, with serial and parallel implementations.

Findings

Effective for Poisson and biharmonic equations on the sphere

Applicable to barotropic vorticity and tidal calculations

Comparable performance with tree code in serial and parallel

Abstract

This work describes a new version of the Fast Multipole Method for summing pairwise particle interactions that arise from discretizing integral transforms and convolutions on the sphere. The kernel approximations use barycentric Lagrange interpolation on a quadtree composed of cubed sphere grid cells. The scheme is kernel-independent and requires kernel evaluations only at points on the sphere. Results are presented for the Poisson and biharmonic equations on the sphere, barotropic vorticity equation on a rotating sphere, and self-attraction and loading potential in tidal calculations. A tree code version is also described for comparison, and both schemes are tested in serial and parallel calculations.