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Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics

Part of: Numerical analysis: Ordinary differential equations Hydrodynamic stability Error analysis and interval analysis

Published online by Cambridge University Press:  01 March 2010

B. Malcolm Brown ,
Matthias Langer ,
Marco Marletta ,
Christiane Tretter  and
Markus Wagenhofer
B. Malcolm Brown
Affiliation:
School of Computer Science, Cardiff University, 5 The Parade, Cardiff CF24 3AA, United Kingdom (email: malcolm.brown@cs.cf.ac.uk)
Matthias Langer
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom (email: m.langer@strath.ac.uk)
Marco Marletta
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, United Kingdom (email: marco.marletta@cs.cf.ac.uk)
Christiane Tretter
Affiliation:
Mathematisches Institut, Universität Bern,, Sidlerstrasse 5, 3012 Bern, Switzerland (email: tretter@math.unibe.ch)
Markus Wagenhofer
Affiliation:
Psylock GmbH, Regerstrasse 4, 93053 Regensburg, Germany (email: markus.wagenhofer@psylock.com)

Abstract

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In this paper we present computer-assisted proofs of a number of results in theoretical fluid dynamics and in quantum mechanics. An algorithm based on interval arithmetic yields provably correct eigenvalue enclosures and exclosures for non-self-adjoint boundary eigenvalue problems, the eigenvalues of which are highly sensitive to perturbations. We apply the algorithm to: the Orr–Sommerfeld equation with Poiseuille profile to prove the existence of an eigenvalue in the classically unstable region for Reynolds number R=5772.221818; the Orr–Sommerfeld equation with Couette profile to prove upper bounds for the imaginary parts of all eigenvalues for fixed R and wave number α; the problem of natural oscillations of an incompressible inviscid fluid in the neighbourhood of an elliptical flow to obtain information about the unstable part of the spectrum off the imaginary axis; Squire’s problem from hydrodynamics; and resonances of one-dimensional Schrödinger operators.

MSC classification

Secondary: 65L15: Eigenvalue problems 65G30: Interval and finite arithmetic 76E99: None of the above, but in this section
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 65 - 81
Copyright
Copyright © London Mathematical Society 2010

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