Open AccessLarge-Scale Eigenvalue Calculations for Stability Analysis of Steady Flows on Massively Parallel Computers
Author(s) -
Richard B Lehoucq,
Andrew G. Salinger
Publication year - 1999
Language(s) - English
Resource type - Reports
DOI - 10.2172/10357
Subject(s) - eigenvalues and eigenvectors , massively parallel , generalized minimal residual method , arnoldi iteration , mathematics , transformation (genetics) , stability (learning theory) , grashof number , krylov subspace , linear stability , iterative method , computer science , mathematical optimization , parallel computing , physics , reynolds number , mechanics , biochemistry , chemistry , turbulence , nusselt number , gene , instability , quantum mechanics , machine learning
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 leads to a generalized eigenvalue problem. The eigenvalues with largest real part are calculated using Arnoldi's iteration driven by a novel implementation of the Cayley transformation to recast the problem as an ordinary eigenvalue problem. The Cayley transformation requires the solution of a linear system at each Arnoldi iteration, which must be done iteratively for the algorithm to scale with problem size. A representative model problem of 3D 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