Computing spectra of linear operators using the Floquet–Fourier–Hill method

In order to establish the stability of an equilibrium solution U   of an infinite-dimensional dynamical system u˙=X(u), one is interested in the spectrum of the linear operator L[U] obtained by linearizing the dynamical system around U. We use a spectrally accurate method for the computation of the spectrum of the maximal extension of the operator L[U]. The method is particularly well-suited to the case of periodic U, although no periodic boundary conditions on the perturbations are imposed. By incorporating the fundamentals of Floquet theory, an almost uniform approximation to the entire spectrum of the maximal extension is obtained, as opposed to an approximation of a few selected elements. The numerical component of the method is limited to: (i) choosing the size of the matrices to be used; and (ii) an eigenvalue solver, such as the QR algorithm. Compared to often-used finite-difference approaches, the method is an order of magnitude faster for comparable accuracy. We illustrate that the method is efficiently extended to problems defined on the whole line.