Collocation Methods for Exploring Perturbations in Linear Stability Analysis
For researchers in fluid dynamics and dynamical systems, this offers a cheaper alternative to pseudo-spectral analysis for detecting transient instabilities missed by eigenvalue analysis.
The paper shows that stochastic collocation methods can inexpensively perform pseudo-spectral analysis to predict transient growth in linearly stable dynamical systems, providing quantitative instability information and insight into unsteady flow behavior.
Eigenvalue analysis is a well-established tool for stability analysis of dynamical systems. However, there are situations where eigenvalues miss some important features of physical models. For example, in models of incompressible fluid dynamics, there are examples where linear stability analysis predicts stability but transient simulations exhibit significant growth of infinitesimal perturbations. This behavior can be predicted by pseudo-spectral analysis. In this study, we show that an approach similar to pseudo-spectral analysis can be performed inexpensively using stochastic collocation methods and the results can be used to provide quantitative information about instability. In addition, we demonstrate that the results of the perturbation analysis provide insight into the behavior of unsteady flow simulations.