Geodesic
A modified Cayley transform for the discretized Navier-Stokes equations
Applications of Mathematics, Tome 38 (1993) no. 4-5, pp. 281-288

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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.
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.
DOI : 10.21136/AM.1993.104556
Classification : 15A18, 65F15, 65F50, 65M12, 76D05, 76M10, 76M25
Keywords: block matrices; eigenvalues; Cayley transform; Navier-Stokes; large sparse generalized eigenvalue problems; Hopf bifurcations 
Cliffe, K. A.; Garratt, T. J.; Spence, A. A modified Cayley transform for the discretized Navier-Stokes equations. Applications of Mathematics, Tome 38 (1993) no. 4-5, pp. 281-288. doi: 10.21136/AM.1993.104556
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[1] W. E. Arnoldi: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math. 9 (1951), 17-29. | DOI | MR | Zbl

[2] K. N. Christodoulou, L.E. Scriven: Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem. J. Sci. Comput. 3 (1988), 355-406. | DOI | Zbl

[3] K. A. Cliffe, T .J. Garratt, and A. Spence: Iterative methods for the detection of hopf bifurcations in finite element discretizations of incompressible flow problems. Submitted SIAM J. Sci. Stat. Соmр. (1992).

[4] K .A. Cliffe T. J. Garratt A. Spence, and K. H. Winters: Eigenvalue calculations for flow over a backward facing step. AEA Decommissioning and Rad waste report, 1992.

[5] J. N. Franklin: Matrix theory. Prentice-Hall, New Jersey, 1968. | MR | Zbl

[6] T. J. Garratt G. Moore, and A. Spence: Two methods for the numerical detection of Hopf bifurcations. In: Bifurcation and chaos: analysis, algorithms, applications (R. Seydel, F.W. Schneider, and H. Troger, eds.), Birkhäuser, 1991, pp. 119-123. | MR

[7] D. K. Gartling: A test problem for outflow boundary conditions - flow over a backward facing step. Int. J. Num. Methods Fluids 11 (1990), 953-967. | DOI

[8] G. H. Golub, C. F. van Loan: Matrix Computations. The Johns Hopkins University Press, Baltimore, Maryland, 1983. | MR | Zbl

[9] C. P. Jackson: A finite element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182 (1987), 23-45. | DOI | Zbl

[10] D. S. Malkus: Eigenproblems associated with the discrete LBB condition for incompressible finite elements. Int. J. for Eng. Sci. 19(1981), 1299-1310. | MR | Zbl

[11] G. Peters, J. H. Wilkinson: Inverse iteration, illconditioned equations and Newton's method. SIAM Rev. 21 (1979), 339-360. | DOI | MR

[12] G. W. Stewart: Simultaneous iteration for computing invariant subspaces of nonhermitian matrices. Numer. Math. 25 (1976), 123-136. | DOI | MR

[13] J. H. Wilkinson: The algebraic eigenvalue problem. Claredon Press, Oxford, 1965. | MR | Zbl

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