Clenshaw-Curtis-Filon-type methods for highly oscillatory Bessel transforms and applications

Xiang, Shuhuang; Cho, Yeol Je; Wang, Haiyong; Brunner, Hermann

doi:10.1093/imanum/drq035

Cited by 98 publications

(29 citation statements)

References 32 publications

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“…. , s, and (1 + cos( kπ N )) for k = 0, 1, · · · , N, and Q N +2s (f ) can be expressed as Here Q N +2s (f ) can be fast calculated based on FFT [27]. The high-order modified FCCs for Eqs.…”

Section: Final Remarksmentioning

confidence: 99%

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

Xiang

Cho

2014

Adv Comput Math

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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.

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Section: Final Remarksmentioning

confidence: 99%

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

Xiang

Cho

2014

Adv Comput Math

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“…We refer to the recent monograph [9] for a comprehensive survey. In recent years, there has been a growing interest in studying numerical quadrature of the finite Hankel transform (1.1); see, for instance, [6,7,18,22,23,27,30,31,32,34]. All these works focus on the simplest oscillator g(x) = x and asymptotic analysis has been carried out either by using the following differential equation [22,32] w ′ (x) = A(ω, x)w(x),…”

Section: Introductionmentioning

confidence: 99%

A unified framework for asymptotic analysis and computation of finite Hankel transform

Wang

2020

Journal of Mathematical Analysis and Applications

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In this paper we present a unified framework for asymptotic analysis and computation of the finite Hankel transform. This framework enables us to derive asymptotic expansions of the transform, including the cases where the oscillator has zeros and stationary points. As a consequence, two efficient and affordable methods for computing the transform numerically are developed and a detailed analysis of their asymptotic error estimate is carried out. Numerical examples are provided to confirm our analysis.

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“…Both of theoretical and numerical results manifested that this method enjoyed high-order convergence rates with respect to the frequency ( [7]). To get stable and fast algorithms, Domínguez,et al ([8]), and Xiang, et al ( [9]), proposed the Clenshaw-Curtis-Filon-type method, respectively, which enjoyed extensive applications at present. Although Filon's methodology leads to many efficient algorithms, most of them suffer to complicate computation of moment integrals.…”

Section: Introductionmentioning

confidence: 99%

On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels

Liu

2018

Symmetry

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Lubich's convolution quadrature rule provides efficient approximations to integrals with special kernels. Particularly, when it is applied to computing highly oscillatory integrals, numerical tests show it does not suffer from fast oscillation. This paper is devoted to studying the convergence property of the convolution quadrature rule for highly oscillatory problems. With the help of operational calculus, the convergence rate of the convolution quadrature rule with respect to the frequency is derived. Furthermore, its application to highly oscillatory integral equations is also investigated. Numerical results are presented to verify the effectiveness of the convolution quadrature rule in solving highly oscillatory problems. It is found from theoretical and numerical results that the convolution quadrature rule for solving highly oscillatory problems is efficient and high-potential.

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Clenshaw-Curtis-Filon-type methods for highly oscillatory Bessel transforms and applications

Cited by 98 publications

References 32 publications

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

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

A unified framework for asymptotic analysis and computation of finite Hankel transform

On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels

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