This work is devoted to computations of deflating subspaces associated with separated groups of finite eigenvalues near specified shifts of large regular matrix pencils. The proposed method is a combination of inexact inverse subspace iteration and Newton’s method. The first one is slow but reliably convergent starting with almost an arbitrary initial subspace and it is used as a preprocessing to obtain a good initial guess for the second method which is fast but only locally convergent. The Newton method necessitates at each iteration the solution of a generalized Sylvester equation and for this task an iterative algorithm based on the preconditioned GMRES method is devised. Numerical properties of the proposed combination are illustrated with a typical hydrodynamic stability problem.
A laminar flow of a viscous incompressible fluid with a constant pressure gradient (Poiseuille flow) is considered in a rectangular duct for different values of cross-sectional aspect ratio. A new method, significantly more efficient than the known ones, is proposed and justified for computing the critical Reynolds number of such a flow. The dependence of the critical Reynolds number on the cross-sectional aspect ratio is numerically studied. A theoretical justification of the obtained dependence is proposed.Analysis of Poiseuille flow 127 contrast to the Poiseuille flow with zero boundary conditions on the lateral walls, this flow admits nonzero perturbations constant in the cross direction. Excluding such perturbations, we obtain an estimate of the value A c for the original flow based on Squire's theorem and the properties of the neutral curve of the Orr-Sommerfeld problem for a plane Poiseuille flow.
This work is devoted to computations of invariant pairs associated with separated groups of finite eigenvalues of large regular non-linear matrix pencils. It is proposed to combine the method of successive linear problems with a Newton-type method designed for partial linear eigenproblems and a deflation procedure. This combination is illustrated with a typical hydrodynamic spatial stability problem.
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