Computing Eigenvalues Occurring in Continuation Methods with the Jacobi–Davidson QZ Method

Abstract

Continuation methods are well-known techniques for computing several stationary solutions of problems involving one or more physical parameters. In order to determine whether a stationary solution is stable, and to detect the bifurcation points of the problem, one has to compute the rightmost eigenvalues of a related, generalized eigenvalue problem. The recently developed Jacobi–Davidson QZ method can be very effective for computing several eigenvalues of a given generalized eigenvalue problem. In this paper we will explain how the Jacobi–Davidson QZ method can be used to compute the eigenvalues needed in the application of continuation methods. As an illustration, the two-dimensional Rayleigh–Bénard problem has been studied, with the Rayleigh number as a physical parameter. We investigated the stability of stationary solutions, and several bifurcation points have been detected. The Jacobi–Davidson QZ method turns out to be very efficient for this problem.