Abstract
A numerical steepest descent method, based on the Laguerre quadrature rule, is developed for integration of one-dimensional highly oscillatory functions on [0, ∞ ) of a general class. It is shown that if the integrand is analytic, then in the absence of stationary points, the method is rapidly convergent. The method is extended to the case when there are a finite number of stationary points in [0, ∞ ). It can be further extended to the case when the integrand is only smooth to some degree (not necessarily analytic). We illustrate the theoretical results using some numerical experiences.
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References
Ablowitz, M.J., Fokas, A.S.: Complex Variables: Introduction and Applications. Cambridge University Press, Cambridge, UK (1997)
Bleistein, N., Handelsman, R.: Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, New York (1975)
Brown, J.W., Churchill, R.V.: Complex Variables and Applications, 6th edn. McGraw-Hill, New York (1996)
Chandler-Wilde, S.N., Hothersall, D.C.: Efficient calculation of the green function for acoustic propagation above a homogeneous impedance plane. J. Sound Vib. 180, 705–724 (1995)
Davies, K., Strayer, M., White, G.: Complex-plane methods for evaluating highly oscillatory integrals in nuclear physics I. J. Phys. G: Nucl. Phys. 14, 961–972 (1988)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, New York (1984)
Deaño, A., Huybrechs, D.: Complex Gaussian quadrature of oscillatory integrals. Numer. Math. 112, 197–219 (2009)
Filon, L.N.G.: On a quadrature formula for trigonometric integrals. Proc. R. Soc. Edinb. 49, 38–47 (1928)
Harris, P.J.: On methods for evaluating a class of highly oscillatory integrals. In: Proceedings of the 8th UK Conference on Boundary Integral Methods. University of Leeds, UK (2011)
Huybrechs, D., Olver, S.: Highly oscillatory quadrature. In: Engquist, B., Fokas, T., Hairer, E., Iserles, A. (eds.) Highly Oscillatory Problems, pp. 25–50. Cambridge University Press (2009)
Huybrechs, D., Vandewalle, S.: On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44, 1026–1048 (2006)
Iserles, A., Nørsett, S.P.: Efficient quadrature of highly oscillatory integrals using derivatives. Proc. R. Soc. Lond. A 461, 1383–1399 (2005)
Levin, D.: Procedure for computing one- and two-dimensional integrals of functions with rapid irregular oscillations. Math. Comput. 38, 531–538 (1982)
Levin, D.: Analysis of a collocation method for integrating rapidly oscillatory functions. J. Comput. Appl. Math. 78, 131–138 (1997)
Luke, Y.L.: On the computation of oscillatory integrals. Proc. Camb. Philos. Soc. 50, 269–277 (1954)
Milovanović, G.V.: Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures. Comput. Math. Appl. 36, 19–39 (1998)
Olver, S.: Moment-free numerical integration of highly oscillatory functions. IMA J. Numer. Anal. 26, 213–227 (2006)
Olver, S.: Moment-free numerical approximation of highly oscillatory integrals with stationary points. Eur. J. Appl. Math. 18, 435–447 (2007)
Olver, S.: Numerical approximation of highly oscillatory integrals. Ph.D. thesis, University of Cambridge (2008)
Sauter, T.: Integration of highly oscillatory functions. Comput. Phys. Commun. 125, 119–126 (2000)
Sidi, A.: A user-friendly extrapolation method for oscillatory infinite integrals. Math. Comput. 51, 249–266 (1988)
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Majidian, H. Numerical approximation of highly oscillatory integrals on semi-finite intervals by steepest descent method. Numer Algor 63, 537–548 (2013). https://doi.org/10.1007/s11075-012-9639-x
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DOI: https://doi.org/10.1007/s11075-012-9639-x