Numerical simulation of propagation problems on the sphere with a compact scheme

We consider propagation problems on the sphere and their approximation by a compact finite difference scheme. The scheme used in this study uses the Cubed Sphere, a particular spherical grid with logically Cartesian structure. A central role is played by the standard one dimensional Hermitian derivative [22]. This compact scheme operates along great circles, thus avoiding any one sided compact scheme. [10, 11]. The scheme is centered. A simple high frequency filter is added to reinforce the stability. The final scheme is reminiscent of compact schemes in Computational Aeroacoustics or in turbulence Direct Numerical Simulation. Numerical results on a broad series of numerical test cases in climatology are presented, including linear convection problems, the linearized shallow water equations and the non linear shallow water equations. The results demonstrate the interest of the present approach in a variety of situations arising in numerical climatology