A modified Cayley transform for the discretized Navier-Stokes equations

K. A. Cliffe; T. J. Garratt; Alastair Spence

Applications of Mathematics (1993)

  • Volume: 38, Issue: 4-5, page 281-288
  • ISSN: 0862-7940

Abstract

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This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the generalized eigenvalue problem which overcomes a drawback of the usual Cayley transform applied to such problems. Standard iterative methods are then applied to the transformed eigenvalue problem. Numerical experiments are performed on large matrices arising from a discretization of the flow over a backward facing step.

How to cite

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Cliffe, K. A., Garratt, T. J., and Spence, Alastair. "A modified Cayley transform for the discretized Navier-Stokes equations." Applications of Mathematics 38.4-5 (1993): 281-288. <http://eudml.org/doc/15755>.

@article{Cliffe1993,
abstract = {This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the generalized eigenvalue problem which overcomes a drawback of the usual Cayley transform applied to such problems. Standard iterative methods are then applied to the transformed eigenvalue problem. Numerical experiments are performed on large matrices arising from a discretization of the flow over a backward facing step.},
author = {Cliffe, K. A., Garratt, T. J., Spence, Alastair},
journal = {Applications of Mathematics},
keywords = {block matrices; eigenvalues; Cayley transform; Navier-Stokes; large sparse generalized eigenvalue problems; Hopf bifurcations; block matrices; large sparse generalized eigenvalue problems; Hopf bifurcations},
language = {eng},
number = {4-5},
pages = {281-288},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A modified Cayley transform for the discretized Navier-Stokes equations},
url = {http://eudml.org/doc/15755},
volume = {38},
year = {1993},
}

TY - JOUR
AU - Cliffe, K. A.
AU - Garratt, T. J.
AU - Spence, Alastair
TI - A modified Cayley transform for the discretized Navier-Stokes equations
JO - Applications of Mathematics
PY - 1993
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 38
IS - 4-5
SP - 281
EP - 288
AB - This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the generalized eigenvalue problem which overcomes a drawback of the usual Cayley transform applied to such problems. Standard iterative methods are then applied to the transformed eigenvalue problem. Numerical experiments are performed on large matrices arising from a discretization of the flow over a backward facing step.
LA - eng
KW - block matrices; eigenvalues; Cayley transform; Navier-Stokes; large sparse generalized eigenvalue problems; Hopf bifurcations; block matrices; large sparse generalized eigenvalue problems; Hopf bifurcations
UR - http://eudml.org/doc/15755
ER -

References

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