Efficient Solution of a Generalized Eigenvalue Problem Arising in a Thermoconvective Instability (Communications in Computational Physics, 2018)
The aim of this paper is to develop an efficient numerical method to compute the eigenvalues of the stability analysis of a problem describing the motion of afluid within a cylindrical container heated non-homogeneously from below. An axisymmetric stationary motion settles in, at certain values of the external parametersappearing in the set of partial differential equations modeling the problem. This basic solution is computed by discretizing the equations with a Chebyshev collocationmethod. Its linear stability is formulated with a generalized eigenvalue problem. Thenumerical approach (generalized Arnoldi method) uses the idea of preconditioningthe eigenvalue problem with a modified Cayley transformation before applying theArnoldi method. Previous works have dealt with transformations requiring regularityto one of the submatrices. In this article we extend those results to the case in whichthat submatrix is singular. This method allows a fast computation of the critical eigenvalues which determine whether the steady flow is stable or unstable. The algorithmbased on this method is compared to the QZ method and is found to be computationally more efficient. The reliability of the computed eigenvalues in terms of stability isconfirmed via pseudospectra calculations.