On error bounds of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals

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.

Levin methods for highly oscillatory integrals with singularities

“…where ξ 1 , ξ 2 ∈ [0, a] depending on the value of x. By the van der Corput-type lemma in [34], there exists a constant C independent of w and n such that…”

A Meshless and Parallelizable Method for Differential Equations with Time-Delay

“…These are highly-oscillatory integrals and can not be computed by usual numerical methods, such as the Gauss-Legendre method; see [12,15,16] for explanations. Here, we study how to use the Filon-Clenshaw-Curtis quadrature [8,9,31] to compute these integrals. For the two integrals in (2.22a), we can directly apply the FCC quadrature, because Y min is only a moderate quantity.…”

“…This error bound implies that for all k, the FCC quadrature is efficient and uniformly convergent for computing the integral [k, F] under the assumption that F possess suitable regularity. Other error bounds which treat the cases that F has algebraic singularities and limited regularity can be found in [9] and [31], respectively. To control the length of the paper, we shall not pursue this here.…”

“…5 on the bottom row. For the purely imaginary case, i.e., an exponential term e ik y instead of e z y is contained in the integrand, such a welcome property is known long in the area of highly-oscillatory quadrature [8,9,15,16,31].…”

“…This issue will be addressed in detail in Section 2. The next step is to discretize the contour integral by the Filon-Clenshaw-Curtis (FCC) quadrature [8,9,31]. This procedure consists of three parts.…”