Adaptive radial basis function and Hermite function pseudospectral methods for computing eigenvalues of the prolate spheroidal wave equation for very large bandwidth parameter

Asymptotic approximations show that the lowest modes of the prolate spheroidal wave equation are concentrated with an O(1/c) length scale where c is the "bandwidth" parameter of the prolate differential equation. Accurate computation of the ground state eigenvalue by the long-known Legendre-Galerkin method requires roughly 3.8c Legendre polynomials. Some studies have therefore applied a grid with 20,000 points in conjunction with high order finite differences to achieve c=10⁷. Here, we show that by adaptively applying either Hermite functions or Gaussian radial basis functions (RBFs), it is never necessary to use more than eighty degrees of freedom to calculate the lowest dozen eigenvalues of each symmetry class. For small c, the eigenmodes are not confined to a small portion of the domain θ∈[-π/2, π/2] in latitude, but are global. We show that by periodizing the basis functions via imbricate series, it is possible to apply Hermite and RBFs even in the limit c→0. (The Legendre method is probably a little more efficient in this limit since the prolate functions tend to Legendre polynomials in this limit.) A "sideband truncation" restricts the discretization to a small block taken from the large Hermite-Galerkin matrix. We show that sideband truncation with blocks as small as 5×5 is a very efficient way to compute high order modes. In an appendix, we prove a rigorous convergence theorem for the periodized Hermite expansion.