On highly oscillatory problems arising in electronic engineering
ESAIM: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 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 : 10.1051/m2an/2009024
Classification : 65L05, 65T99
Mots-clés : high oscillation, quadrature, ordinary differential equations
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Condon, Marissa; Deaño, Alfredo; Iserles, Arieh. On highly oscillatory problems arising in electronic engineering. ESAIM: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 785-804. doi : 10.1051/m2an/2009024. http://www.numdam.org/articles/10.1051/m2an/2009024/

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

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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  • Zaman, Sakhi; Khan, Latif Ullah; Hussain, Irshad; Mihet-Popa, Lucian Fast Computation of Highly Oscillatory ODE Problems: Applications in High-Frequency Communication Circuits, Symmetry, Volume 14 (2022) no. 1, p. 115 | DOI:10.3390/sym14010115
  • Kalita, Hemanta Why we need Non absolute integral in place of Lebesgue integral?, Dera Natung Government College Research Journal, Volume 6 (2021) no. 1, p. 86 | DOI:10.56405/dngcrj.2021.06.01.09
  • Wu, Y.-K.; Liu, Z.-D.; Zhao, W.-D.; Duan, L.-M. High-fidelity entangling gates in a three-dimensional ion crystal under micromotion, Physical Review A, Volume 103 (2021) no. 2 | DOI:10.1103/physreva.103.022419
  • Pérez-Becerra, T.; Sánchez-Perales, S.; Oliveros-Oliveros, J.J. The HK-Sobolev space and applications to one-dimensional boundary value problems, Journal of King Saud University - Science, Volume 32 (2020) no. 6, p. 2790 | DOI:10.1016/j.jksus.2020.06.016
  • Li, Bin; Xiang, Shuhuang Efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels, Applied Mathematics and Computation, Volume 362 (2019), p. 14 (Id/No 124499) | DOI:10.1016/j.amc.2019.06.013 | Zbl:1433.65034
  • Liu, Zhongli; Tian, Tianhai; Tian, Hongjiong Asymptotic-numerical solvers for highly oscillatory second-order differential equations, Applied Numerical Mathematics, Volume 137 (2019), pp. 184-202 | DOI:10.1016/j.apnum.2018.11.004 | Zbl:1410.34057
  • Spigler, Renato Asymptotic-numerical approximations for highly oscillatory second-order differential equations by the phase function method, Journal of Mathematical Analysis and Applications, Volume 463 (2018) no. 1, pp. 318-344 | DOI:10.1016/j.jmaa.2018.03.027 | Zbl:1404.65059
  • Zhao, Xiaofei Uniformly accurate multiscale time integrators for second order oscillatory differential equations with large initial data, BIT, Volume 57 (2017) no. 3, pp. 649-683 | DOI:10.1007/s10543-017-0646-0 | Zbl:1377.65087
  • Monzón, Lucas; Beylkin, Gregory Efficient representation and accurate evaluation of oscillatory integrals and functions, Discrete and Continuous Dynamical Systems, Volume 36 (2016) no. 8, pp. 4077-4100 | DOI:10.3934/dcds.2016.36.4077 | Zbl:1333.65020
  • Sheng, Qin; Sun, Hai-wei Stability of a modified Peaceman-Rachford method for the paraxial Helmholtz equation on adaptive grids, Journal of Computational Physics, Volume 325 (2016), pp. 259-271 | DOI:10.1016/j.jcp.2016.08.040 | Zbl:1375.78021
  • Wang, Bin; Li, Guolong Bounds on asymptotic-numerical solvers for ordinary differential equations with extrinsic oscillation, Applied Mathematical Modelling, Volume 39 (2015) no. 9, pp. 2528-2538 | DOI:10.1016/j.apm.2014.10.054 | Zbl:1443.65098
  • Bunder, J. E.; Roberts, A. J. Numerical integration of ordinary differential equations with rapidly oscillatory factors, Journal of Computational and Applied Mathematics, Volume 282 (2015), pp. 54-70 | DOI:10.1016/j.cam.2014.12.033 | Zbl:1309.65078
  • Kuehn, Christian Numerical Methods, Multiple Time Scale Dynamics, Volume 191 (2015), p. 295 | DOI:10.1007/978-3-319-12316-5_10
  • Sheng, Qin; Sun, Hai-Wei Exponential splitting for n-dimensional paraxial Helmholtz equation with high wavenumbers, Computers Mathematics with Applications, Volume 68 (2014) no. 10, pp. 1341-1354 | DOI:10.1016/j.camwa.2014.09.005 | Zbl:1367.35069
  • Sheng, Qin; Sun, Hai-Wei Asymptotic Stability of an Eikonal Transformation Based ADI Method for the Paraxial Helmholtz Equation at High Wave Numbers, Communications in Computational Physics, Volume 12 (2012) no. 4, p. 1275 | DOI:10.4208/cicp.100811.090112a
  • Hairer, E.; Lubich, Ch. Modulated Fourier Expansions for Continuous and Discrete Oscillatory Systems, Foundations of Computational Mathematics, Budapest 2011 (2012), p. 113 | DOI:10.1017/cbo9781139095402.007
  • Spigler, Renato; Vianello, Marco The “phase function” method to solve second-order asymptotically polynomial differential equations, Numerische Mathematik, Volume 121 (2012) no. 3, pp. 565-586 | DOI:10.1007/s00211-011-0441-9 | Zbl:1256.65080
  • Sheng, Qin; Guha, Shekhar; Gonzalez, Leonel P. An exponential transformation based splitting method for fast computations of highly oscillatory solutions, Journal of Computational and Applied Mathematics, Volume 235 (2011) no. 15, pp. 4452-4463 | DOI:10.1016/j.cam.2011.04.013 | Zbl:1216.78008
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  • Condon, Marissa; Deaño, Alfredo; Iserles, Arieh Asymptotic solvers for oscillatory systems of differential equations, SeMA Journal, Volume 53 (2011), pp. 79-101 | DOI:10.1007/bf03322583 | Zbl:1242.65132
  • Hochbruck, Marlis; Ostermann, Alexander Exponential integrators, Acta Numerica, Volume 19 (2010), p. 209 | DOI:10.1017/s0962492910000048
  • Tao, Molei; Owhadi, Houman; Marsden, Jerrold E. Nonintrusive and Structure Preserving Multiscale Integration of Stiff ODEs, SDEs, and Hamiltonian Systems with Hidden Slow Dynamics via Flow Averaging, Multiscale Modeling Simulation, Volume 8 (2010) no. 4, p. 1269 | DOI:10.1137/090771648
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  • Harris, Paul J.; Chen, Ke An efficient method for evaluating the integral of a class of highly oscillatory functions, Journal of Computational and Applied Mathematics, Volume 230 (2009) no. 2, pp. 433-442 | DOI:10.1016/j.cam.2008.12.026 | Zbl:1168.65013

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