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The superconvergence of composite Newton–Cotes rules for Hadamard finite-part integral on a circle

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Abstract

We study the general (composite) Newton–Cotes rules for the computation of Hadamard finite-part integral on a circle with the hypersingular kernel \({\sin^{-2}\frac{x-s}2 }\) and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton–Cotes rules occurs at the zeros of a special function \({\Phi_k(\tau)}\) and prove the existence of the superconvergence points. The relation between \({\Phi_k(\tau)}\) and \({\mathcal{S}_k(\tau)}\) defined in Wu and Sun (Numer Math 109:143–165, 2008) is established, and the efficient calculation of Cotes coefficients is also discussed. Several numerical examples are provided to validate the theoretical analysis.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, 430072, Wuhan, China

    ** Zhang

  2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, 100088, Bei**g, People’s Republic of China

    Jiming Wu

  3. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, CAS, 100190, Bei**g, People’s Republic of China

    ** Zhang & Dehao Yu

Authors
  1. ** Zhang
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  2. Jiming Wu
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  3. Dehao Yu
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Correspondence to ** Zhang.

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Communicated by W. Hackbusch.

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Zhang, X., Wu, J. & Yu, D. The superconvergence of composite Newton–Cotes rules for Hadamard finite-part integral on a circle. Computing 85, 219–244 (2009). https://doi.org/10.1007/s00607-009-0048-5

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  • DOI: https://doi.org/10.1007/s00607-009-0048-5

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