LMS Journal of Computation and Mathematics

Research Article

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

B. Malcolm Browna1, Matthias Langera2, Marco Marlettaa3, Christiane Trettera4 and Markus Wagenhofera5

a1 School of Computer Science, Cardiff University, 5 The Parade, Cardiff CF24 3AA, United Kingdom (email: malcolm.brown@cs.cf.ac.uk)

a2 Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom (email: m.langer@strath.ac.uk)

a3 School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, United Kingdom (email: marco.marletta@cs.cf.ac.uk)

a4 Mathematisches Institut, Universität Bern,, Sidlerstrasse 5, 3012 Bern, Switzerland (email: tretter@math.unibe.ch)

a5 Psylock GmbH, Regerstrasse 4, 93053 Regensburg, Germany (email: markus.wagenhofer@psylock.com)

Abstract

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.

(Received December 22 2008)

(Revised May 06 2009)

(Online publication March 2010)

2000 Mathematics Subject Classification

  • 65L15;
  • 65G30 (primary);
  • 76E99 (secondary)