An Order N log ₂(N) Parallel Solver for Time Spectral Problems

Abstract

The time-spectral (TS) method is a fast and efficient scheme for computing the solution to temporal periodic problems. Compared to traditional backward difference timeimplicit methods, time-spectral methods incur significant computational savings by taking advantage of the temporal Fourier representation of the time discretization. However, the computational cost for current time-spectral solver implementations, which are based on the discrete Fourier transform (DFT), scales as O(N 2) where N represents the number of time instances. For parallel implementations, where each time instance is assigned to an individual processor, the wall clock time necessary to converge time-spectral solutions is of order O(N), whereas, the wall clock time for an optimal solver in this weak scaling approach is expected to be on the order of O(1). The present paper is focused on making the current TS method more efficient by reducing the computational cost to O(N log 2 N) for even and O(2N log 3 N) for odd numbers of samples (where N is a power of 3) and hence for parallel implementations, achieving a wall clock time on the order of O(log 2 N) for even and O(log 3 N) for odd numbers of samples. This is achieved by developing a parallel fast Fourier transform (FFT) implementation instead of the current discrete Fourier transform approach for evaluating and solving the temporal derivative terms, which significantly decreases the computational cost and results in notably shorter wall clock time.