A Cubed Sphere Fast Multipole Method

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.

Figures (12)

• Figure 1: Partitions of the sphere showing grid cells and cell centers, (a) icosahedral grid, hexagonal cells (aside from 12 pentagons), cell centers are vertices of the icosahedral triangulation, (b) cubed sphere grid, cell centers are mapped to the sphere from uniform grid points on the faces of the inscribed cube, (c) latitude-longitude grid, cell center coordinates are given by averaging the latitude and longitude of cell edges.
• Figure 1: Quadtree composed of equiangular gnomonic cubed sphere grid cells, (a) level 0, (b) level 1, (c) level 2, CSFMM and CSTC use barycentric Lagrange interpolation in cells, example of $5 \times 5$ Chebyshev points ($\bullet$) in each cell.
• Figure 1: Four types of interactions, in each case the target cluster $C_t$ is on the left (target particles in blue $\bullet$) and the source cluster $C_s$ is on the right (source particles in red $\bullet$), sample $5 \times 5$ grid of Chebyshev proxy particles is shown (black $\square$), $R$ = distance between cluster centers, $r_t,r_s$ = radii of $C_t, C_s$.
• Figure 1: Poisson equation \ref{['eq:Poisson_equation']}, the integral in \ref{['eq:G_L_1']} is computed for real spherical harmonics (a) $Y_{4,3}$, (e) $Y_{8,5}$, icosahedral grid with $N=163842$ points, (b,f) discretization error $|\phi_{\rm DS}-\phi_{\rm EX}|$, (c,g) CSFMM approximation error $|\phi_{\rm FS}-\phi_{\rm DS}|$, (d,h) CSFMM solution error $|\phi_{\rm FS}-\phi_{\rm EX})$, CSFMM with MAC = 0.7, degree $n=6$.
• Figure 1: Serial runtime (s) versus particle count $N$ for $N$-body sum \ref{['eq:nbodysum']} with Laplace kernel \ref{['eq:G_L']}, target and source particles on icosahedral grid, direct sum, CSFMM and CSTC with $\mathrm{MAC}=0.7$, degree $n=6$.