On error bounds of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals
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On error bounds of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals Shuhuang Xiang · Guo He · Yeol Je Cho
Received: 17 February 2014 / Accepted: 29 August 2014 © Springer Science+Business Media New York 2014
Abstract In this paper, we aim to derive some error bounds for Filon-ClenshawCurtis quadrature for highly oscillatory integrals. Thanks to the asymptotics of the coefficients in the Chebyshev series expansions of analytic functions or functions of limited regularities, these bounds are established by the aliasings of Fourier transforms on Chebyshev polynomials together with van der Corput-type lemmas. These errors share the property that the errors decrease with the increase of the frequency ω. Moreover, for fixed ω, the order of the error bound related to the number of interpolation nodes N is attainable, while for fixed N , the order of the error on ω is attainable too, which is verified by some functions of limited regularities. In particular, if the functions are analytic in Bernstein ellipses, then the errors decay exponentially. Furthermore, for large values of ω, the accuracy can be further improved by applying a special Hermite interpolants in the Filon-Clenshaw-Curtis quadrature, which can be efficiently evaluated by the Fast Fourier Transform (FFT) techniques. Keywords Filon-Clenshaw-Curtis quadrature · Oscillatory integral · Error bound Mathematics Subject Classifications (2010) 65D32 · 65D30
Communicated by: A. Iserles This paper was supported partly by NSF of China (No.11371376) and partly by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (KRF-2013053358). S. Xiang () · G. He () Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 410083, China e-mail: [email protected] e-mail: [email protected] Y. J. Cho Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju, Korea
S. Xiang et al.
1 Introduction b The computation of a u(x)eiωg(x)dx occurs in a wide range of practical problems and applications such as nonlinear optics, fluid dynamics, computerized tomography, celestial mechanics, electromagnetics, acoustic scattering, etc. (see [4, ChandlerWilde et al.] and [5, Colton and Kress], etc.). By an inverse transformation, the integral can be transferred into the form of 1 Iω (f ) = f (x)eiωx dx, (1.1) −1
where f (x) could be algebraically or logarithmically singular (see [3, Bruno and Haslam], [7, 8, Dom´ınguez et al.], [9, Dom´ınguez et al.] and [19, Spence et al.]). Without loss of generality, we assume ω > 0 in this paper. In the case that f is suitably smooth, Dom´ınguez, Graham and Smyshlyaev [8] proposed an efficient and stable implementation of a Filon-like approach— Filon-Clenshaw-Curtis quadrature (FCC) Iω,N (f ), in which f is replaced by its interpolating polynomial QN (f )(x) of degree N at the Clenshaw-Curtis points tj,N = cos(j π/N ) for each j = 0, 1, . . . , N , 1 Iω,N (f ) = QN (f
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