An adaptive wavelet collocation method for the solution of partial differential equations on the sphere

A dynamic adaptive numerical method for solving partial differential equations on the sphere is developed. The method is based on second generation spherical wavelets on almost uniform nested spherical triangular grids, and is an extension of the adaptive wavelet collocation method to curved manifolds. Wavelet decomposition is used for grid adaption and interpolation. An O(N) hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate Laplace–Beltrami, Jacobian and flux-divergence operators. The accuracy and efficiency of the method is demonstrated using linear and nonlinear examples relevant to geophysical flows. Although the present paper considers only the sphere, the strength of this new method is that it can be extended easily to other curved manifolds by considering appropriate coarse approximations to the desired manifold (here we used the icosahedral approximation to the sphere at the coarsest level).