Fast integration of highly oscillatory integrals with exotic oscillators

Xiang, Shuhuang; Wang, Haiyong

doi:10.1090/S0025-5718-09-02279-0

Fast integration of highly oscillatory integrals with exotic oscillators
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Math. Comp. 79 (2010), 829-844 Request permission

Abstract:

In this paper, we present an efficient Filon-type method for the integration of systems containing Bessel functions with exotic oscillators based on a diffeomorphism transformation and give applications to Airy transforms. Preliminary numerical results show the effectiveness and accuracy of the quadrature for large arguments of integral systems.

References

Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
K. C. Chung, G. A. Evans, and J. R. Webster, A method to generate generalized quadrature rules for oscillatory integrals, Appl. Numer. Math. 34 (2000), no. 1, 85–93. MR 1755695, DOI 10.1016/S0168-9274(99)00033-1
G. A. Evans and K. C. Chung, Some theoretical aspects of generalised quadrature methods, J. Complexity 19 (2003), no. 3, 272–285. Numerical integration and its complexity (Oberwolfach, 2001). MR 1984114, DOI 10.1016/S0885-064X(03)00004-9
L. N. G. Filon, On a quadrature formula for trigonometric integrals, Proc. Royal Soc. Edinburgh 49 (1928), 38-47.
E. A. Flinn, A modification of Filon’s method of numerical integration, J. Assoc. Comput. Mach. 7 (1960), 181–184. MR 114298, DOI 10.1145/321021.321029
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 5th ed., Academic Press, Inc., Boston, MA, 1994. Translation edited and with a preface by Alan Jeffrey. MR 1243179
Daan Huybrechs and Stefan Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal. 44 (2006), no. 3, 1026–1048. MR 2231854, DOI 10.1137/050636814
Daan Huybrechs and Stefan Vandewalle, A sparse discretization for integral equation formulations of high frequency scattering problems, SIAM J. Sci. Comput. 29 (2007), no. 6, 2305–2328. MR 2357616, DOI 10.1137/060651525
A. Iserles and S. P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT 44 (2004), no. 4, 755–772. MR 2211043, DOI 10.1007/s10543-004-5243-3
Arieh Iserles and Syvert P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2057, 1383–1399. MR 2147752, DOI 10.1098/rspa.2004.1401
A. Iserles, S. P. Nørsett, and S. Olver, Highly oscillatory quadrature: the story so far, Numerical mathematics and advanced applications, Springer, Berlin, 2006, pp. 97–118. MR 2303638, DOI 10.1007/978-3-540-34288-5_{6}
A. Iserles and S. P. Nørsett, Highly oscillatory quadrature and its applications, http://handle.dtic.mil/100. 2/ADA433730, Defense Technical Information Center, 2005.
David Levin, Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations, Math. Comp. 38 (1982), no. 158, 531–538. MR 645668, DOI 10.1090/S0025-5718-1982-0645668-7
David Levin, Fast integration of rapidly oscillatory functions, J. Comput. Appl. Math. 67 (1996), no. 1, 95–101. MR 1388139, DOI 10.1016/0377-0427(94)00118-9
Yudell L. Luke, On the computation of oscillatory integrals, Proc. Cambridge Philos. Soc. 50 (1954), 269–277. MR 62518
Sheehan Olver, Moment-free numerical integration of highly oscillatory functions, IMA J. Numer. Anal. 26 (2006), no. 2, 213–227. MR 2218631, DOI 10.1093/imanum/dri040
S. Olver, Numerical approximation of vector-valued highly oscillatory integrals, BIT 47 (2007), no. 3, 637–655. MR 2338536, DOI 10.1007/s10543-007-0137-9
Sheehan Olver, Moment-free numerical approximation of highly oscillatory integrals with stationary points, European J. Appl. Math. 18 (2007), no. 4, 435–447. MR 2344314, DOI 10.1017/S0956792507007012
R. Piessens, Automatic computation of Bessel function integrals, Comput. Phys. Commun. 25 (1982), 289-295.
Robert Piessens and Maria Branders, Modified Clenshaw-Curtis method for the computation of Bessel function integrals, BIT 23 (1983), no. 3, 370–381. MR 705003, DOI 10.1007/BF01934465
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Lloyd N. Trefethen, Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev. 50 (2008), no. 1, 67–87. MR 2403058, DOI 10.1137/060659831
G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
Shuhuang Xiang, Efficient Filon-type methods for $\int ^b_af(x)e^{i\omega g(x)}dx$, Numer. Math. 105 (2007), no. 4, 633–658. MR 2276763, DOI 10.1007/s00211-006-0051-0
Shuhuang Xiang, Numerical analysis of a fast integration method for highly oscillatory functions, BIT 47 (2007), no. 2, 469–482. MR 2334051, DOI 10.1007/s10543-007-0127-y
Shuhuang Xiang, Weihua Gui, and Pinghua Mo, Numerical quadrature for Bessel transformations, Appl. Numer. Math. 58 (2008), no. 9, 1247–1261. MR 2444255, DOI 10.1016/j.apnum.2007.07.002
Shuhuang Xiang and Weihua Gui, On generalized quadrature rules for fast oscillatory integrals, Appl. Math. Comput. 197 (2008), no. 1, 60–75. MR 2396291, DOI 10.1016/j.amc.2007.07.052