Iterative Solution of the Eigenvalue Problem in Hopf Bifurcation for the Boussinesq Equations

Abstract

Some of the most challenging eigenvalue problems arise in the stability analysis of solutions to parameter-dependent nonlinear partial differential equations. Linearized stability analysis requires the computation of a certain purely imaginary eigenvalue pair of a very large, sparse complex matrix pencil. A computational strategy, the core of which is a method of inverse iteration type with preconditioned conjugate gradients, is used to solve this problem for the stability of thermocapillary convection. This convection arises in the float-zone model of crystal growth governed by the Boussinesq equations. The results obtained complete the stability picture augmenting the energy stability results [Mittelmann, et al., SIAM J. Sci. Statist. Comput., 13 (1992), pp. 411–424] and recent experimental results. Here a real eigenvalue of a Hermitian eigenvalue problem had to be determined.