On highly oscillatory problems arising in electronic engineering
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 4, p. 785-804

In this paper, we consider linear ordinary differential equations originating in electronic engineering, which exhibit exceedingly rapid oscillation. Moreover, the oscillation model is completely different from the familiar framework of asymptotic analysis of highly oscillatory integrals. Using a Bessel-function identity, we expand the oscillator into asymptotic series, and this allows us to extend Filon-type approach to this setting. The outcome is a time-stepping method that guarantees high accuracy regardless of the rate of oscillation.

DOI : https://doi.org/10.1051/m2an/2009024
Classification:  65L05,  65T99
Keywords: high oscillation, quadrature, ordinary differential equations
@article{M2AN_2009__43_4_785_0,
     author = {Condon, Marissa and Dea\~no, Alfredo and Iserles, Arieh},
     title = {On highly oscillatory problems arising in electronic engineering},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {4},
     year = {2009},
     pages = {785-804},
     doi = {10.1051/m2an/2009024},
     zbl = {1172.78009},
     mrnumber = {2542877},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2009__43_4_785_0}
}
Condon, Marissa; Deaño, Alfredo; Iserles, Arieh. On highly oscillatory problems arising in electronic engineering. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 4, pp. 785-804. doi : 10.1051/m2an/2009024. http://www.numdam.org/item/M2AN_2009__43_4_785_0/

[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC, (1964). | MR 167642

[2] D. Cohen, T. Jahnke, K. Lorenz and C. Lubich, Numerical integrators for highly oscillatory Hamiltonian systems: a review, in Analysis, Modeling and Simulation of Multiscale Problems, A. Mielke Ed., Springer-Verlag (2006) 553-576. | MR 2275175

[3] E. Dautbegovic, M. Condon and C. Brennan, An efficient nonlinear circuit simulation technique. IEEE Trans. Microwave Theory Tech. 53 (2005) 548-555.

[4] P.J. Davis and P. Rabinowitz, Methods of Numerical Integration. Second Edition, Academic Press, Orlando, USA (1984). | MR 760629 | Zbl 0537.65020

[5] V. Grimm and M. Hochbruck, Error analysis of exponential integrators for oscillatory second-order differential equations. J. Phys. A: Math. Gen. 39 (2006) 5495-5507. | MR 2220772 | Zbl 1093.65078

[6] S. Haykin, Communications Systems. Fourth Edition, John Wiley, New York, USA (2001).

[7] D. Huybrechs and S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44 (2006) 1026-1048. | MR 2231854 | Zbl 1123.65017

[8] A. Iserles, On the global error of discretization methods for highly-oscillatory ordinary differential equations. BIT 42 (2002a) 561-599. | MR 1931887 | Zbl 1027.65107

[9] A. Iserles, Think globally, act locally: solving highly-oscillatory ordinary differential equations. Appl. Num. Anal. 43 (2002b) 145-160. | MR 1936107 | Zbl 1016.65050

[10] A. Iserles and S.P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation. BIT 44 (2004) 755-772. | MR 2211043 | Zbl 1076.65025

[11] A. Iserles and S.P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives. Proc. Royal Soc. A 461 (2005) 1383-1399. | MR 2147752 | Zbl 1145.65309

[12] A. Iserles and S.P. Nørsett, From high oscillation to rapid approximation I: Modified Fourier expansions. IMA J. Num. Anal. 28 (2008) 862-887. | MR 2457350 | Zbl pre05377087

[13] M.C. Jeruchim, P. Balaban and K.S. Shanmugan, Simulation of Communication Systems, Modeling, Methodology and Techniques. Second Edition, Kluwer Academic/Plenum Publishers, New York, USA (2000).

[14] M. Khanamirian, Quadrature methods for systems of highly oscillatory ODEs. Part I. BIT 48 (2008) 743-761. | MR 2465701 | Zbl 1167.65040

[15] C.A. Micchelli and T.J. Rivlin, Quadrature formulæ and Hermite-Birkhoff interpolation. Adv. Maths 11 (1973) 93-112. | MR 318743 | Zbl 0259.41016

[16] S. Olver, Moment-free numerical integration of highly oscillatory functions. IMA J. Num. Anal. 26 (2006) 213-227. | MR 2218631 | Zbl 1106.65021

[17] R. Pulch, Multi-time scale differential equations for simulating frequency modulated signals. Appl. Numer. Math. 53 (2005) 421-436. | MR 2128535 | Zbl 1069.65103

[18] J. Roychowdhury, Analysing circuits with widely separated time scales using numerical PDE methods. IEEE Trans. Circuits Sys. I, Fund. Theory Appl. 48 (2001) 578-594. | MR 1854953 | Zbl 1001.94060

[19] C.J. Weisman, The Essential Guide to RF and Wireless. Second Edition, Prentice-Hall, Englewood Cliffs, USA (2002).

[20] R. Wong, Asymptotic Approximations of Integrals. SIAM, Philadelphia (2001). | MR 1851050 | Zbl 1078.41001