An order N log N parallel solver for time-spectral problems

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The time-spectral method is a fast and efficient scheme for computing the solution to temporal periodic problems. Compared to traditional backward difference implicit time-stepping methods, time-spectral methods incur significant computational savings by using a temporal Fourier representation of the time discretization and solving the periodic problem directly. In the time-spectral discretization, all time instances are fully coupled to each other, resulting in a dense temporal discretization matrix, the evaluation of which scales as O(N2), where N denotes the number of time instances. However, by implementing the time-spectral method based on the fast Fourier transform (FFT) the computational cost can be reduced to O(Nlog⁡N). Furthermore, in parallel implementations, where each time instance is assigned to an individual processor, the wall-clock time necessary to solve time-spectral problems is reduced to O(log⁡N) using the FFT-based approach, as opposed to the O(N) weak scaling incurred by previous dense matrix or discrete Fourier transform (DFT) parallel time-spectral solver implementations. In this work, first an FFT-based approximate factorization (AF) scheme is used to solve time-spectral problems with large numbers of time instances. Subsequently, this solution strategy is reformulated as a preconditioner to be used in the context of a Newton-Krylov method applied directly to the complete non-linear space-time time-spectral residual. The use of the Generalized Minimal Residual Method (GMRES) Krylov method enables additional coupling between the various time instances running on different processors resulting in faster overall convergence. The GMRES/AF scheme is shown to produce significantly faster convergence than the AF scheme used as a solver alone, and achieves orders of magnitude gains in efficiency compared to previous DFT-based implementations of similar solution strategies for large numbers of time instances.