Prolate spheroidal wavefunctions as an alternative to Chebyshev and Legendre polynomials for spectral element and pseudospectral algorithms

Abstract

Prolate spheroidal functions of order zero are generalizations of Legendre polynomials which, when the “bandwidth parameter” c>0, oscillate more uniformly on x∈[−1,1] than either Chebyshev or Legendre polynomials. This suggests that, compared to these polynomials, prolate functions give more uniform spatial resolution. Further, when used as the spatial discretization for time-dependent partial differential equations in combination with explicit time-marching, prolate functions allow a longer stable timestep than Legendre polynomials. We show that these advantages are real and further, that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis. However, improvements in spatial resolution are at most a factor of π/2, approached slowly as N→∞. The timestep can be lengthened by several times, but not by a factor that grows rapidly with N. The prolate basis is not likely to radically expand the range of problems that can be done on a workstation. However, for production runs on the “bleeding edge” edge of arithmurgy, such as numerical weather prediction, the rewards for switching to a prolate basis may be considerable.