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

We present an approach for determining the linear stability of steady-states of PDEs on massively parallel computers. Linearizing the transient behavior around a steady-state solution leads to an eigen-value problem. The eigenvalues with largest real part are calculated using Arnoldi's iteration driven by a novel implementation of the Cay-ley 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 3D incompressible ow and heat transfer in a rotating disk reactor is used to analyze the eect 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 bifur-cation.