Abstract

The Cubed-Sphere is a spherical grid made of six quasi-cartesian square like patches. It was originally introduced by Sadourny some forty years ago. We extend to this grid the design of high-order finite difference compact operators. Such discrete operators are used in Computational Fluid Dynamics on structured grids for applications such as Direct Numerical Simulation of turbulent flows, or aeroacoustics problems. We consider in this work the design of a uniformly fourth-order accurate spherical gradient. The main approximation principle consists in defining a network of great circles covering the Cubed-Sphere along which a high-order hermitian gradient can be calculated. This procedure allows a natural treatment at the interface of the six patches. The main interest of the approach is a fully symmetric approximation system on the Cubed-Sphere. We numerically demonstrate the accuracy of the approximate gradient on several test problems, in particular the cosine-bell test-case of Williamson et al. for climatology.