Quadrature and symmetry on the Cubed Sphere

Abstract

In the companion paper [6], a spherical harmonics subspace associated to the Cubed Sphere has been introduced. This subspace is further analyzed here. In particular, it permits to dene a new Cubed Sphere based quadrature. This quadrature inherits the rotationally invariant properties of the spherical harmonics subspace. Contrary to Gauss quadratures, where the set of nodes and weights is solution of a nonlinear system, only the weights are unknown here. Despite this conceptual simplicity, the new quadrature displays an accuracy comparable to optimal quadratures, such as the Lebedev rules.

Symmetry group of the equiangular cubed sphere

“…Our main result shows that the group of the cube is the suitable symmetry group for the determination of quadrature weights on the cubed sphere. This background supports a quadrature rule of ongoing works [2]. Moreover, for quadrature rules, the geometric distribution of the nodes is often examinated.…”

Least squares spherical harmonics approximation on the Cubed Sphere

“…Here we consider the important case of a weight w(x ) with cubic symmetry. This property has been considered in [5,9]. Proof.…”

“…This interpolation framework has been in particular used in [5] to define new spherical quadrature rules of accuracy comparable to optimal ones (Lebedev rules). Here, we show that the first subspace Y 2N −1 is a suitable choice if one wants a least squares approximant instead of an interpolant.…”