On the numerical quadrature of highly-oscillating integrals I: Fourier transforms
- 1 July 2004
- journal article
- research article
- Published by Oxford University Press (OUP) in IMA Journal of Numerical Analysis
- Vol. 24 (3) , 365-391
- https://doi.org/10.1093/imanum/24.3.365
Abstract
Highly‐oscillatory integrals are allegedly difficult to calculate. The main assertion of this paper is that that impression is incorrect. As long as appropriate quadrature methods are used, their accuracy increases when oscillation becomes faster and suitable choice of quadrature points renders this welcome phenomenon more pronounced. We focus our analysis on Filon‐type quadrature and analyse its behaviour in a range of frequency regimes for integrals of the form ∫0hf(x)ei ω xw(x)d x, where h>0 is small and |ω | large. Our analysis is applied to modified Magnus methods for highly‐oscillatory ordinary differential equations.Keywords
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