Multilevel approximation of the gradient operator on an adaptive spherical geodesic grid

This work presents a new adaptive multilevel approximation of the gradient operator on a recursively refined spherical geodesic grid. The multilevel structure provides a simple way to adapt the computation to the local structure of the gradient operator so that high resolution computations are performed only in regions where singularities or sharp transitions occur. This multilevel approximation of the gradient operator is used to solve the linear spherical advection equation for both time-independent and time-dependent wind field geophysical test cases. In contrast with other approximation schemes, this approach can be extended easily to other curved manifolds by choosing an appropriate coarse approximation and using recursive surface subdivision. The results indicate that the adaptive gradient calculation and the solution of spherical advection equation accurate, efficient and free of numerical dispersion.