Sixth-order spherical quadrature rule based on the trapezoidal rule on the Cubed Sphere

The Cubed Sphere is a spherical grid based on cartesian grids on the faces of a cube, widely used since its advent in the early seventies. The derivation of quadrature rules on this grid is still an active subject. Among the simplest rules, a bivariate trapezoidal rule on each face of the cube can be used. This strategy converges to any order for smooth functions supported inside the faces of the cube; more generally, it is only second-order accurate. However, J.-P.~Croisille introduced about fifteen years ago an ingenious modification of the weight on the eight corners, and observed the fourth-order accuracy; an accuracy which has not been theoretically proved so far. The first main result of this paper solves this open question. Next, a deeper analysis of the remainder permits to identify a further correction at the corners, yielding the second main result: a new quadrature rule which is sixth-order accurate. Numerical experiments are in agreement with this order of accuracy, and pave the way towards corrections of higher orders, based on values near the corners only.