Computing eigenvalues occuring in continuation methods with the Jacobi-Davidson QZ method

Abstract

Continuation methods are a well-known technique 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 eective 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-Benard 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 ecient for this problem.