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Large Scale Eigenvalue Calculations for Computing the Stability of Buoyancy Driven Flows

We present results for large scale linear stability analysis of buoyancy driven fluid flows using a parallel finite element CFD code (MPSalsa) along with a general purpose eigensolver (ARPACK). The goal of this paper is to examine both the capabilities and limitations of such an approach, with particular focus on solving large problems on massively parallel computers using iterative methods. We accomplish our goal by solving a large variety of two and three dimensional problems of varying difficulty, comparing our results (whenever possible) to semi-analytical results. We also carefully explain how we successfully combined Cayley transformations with an Arnoldi based eigensolver and preconditioned Krylov methods for the necessary linear solves. For problems where the advective terms are not significant, we achieve excellent convergence of the computed eigenvalues as we refine the finite element mesh. We also successfully solve advectively dominated problems, but the convergence is slower. We believe that the main difficulties arise not from problems with the eigensolver, but from the accuracy of the finite element discretization. Therefore, we believe that our results are as reliable as using transient integration but are more efficiently computed. The largest eigenvalue problem we solve has over 16 million unknowns on 2048 processors.