Efficient calculation and asymptotic expansions of many different oscillatory infinite integrals
Efficient calculation and asymptotic expansions of many different oscillatory infinite integrals
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DOI：
10.1016/j.amc.2018.09.066
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发表时间：
2019-04
期刊：
影响因子：
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通讯作者：
Hongchao Kang
中科院分区：
文献类型：
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作者：
Hongchao Kang
本文介绍并分析了几个高振荡无穷积分的求积法则和渐近展开式。我们首先导出一系列有用的频率参数ω的逆幂的渐近展开式，阐明了这些积分的大ω行为。然后，基于所得的渐近展开式，给出了两种不同的插值求积规则。一种是基于等距节点处被积函数的非振荡和非奇异部分的标准Hermite插值的所谓Filon型方法。另一种是Filon-Clenshaw-Curtis型方法（FCC），该方法在N+ 1个Clenshaw-Curtis点处使用特殊的Hermite插值并快速计算修正矩。FCC方法中所需的插值系数可以通过基于快速傅里叶变换（FFT）的O（NlogN）运算的数值稳定算法来计算。所需的修正矩，可以准确和有效地计算出一些递推关系式。此外，对于这些正交规则，其误差分析的频率ω，的倒数幂。所提出的方法共享的有利性质，即对于固定的N，随着ω的增加，精度大大提高。数值算例表明了所提方法的准确性和有效性。
This paper introduces and analyzes quadrature rules and asymptotic expansions of a few highly oscillatory infinite integrals. We first derive a series of useful asymptotic expansions in inverse powers of the frequency parameter ω, which clarify the large ω behavior of these integrals. Then, based on the resulting asymptotic expansions, two different interpolatory quadrature rules are given. One is the so-called Filon-type methods based on standard Hermite interpolation of the non-oscillatory and non-singular part of the integrands at equidistant nodes. The other is the Filon–Clenshaw–Curtis-type method (FCC) by using special Hermite interpolation at N+ 1 Clenshaw–Curtis points and the fast computation of modified moments. The interpolation coefficients needed in the FCC method, can be computed by a numerically stable algorithm in O (Nlog N) operations based on fast Fourier transform (FFT). The required modified moments, can be accurately and efficiently calculated by some recurrence relation formulae. Moreover, for these quadrature rules, their error analyses in inverse powers of the frequency ω, are provided. The presented methods share the advantageous property that the accuracy improves greatly, for fixed N, as ω increases. Numerical examples show the accuracy and efficiency of the proposed methods.