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Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering*

Published online by Cambridge University Press:  19 April 2012

Simon N. Chandler-Wilde
Affiliation:
Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, UK E-mail: S.N.Chandler-Wilde@reading.ac.uk, S.Langdon@reading.ac.uk
Ivan G. Graham
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK E-mail: I.G.Graham@bath.ac.uk, E.A.Spence@bath.ac.uk
Stephen Langdon
Affiliation:
Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, UK E-mail: S.N.Chandler-Wilde@reading.ac.uk, S.Langdon@reading.ac.uk
Euan A. Spence
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK E-mail: I.G.Graham@bath.ac.uk, E.A.Spence@bath.ac.uk

Abstract

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2012

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References

* Colour online for monochrome figures available at journals.cambridge.org/anu.