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In this work, we develop a high-order collocation method using radial basisfunction (RBF) for the incompressible Navier-Stokes equation (NSE) on therotating sphere. The method is based on solving the projection of the NSE onthe space of divergence-free functions. For that, we use matrix valued kernelfunctions which allow an analytically divergence-free approximation of thevelocity field. Using kernel functions which lead to rotation-freeapproximations, the pressure can be recovered by a simple kernel exchange inone of the occurring approximations, without solving an additional Poissonproblem. We establish precise error estimates for the velocity and the pressurefunctions for the semi-discretised solution. In the end, we give a shortestimate of the numerical cost and apply the new method to an experimental testcase.

An analytically divergence-free collocation method for the incompressible Navier-Stokes equations on the rotating sphere