Spectral Elements with Prolate Spheroidal Basis Set

Prolate spheroidal wavefunctions of order zero have been proposed by Xiao, Rokhlin and Yarvin as an alternative to the usual Legendre polynomial basis for implementing spectral and pseudospectral methods for solving differential equations. The rewards for the change of basis are two-fold. First, when a spectral spatial discretization is combined with explicit time-marching, the Courant-Friedrichs-Lewy (CFL) timestep limit is increased by $O(\sqrt{N})$ where $N$ is the number of degrees of freedom by the change of expansion functions. Second, the effective resolution is increased by a factor of $\pi/2$ per spatial dimension, or a factor of almost four in three dimensions. We show that although the practical rewards for changing basis are always less than the theoretical imits, it is nevertheless possible to ``turbocharge'' spectral methods by using a prolate spheroidal basis.