Many laboratory experiments have been performed with the differentially heated rotating fluid annulus, where various flow patterns are observed as the rotation and temperature difference between the annulus walls are varied. A rich variety of dynamical behaviour has been observed in the experiments that has not been theoretically explained. Bifurcation analysis is a promising means of gaining insight into these unexplained dynamics. For example, an analysis of the double Hopf bifurcations that occur in a mathematical model of the annulus along the transition from steady flow to wave motion indicates the mechanism by which the experimentally observed hysteresis of the wave motion occurs.

To perform such a bifurcation analysis, the boundaries of the region of linear stability of a steady solution of the dynamical equations must be found. This involves the computation of certain eigenvalues at many locations in the space of parameters. For the present model of the differentially heated rotating annulus, the eigenvalues (and eigenfunctions) satisfy a partial differential generalized eigenvalue problem (in two spatial dimensions) that cannot be solved analytically. Discretization leads to a generalized matrix eigenvalue problem, with large sparse matrices, from which approximations of the eigenvalues of the continuous problem may be found.

Because the approximation of the eigenvalues is the most computationally intensive step in the bifurcation analysis, much can be gained by increasing the efficiency of these approximations. I will discuss the eigenvalue approximation in the context of the differentially heated rotating annulus and the possibility of using the nature of the problem to make improvements. Such improvements should be directly applicable to many other systems.