# Stability and error estimates for Filon-Clenshaw-Curtis rules for highly-oscillatory integrals

Victor Dominguez, Ivan Graham, Valery Smyshlyaev

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59 Citations (Scopus)

### Abstract

In this paper we obtain new results on Filon-type methods for computing oscillatory integrals of the form $\int_{-1}^1 f(s) \exp({\rm i}ks) \ {\rm d}s$. We use a Filon approach based on interpolating $f$ at the classical Clenshaw-Curtis points $\cos(j\pi/N), \ j = 0, \ldots , N$. The rule may be implemented in $\mathcal{O}(N \log N)$ operations. We prove error estimates which show explicitly how the error depends both on the parameters $k$ and $N$ and on the Sobolev regularity of $f$. In particular we identify the regularity of $f$ required to ensure the maximum rate of decay of the error as $k \rightarrow \infty$. We also describe a method for implementating the method and prove its stability both when $N \leq k$ and $N>k$. Numerical experiments illustrate both the stability of the algorithm and the sharpness of the error estimates.
Original language English 1253-1280 28 IMA Journal of Numerical Analysis 31 4 https://doi.org/10.1093/imanum/drq036 Published - 2011

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Oscillatory Integrals
Stability Estimates
Error Estimates
Regularity
Sharpness
Pi
Numerical Experiment
Decay
Computing
Experiments

### Cite this

Stability and error estimates for Filon-Clenshaw-Curtis rules for highly-oscillatory integrals. / Dominguez, Victor; Graham, Ivan; Smyshlyaev, Valery.

In: IMA Journal of Numerical Analysis, Vol. 31, No. 4, 2011, p. 1253-1280.

Research output: Contribution to journalArticle

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AB - In this paper we obtain new results on Filon-type methods for computing oscillatory integrals of the form $\int_{-1}^1 f(s) \exp({\rm i}ks) \ {\rm d}s$. We use a Filon approach based on interpolating $f$ at the classical Clenshaw-Curtis points $\cos(j\pi/N), \ j = 0, \ldots , N$. The rule may be implemented in $\mathcal{O}(N \log N)$ operations. We prove error estimates which show explicitly how the error depends both on the parameters $k$ and $N$ and on the Sobolev regularity of $f$. In particular we identify the regularity of $f$ required to ensure the maximum rate of decay of the error as $k \rightarrow \infty$. We also describe a method for implementating the method and prove its stability both when $N \leq k$ and $N>k$. Numerical experiments illustrate both the stability of the algorithm and the sharpness of the error estimates.

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