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

摘要

We study the general (composite) Newton-Cotes rules for the computation of Hadamard finite-part integral on a circle with the hypersingular kernel sin^sup -2^(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 Φ^sup k^(τ) and prove the existence of the superconvergence points. The relation between Φ^sup k^(τ) and S^sup k^(τ) 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.