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IMA Journal of Numerical Analysis Advance Access originally published online on December 14, 2007
IMA Journal of Numerical Analysis 2008 28(3):580-597; doi:10.1093/imanum/drm037
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Interpolatory quadrature rules for Hadamard finite-part integrals and their superconvergence

Weiwei Sun{dagger}

Department of Mathematics, City University of Hong Kong, Hong Kong

Jiming Wu{ddagger}

Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, People's Republic of China

{dagger} Email: maweiw{at}cpmaspc07.cityu.edu.hk

{ddagger} Email: wu_jiming{at}iapcm.ac.cn

Received on 31 August 2006. Accepted for publication 10 August 2007.


  Abstract

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

Key Words: finite-part integral; interpolatory quadrature rule; superconvergence


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