Hermitian Compact Interpolation on the Cubed-Sphere Grid

Abstract

The cubed-sphere grid is a spherical grid made of six quasi-cartesian square-like patches. It was originally introduced in Sadourny (Mon Weather Rev 100:136---144, 1972). We extend to this grid the design of high-order finite-difference compact operators (Collatz, The numerical treatment of differential equations. Springer, Berlin, 1960; Lele, J Comput Phys 103:16---42, 1992). The present work is limitated to the design of a fourth-order accurate spherical gradient. The treatment at the interface of the six patches relies on a specific interpolation system which is based on using great circles in an essential way. The main interest of the approach is a fully symmetric treatment of the sphere. We numerically demonstrate the accuracy of the approximate gradient on several test problems, including the cosine-bell test-case of Williamson et al. (J Comput Phys 102:211---224, 1992) and a deformational test-case reported in Nair and Lauritzen (J Comput Phys 229:8868---8887, 2010).