Large-scale eigenvalue calculations for stability analysis of steady flows on massively parallel computers

This paper presents an approach for determining the linear stability of steady states of partial differential equations (PDEs) on massively parallel computers. Linearizing the transient behavior around a steady state solution leads to an eigenvalue problem. The eigenvalues with the largest real part are calculated using Arnoldi's iteration driven by a novel implementation of the Cayley transformation. The Cayley transformation requires the solution of a linear system at each Arnoldi iteration. This is done iteratively so that the algorithm scales with problem size. A representative model problem of three-dimensional incompressible flow and heat transfer in a rotating disk reactor is used to analyze the effect of algorithmic parameters on the performance of the eigenvalue algorithm. Successful calculations of leading eigenvalues for matrix systems of order up to 4 million were performed, identifying the critical Grashof number for a Hopf bifurcation. Copyright (C) 2001 John Wiley & Sons, Ltd.