### Abstract

Language | English |
---|---|

Pages | 1253-1280 |

Number of pages | 28 |

Journal | IMA Journal of Numerical Analysis |

Volume | 31 |

Issue number | 4 |

DOIs | |

Status | Published - 2011 |

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### Cite this

*IMA Journal of Numerical Analysis*,

*31*(4), 1253-1280. https://doi.org/10.1093/imanum/drq036

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

Research output: Contribution to journal › Article

*IMA Journal of Numerical Analysis*, vol. 31, no. 4, pp. 1253-1280. https://doi.org/10.1093/imanum/drq036

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TY - JOUR

T1 - Stability and error estimates for Filon-Clenshaw-Curtis rules for highly-oscillatory integrals

AU - Dominguez, Victor

AU - Graham, Ivan

AU - Smyshlyaev, Valery

PY - 2011

Y1 - 2011

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=80054705865&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1093/imanum/drq036

U2 - 10.1093/imanum/drq036

DO - 10.1093/imanum/drq036

M3 - Article

VL - 31

SP - 1253

EP - 1280

JO - IMA Journal of Numerical Analysis

T2 - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

SN - 0272-4979

IS - 4

ER -