Efficient computation of highly oscillatory integrals with weak singularities by Gauss-type method

In this paper, we consider the numerical computation of Hankel transform with weak singularities at the endpoints. By using the analytic continuation, we transform the integral into two line integrals in complex plane, which can be efficiently evaluated by some proper Gauss quadrature rules. The asymptotic orders of the errors on k and ω are derived. These asymptotic orders are demonstrated by several numerical experiments either for fixed k or for fixed ω, which show that the efficiency and accuracy of this method significantly improve as the frequencies increase.

[1]  Ivan G. Graham,et al.  A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering , 2007, Numerische Mathematik.

[2]  Ivan G. Graham,et al.  Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals , 2011 .

[3]  Thorsten Gerber,et al.  Handbook Of Mathematical Functions , 2016 .

[4]  Vladimir Rokhlin,et al.  A Fast Algorithm for the Calculation of the Roots of Special Functions , 2007, SIAM J. Sci. Comput..

[5]  S. Xiang,et al.  On the Levin iterative method for oscillatory integrals , 2008 .

[6]  Daan Huybrechs,et al.  Complex Gaussian quadrature for oscillatory integral transforms , 2013 .

[7]  Nelson H. F. Beebe,et al.  Efficient Gauss-related quadrature for two classes of logarithmic weight functions , 2007, TOMS.

[8]  Roderick Wong,et al.  Quadrature formulas for oscillatory integral transforms , 1982 .

[9]  Congpei An,et al.  On evaluation of Bessel transforms with oscillatory and algebraic singular integrands , 2014, J. Comput. Appl. Math..

[10]  Daan Huybrechs,et al.  A Sparse Discretization for Integral Equation Formulations of High Frequency Scattering Problems , 2007, SIAM J. Sci. Comput..

[11]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[12]  David Levin,et al.  Fast integration of rapidly oscillatory functions , 1996 .

[13]  Gene H. Golub,et al.  Calculation of Gauss quadrature rules , 1967, Milestones in Matrix Computation.

[14]  David Levin,et al.  Analysis of a collocation method for integrating rapidly oscillatory functions , 1997 .

[15]  Yanfei Gao,et al.  AN APPROXIMATE FORMULATION OF THE EFFECTIVE INDENTATION MODULUS OF ELASTICALLY ANISOTROPIC FILM-ON-SUBSTRATE SYSTEMS , 2009 .

[16]  W. Gautschi.  Orthogonal Polynomials: Computation and Approximation , 2004 .

[17]  A. Iserles,et al.  Efficient quadrature of highly oscillatory integrals using derivatives , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[18]  Walter Gautschi.  Gauss quadrature routines for two classes of logarithmic weight functions , 2010, Numerical Algorithms.

[19]  Stephen Langdon,et al.  Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering* , 2012, Acta Numerica.

[20]  Arieh Iserles,et al.  Quadrature methods for multivariate highly oscillatory integrals using derivatives , 2006, Math. Comput..

[21]  Víctor Domínguez,et al.  Filon-Clenshaw-Curtis rules for a class of highly-oscillatory integrals with logarithmic singularities , 2013, J. Comput. Appl. Math..

[22]  Siraj-ul-Islam,et al.  Meshless and wavelets based complex quadrature of highly oscillatory integrals and the integrals with stationary points , 2013 .

[23]  Siraj-ul-Islam,et al.  New quadrature rules for highly oscillatory integrals with stationary points , 2015, J. Comput. Appl. Math..

[24]  Daan Huybrechs,et al.  On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation , 2006, SIAM J. Numer. Anal..

[25]  Shuhuang Xiang,et al.  On the Filon and Levin methods for highly oscillatory integral , 2007 .

[26]  Simon N. Chandler-Wilde,et al.  A collocation method for high-frequency scattering by convex polygons , 2007 .

[27]  Stephen Langdon,et al.  A Galerkin Boundary Element Method for High Frequency Scattering by Convex Polygons , 2007, SIAM J. Numer. Anal..

[28]  T. Kim,et al.  Filon-Clenshaw-Curtis Rules for Highly Oscillatory Integrals with Algebraic Singularities and Stationary Points , 2012, SIAM J. Numer. Anal..

[29]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[30]  Robert Piessens,et al.  On the computation of Fourier transforms of singular functions , 1992 .

[31]  Shuhuang Xiang,et al.  Computation of integrals with oscillatory and singular integrands using Chebyshev expansions , 2013, J. Comput. Appl. Math..

[32]  Shuhuang Xiang,et al.  Clenshaw–Curtis–Filon-type methods for highly oscillatory Bessel transforms and applications , 2011 .

[33]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[34]  Shuhuang Xiang,et al.  On the evaluation of Cauchy principal value integrals of oscillatory functions , 2010, J. Comput. Appl. Math..

[35]  Shuhuang Xiang,et al.  On error bounds of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals , 2015, Adv. Comput. Math..

[36]  David Levin,et al.  Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations , 1982 .

[37]  Siraj-ul-Islam,et al.  Numerical integration of multi-dimensional highly oscillatory, gentle oscillatory and non-oscillatory integrands based on wavelets and radial basis functions , 2012 .

[38]  Weiwei Sun,et al.  A Fast Algorithm for the Electromagnetic Scattering from a Large Cavity , 2005, SIAM J. Sci. Comput..

[39]  Daan Huybrechs,et al.  Complex Gaussian quadrature of oscillatory integrals , 2009, Numerische Mathematik.