Asymptotic expansion and quadrature of composite highly oscillatory integrals
HTML articles powered by AMS MathViewer
- by Arieh Iserles and David Levin PDF
- Math. Comp. 80 (2011), 279-296 Request permission
Abstract:
We consider in this paper asymptotic and numerical aspects of highly oscillatory integrals of the form $\int _a^b f(x) g(\sin [\omega \theta (x)])\mathrm {d} x$, where $\omega \gg 1$. Such integrals occur in the simulation of electronic circuits, but they are also of independent mathematical interest.
The integral is expanded in asymptotic series in inverse powers of $\omega$. This expansion clarifies the behaviour for large $\omega$ and also provides a powerful means to design effective computational algorithms. In particular, we introduce and analyse Filon-type methods for this integral.
References
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
- Marissa Condon, Alfredo Deaño, Arieh Iserles, Kornel Maczyński, and Tao Xu, On numerical methods for highly oscillatory problems in circuit simulation, COMPEL 28 (2009), no. 6, 1607–1618. MR 2597311, DOI 10.1108/03321640910999897
- Marissa Condon, Alfredo Deaño, and Arieh Iserles, On highly oscillatory problems arising in electronic engineering, M2AN Math. Model. Numer. Anal. 43 (2009), no. 4, 785–804. MR 2542877, DOI 10.1051/m2an/2009024
- E. Dautbegovic, M. Condon, and C. Brennan, An efficient nonlinear circuit simulation technique, IEEE Trans. Microwave Theory & Techniques 53 (2005), 548–555.
- 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
- 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
- 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
- Arieh Iserles and Syvert P. Nørsett, From high oscillation to rapid approximation. I. Modified Fourier expansions, IMA J. Numer. Anal. 28 (2008), no. 4, 862–887. MR 2457350, DOI 10.1093/imanum/drn006
- 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
- F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
- 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
- —, Numerical approximation of highly oscillatory integrals, Ph.D. thesis, DAMTP, University of Cambridge, 2008.
- Earl D. Rainville, Special functions, The Macmillan Company, New York, 1960. MR 0107725
- 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
- R. Wong, Asymptotic approximations of integrals, Classics in Applied Mathematics, vol. 34, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Corrected reprint of the 1989 original. MR 1851050, DOI 10.1137/1.9780898719260
Additional Information
- Arieh Iserles
- Affiliation: Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
- Email: ai@damtp.cam.ac.uk
- David Levin
- Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
- Email: levin@tau.ac.il
- Received by editor(s): October 30, 2008
- Received by editor(s) in revised form: August 21, 2009
- Published electronically: June 7, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 80 (2011), 279-296
- MSC (2010): Primary 65D30; Secondary 41A55
- DOI: https://doi.org/10.1090/S0025-5718-2010-02386-5
- MathSciNet review: 2728980