# When eigenvalues aren't enough: problems with linear stability analysis

#### When eigenvalues aren't enough: problems with linear stability analysis

- Event time: 11:30am
- Event date: 27th March 2013
- Speaker: Lawrence Mitchell (EPCC)
- Location: Room 2511, James Clerk Maxwell Building (JCMB) James Clerk Maxwell Building Peter Guthrie Tait Road Edinburgh EH9 3FD GB

### Event details

We are often interested in how and when a stable solution to a problem becomes unstable. For example, in laminar fluid flows, at what Reynolds number does the flow become unstable? The typical technique to address this is to linearise the problem about the stable solution and the consider the eigenvalue spectrum of the linearised operator looking for "unstable" eigenvalues (values for which the corresponding eigenvector grows exponentially as a function of time). In some cases, when the eigenvectors of the operator are not orthogonal, this method fails to give accurate predictions. During the 1980s and 90s, these problems were addressed through the technique of pseudospectral analysis leading to the development of "generalised stability theory". I shall give a brief overview of these techniques, and describe likely reasons as to why the method is not often used, despite its efficacy. Finally I shall describe some recent advances that make the implementation of generalised stability analysis in computational finite element models a trivial procedure for modellers.

Associated papers:

Trefethen, Trefethen, Reddy, Driscoll. Science 261:578 (1993)

Farrell, Ioannou. Journal of the Atmospheric Sciences 53(14):2025 (1996)

Farrell, Ioannou. Journal of the Atmospheric Sciences 53(14):2041 (1996)

Farrell, Cotter, Funke. arXiv:1211.6989 [cs.MS]

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