Hybrid prolate pseudospectral methods on Chebyshev and Legendre grids I - A preview

The use of prolate spheroidal wave functions as basis functions in the pseudospectral method has been shown to possess strong advantages over the orthogonal polynomials and related functions of conventional pseudospectral methods. The most notable of these is higher accuracy and a smaller number of unknowns for problems involving bandlimited functions. However, when the bandwidth parameter c and unknown number N are large, significant pre-processing effort is involved in computing the prolate collocation grid positions. In this paper, we introduce hybrid prolate-Chebyshev and prolate-Legendre pseudospectral schemes which are based on prolate spheroidal wave functions but employ classic Chebyshev and Legendre Gauss-Lobatto grids for collocation. The hybrid schemes afford less pre-computation overhead compared to pure prolate pseudospectral methods, yielding solutions of higher accuracy than conventional Chebyshev or Legendre pseudospectral methods. The utility of the hybrid algorithms is demonstrated through several numerical examples.