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. A recently developed quasi-periodic time-spectral (BDFTS) method extends the time-spectral method to problems with fast periodic content and slow mean flow transients, which should lead to faster solution of these types of problems as well. However, attaining superior efficiency with TS or BDFTS methods over traditional time-implicit methods hinges on the ability to rapidly solve the large non-linear system resulting from TS 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 TS and BDFTS methods are reworked such that 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 these 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.
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