Approximation of the differential operators on an adaptive spherical geodesic grid using spherical wavelets

In this work, a new adaptive multi-level approximation of surface divergence and scalar-valued surface curl operator on a recursively refined spherical geodesic grid is presented. A hierarchical finite volume scheme based on the wavelet multi-level decomposition is used to approximate the surface divergence and scalar-valued surface curl operator. The multi-level structure provides a simple way to adapt the computation to the local structure of the surface divergence and scalar-value surface curl operator so that the high resolution computations are performed only in regions where singularities or sharp transitions occur. This multi-level approximation of the surface divergence operator is then used in an adaptive wavelet collocation method (AWCM) to solve two standard advection tests, solid-body rotation and divergent flow on the sphere. In contrast with other approximate schemes, this approach can be extended easily to other curved manifolds by considering appropriate coarse approximation to the desired manifold (here we used the icosahedral approximation to the sphere at the coarsest level) and using recursive surface subdivision.