A Performance Comparison of Associated Legendre Projections

The associated Legendre projection provides a means for accelerating the dynamical core of a global weather or climate model. Therefore, the goal is to determine the fastest possible projection algorithm, of which this paper compares six: the standard method, which computes the projection using a forward and backward associated Legendre transform; the direct method, which uses a single projection matrix when this approach results in fewer operations; the fast multipole method; the weighted orthogonal complement method; the seminaive method; and a projection based on transforms proposed by Mohlenkamp. Timing results indicate that all the projections behave like O(N3) algorithms up to at least N=200 spectral truncation. For this range of resolutions, the weighted orthogonal complement has the lowest operation count, best cache utilization, and best overall timings.