Wave-number Independent Preconditioning for GMRES Time-spectral Solvers

Abstract

The time-spectral method applied to the Euler equations theoretically offers significant computational savings for purely periodic problems when compared to standard time-implicit methods. However, attaining superior efficiency with time-spectral methods over traditional time-implicit methods hinges on the ability rapidly to solve the large non-linear system resulting from time-spectral discretizations which become larger and stiffer as more time instances are employed. In order to increase the efficiency of these solvers, and to improve robustness, particularly for large numbers of time instances, the Generalized Minimal Residual Method (GMRES) is used to solve the implicit linear system over all coupled time instances. The use of GMRES as the linear solver makes the time-spectral methods more robust, allows them to be applied to a far greater subset of time-accurate problems, including those with a broad range of harmonic content, and vastly improves the efficiency of time-spectral methods. However, it has been shown in previous work that when the number of time instances and/or the reduced frequency of motion increases (i.e. the maximum resolvable wave-number increases), the convergence degrades rapidly, requiring many more total preconditioning iterations to reach a converged solution. To alleviate this convergence degradation, this work formulates a wave-number independent preconditioner by inverting the spatial-temporal diagonal blocks in the preconditioner instead of the spatial diagonal blocks individually, as has been done previously. The solver utilizing this wave-number independent preconditioner is shown to be more efficient than past solvers under all conditions, but especially for high reduced frequencies and large number of time instances, i.e. conditions with high maximum wave numbers.