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Published in: BIT Numerical Mathematics 2/2015

01-06-2015

On the application of GMRES to oscillatory singular integral equations

Author: Thomas Trogdon

Published in: BIT Numerical Mathematics | Issue 2/2015

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Abstract

We present a new method for the numerical solution of singular integral equations on the real axis. The method’s value stems from a new formula for the Cauchy integral of a rational function with an oscillatory exponential factor. The inner product of such functions is also computed explicitly. With these tools in hand, the GMRES algorithm is applied to both non-oscillatory and oscillatory singular integral equations. In specific cases, ideas from Fredholm theory and Riemann–Hilbert problems are used to motivate preconditioners for these singular integral equations. A significant acceleration in convergence is realized for these examples. This presents a useful link between the theory of singular integral equations and the numerical analysis of such equations. Furthermore, this method presents a first step towards a solver for the inverse scattering transform that does not require the deformation of a Riemann–Hilbert problem.

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Metadata
Title
On the application of GMRES to oscillatory singular integral equations
Author
Thomas Trogdon
Publication date
01-06-2015
Publisher
Springer Netherlands
Published in
BIT Numerical Mathematics / Issue 2/2015
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-014-0502-4

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