Krylov methods for large-scale dynamical systems: Application in fluid dynamics

TL;DR

This paper reviews Krylov methods for analyzing large-scale fluid dynamics systems, emphasizing stability analysis and bifurcation detection using matrix-free approaches integrated into open-source tools.

Contribution

It introduces a practical framework employing Krylov methods for stability analysis of high-dimensional fluid systems, extending existing tools and demonstrating their effectiveness.

Findings

Efficient stability analysis of Navier-Stokes discretizations

Implementation of methods in open-source toolbox nekStab

Validation on standard fluid mechanics benchmarks

Abstract

In fluid dynamics, predicting and characterizing bifurcations, from the onset of unsteadiness to the transition to turbulence, is of critical importance for both academic and industrial applications. Different tools from dynamical systems theory can be used for this purpose. In this review, we present a concise theoretical and numerical framework focusing on practical aspects of the computation and stability analyses of steady and time-periodic solutions, with emphasis on high-dimensional systems such as those arising from the spatial discretization of the Navier-Stokes equations. Using a matrix-free approach based on Krylov methods, we extend the capabilities of the open-source high-performance spectral element-based time-stepper Nek5000. The numerical methods discussed are implemented in nekStab, an open-source and user-friendly add-on toolbox dedicated to the study of stability properties of flows in complex three-dimensional geometries. The performance and accuracy of the methods are illustrated and examined using standard benchmarks from the fluid mechanics literature. Thanks to 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 mechanics.