Efficient quadrature of highly oscillatory integrals with algebraic singularities

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In this paper we are concerned with the numerical evaluation of a class of highly oscillatory integrals containing algebraic singularities. First, we expand such integrals derived by two transformations t=x−β,β>0,t=21+z,−1≤z≤1, into asymptotic series in inverse powers of the frequency ω. Then, based the asymptotic series, two methods are presented. One is the Filon-type method. The other is the Clenshaw–Curtis–Filon-type method which is based on a special Hermite interpolation polynomial in the Clenshaw–Curtis points and can be evaluated efficiently in O(NlogN) operations, where N+1 is the number of Clenshaw–Curtis points in the interval of integration. Some error and convergence analysis and robust numerical examples are used to demonstrate the accuracy and effectiveness of the proposed approaches for approximating the class of highly oscillatory singular integrals.

论文关键词:65D32,65D30,Oscillatory integrals,Asymptotic expansion,Filon-type method,FFT,Clenshaw–Curtis–Filon-type method

论文评审过程:Received 7 January 2012, Revised 23 May 2012, Available online 23 June 2012.

论文官网地址:https://doi.org/10.1016/j.cam.2012.06.030