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Title: Large-Scale Eigenvalue Calculations for Stability Analysis of Steady Flows on Massively Parallel Computers

Abstract

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.

Authors:
;
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Sponsoring Org.:
US Department of Energy (US)
OSTI Identifier:
10357
Report Number(s):
SAND99-1593
TRN: AH200125%%326
DOE Contract Number:  
AC04-94AL85000
Resource Type:
Technical Report
Resource Relation:
Other Information: PBD: 1 Aug 1999
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ALGORITHMS; BIFURCATION; EIGENVALUES; GRASHOF NUMBER; HEAT TRANSFER; INCOMPRESSIBLE FLOW; STEADY FLOW; PARALLEL PROCESSING

Citation Formats

Lehoucq, Richard B, and Salinger, Andrew G. Large-Scale Eigenvalue Calculations for Stability Analysis of Steady Flows on Massively Parallel Computers. United States: N. p., 1999. Web. doi:10.2172/10357.
Lehoucq, Richard B, & Salinger, Andrew G. Large-Scale Eigenvalue Calculations for Stability Analysis of Steady Flows on Massively Parallel Computers. United States. https://doi.org/10.2172/10357
Lehoucq, Richard B, and Salinger, Andrew G. 1999. "Large-Scale Eigenvalue Calculations for Stability Analysis of Steady Flows on Massively Parallel Computers". United States. https://doi.org/10.2172/10357. https://www.osti.gov/servlets/purl/10357.
@article{osti_10357,
title = {Large-Scale Eigenvalue Calculations for Stability Analysis of Steady Flows on Massively Parallel Computers},
author = {Lehoucq, Richard B and Salinger, Andrew G},
abstractNote = {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.},
doi = {10.2172/10357},
url = {https://www.osti.gov/biblio/10357}, journal = {},
number = ,
volume = ,
place = {United States},
year = {Sun Aug 01 00:00:00 EDT 1999},
month = {Sun Aug 01 00:00:00 EDT 1999}
}