Iterative solvers for the thermoacoustic nonlinear eigenvalue problem and their convergence properties

The spectrum of the thermoacoustic operator is governed by a nonlinear eigenvalue problem. A few different strategies have been proposed by the thermoacoustic community to tackle it and identify the frequencies and growth rates of thermoacoustic eigenmodes. These strategies typically require the use of iterative algorithms, which need an initial guess and are not necessarily guaranteed to converge to an eigenvalue. A quantitative comparison between the convergence properties of these methods has however never been addressed. By using adjoint-based sensitivity, in this study we derive an explicit formula that can be used to quantify the behaviour of an iterative method in the vicinity of an eigenvalue. In particular, we employ Banach’s fixed-point theorem to demonstrate that there exist thermoacoustic eigenvalues that cannot be identified by some of the iterative methods proposed in the literature, in particular fixed-point iterations, regardless of the accuracy of the initial guess provided. We then introduce a family of iterative methods known as Householder’s methods, of which Newton’s method is a special case. The coefficients needed to use these methods are explicitly derived by means of high-order adjoint-based perturbation theory. We demonstrate how these methods are always guaranteed to converge to the closest eigenvalue, provided that the initial guess is accurate enough. We also show numerically how the basin of attraction of the eigenvalues varies with the order of the employed Householder’s method.