A Prolate-Element Method for Nonlinear PDEs on the Sphere

A p-type spectral-element method using prolate spheroidal wave functions (PSWFs) as basis functions, termed as the prolate-element method, is developed for solving partial differential equations (PDEs) on the sphere. The gridding on the sphere is based on a projection of the prolate-Gauss-Lobatto points by using the cube-sphere transform, which is free of singularity and leads to quasi-uniform grids. Various numerical results demonstrate that the proposed prolate-element method enjoys some remarkable advantages over the polynomial-based element method: (i) it can significantly relax the time step size constraint of an explicit time-marching scheme, and (ii) it can increase the accuracy and enhance the resolution.