Krylov Methods for Large-Scale Dynamical Systems: Application in Fluid Dynamics

The ability to predict and characterize bifurcations from the onset of unsteadiness to the transition to turbulence is of critical importance in both academic and industrial applications. Numerous tools from dynamical system theory can be employed for that purpose. In this review, we focus on the practical computation and stability analyses of steady and time-periodic solutions with a particular emphasis on very high-dimensional systems such as those resulting from the discrete Navier-Stokes equations. In addition to a didactically concise theoretical framework, we introduce nekStab, an open source and user-friendly toolbox dedicated to such analyses using the spectral element solver Nek5000. Relying on Krylov methods and a time-stepper formulation, nekStab inherits the flexibility and high performance capabilities of Nek5000 and can be used to study the stability properties of flows in complex three-dimensional geometries. The performances and accuracy of nekStab are presented on the basis of standard benchmarks from the literature. For the sake of pedagogy and clarity, most of the algorithms implemented in nekStab are presented herein using Python pseudocode. Because of its flexibility and domain-agnostic nature, the methodology presented in this work can be applied to develop similar toolboxes for other solvers, most importantly outside the field of fluid dynamics.