# Collocation Methods for Exploring Perturbations in Linear Stability Analysis

## TL;DR

This paper introduces a stochastic collocation approach to pseudo-spectral analysis for better understanding of instabilities in fluid dynamics, revealing features missed by traditional eigenvalue analysis.

## Contribution

It presents a novel, cost-effective method using collocation techniques to analyze perturbations and instability in dynamical systems beyond eigenvalue analysis.

## Key findings

- Stochastic collocation methods can predict transient growth of perturbations.
- The approach provides insights into unsteady flow behaviors.
- It offers a quantitative tool for stability analysis in fluid dynamics.

## Abstract

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.