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Large‐scale eigenvalue calculations for stability analysis of steady flows on massively parallel computers
Author(s) -
Lehoucq Richard B.,
Salinger Andrew G.
Publication year - 2001
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.135
Subject(s) - eigenvalues and eigenvectors , massively parallel , generalized minimal residual method , mathematics , arnoldi iteration , partial differential equation , krylov subspace , grashof number , stability (learning theory) , transformation (genetics) , linear stability , iterative method , computer science , mathematical optimization , mathematical analysis , reynolds number , parallel computing , physics , nonlinear system , mechanics , biochemistry , chemistry , quantum mechanics , machine learning , turbulence , nusselt number , gene
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 © 2001 John Wiley & Sons, Ltd.