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 eigenvalue problem. The eigenvalues with 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 3D incompressible ow and heat transfer in a rotating disk reactor is used to analyze the e ect 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-