Interpolatory quadrature rules for Hadamard finite-part integrals and their superconvergence
Interpolatory quadrature rules for Hadamard finite-part integrals and their superconvergence
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Hadamard 有限部分积分的插值求积规则及其超收敛
DOI：
10.1093/imanum/drm037
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发表时间：
2007-11
影响因子：
2.1
通讯作者：
Wu, Jiming
中科院分区：
文献类型：
--
作者：
Sun, Weiwei;Wu, Jiming
在本文中，我们提出了一个一般的框架内插求积规则的阿达玛有限部分积分与二阶奇异性。高斯求积规则被看作是一个特殊的情况下，许多有趣的功能可以很容易地从框架。从理论上证明了现有的一些用不同方法得到的公式的等价性。我们证明了这些插值求积规则的逐点超收敛性，即当奇点与某些先验已知点重合时，其精度比一般可能的要好。推广了柯西主值积分中一个常用的插值求积法则。本文提出了一种新的高斯型求积法则，用于计算同时包含不同类型奇点的积分。数值例子证实了我们的理论结果。
In this paper, we present a general framework for interpolatory quadrature rules for Hadamard finite-part integrals with a second-order singularity. Gaussian quadrature rules are viewed as a special case and many interesting features can be obtained easily from the framework. We prove theoretically the equivalence of some existing formulas which were obtained in different ways. We show the point-wise superconvergence of these interpolatory quadrature rules, i.e. when the singular point coincides with certain a priori known points, the accuracy is better than what is generally possible. The extension of a popular interpolatory quadrature rule for Cauchy principal value integrals is presented. A new quadrature rule of Gaussian type is proposed for the evaluation of integrals simultaneously involving different types of singularities. Numerical examples confirm our theoretical results.