Efficient computation of delay differential equations with highly oscillatory terms
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 6, p. 1407-1420

This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation.

DOI : https://doi.org/10.1051/m2an/2012004
Classification:  34E05,  34E99,  42A99,  34K28
Keywords: Delay differential equations, asymptotic expansions, modulated Fourier expansions, numerical analysis
@article{M2AN_2012__46_6_1407_0,
     author = {Condon, Marissa and Dea\~no, Alfredo and Iserles, Arieh and Kropielnicka, Karolina},
     title = {Efficient computation of delay differential equations with highly oscillatory terms},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {6},
     year = {2012},
     pages = {1407-1420},
     doi = {10.1051/m2an/2012004},
     zbl = {1270.65032},
     mrnumber = {2996333},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2012__46_6_1407_0}
}
Condon, Marissa; Deaño, Alfredo; Iserles, Arieh; Kropielnicka, Karolina. Efficient computation of delay differential equations with highly oscillatory terms. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 6, pp. 1407-1420. doi : 10.1051/m2an/2012004. http://www.numdam.org/item/M2AN_2012__46_6_1407_0/

[1] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford, UK (2003). | MR 1997488 | Zbl 1255.65129

[2] M.P. Calvo and J.M. Sanz-Serna, Heterogeneous multiscale methods for mechanical systems with vibrations. SIAM J. Sci. Comput. 32 (2010) 2029-2046. | MR 2678090 | Zbl 1241.65071

[3] P. Chartier, A. Murua and J.M. Sanz-Serna, Higher-order averaging, formal series and numerical integration I : B-series. Found. Comput. Math. 10 (2010) 695-727. | MR 2728427 | Zbl 1211.34052

[4] Y.K. Chembo, L. Larger and P. Colet, Nonlinear dynamics and spectral stability of optoelectronic microwave oscillators. IEEE J. Quant. Electron. 44 (2008) 858-866.

[5] D. Cohen, E. Hairer and C. Lubich, Modulated Fourier expansions of highly oscillatory differential equations. Found. Comput. Math. 3 (2005) 327-450. | MR 2009682 | Zbl 1056.34005

[6] M. Condon, A. Deaño and A. Iserles, On second order differential equations with highly oscillatory forcing terms. Proc. Roy. Soc. A 466 (2010) 1809-1828. | MR 2639052 | Zbl 1194.34019

[7] M. Condon, A. Deaño and A. Iserles, On systems of differential equations with extrinsic oscillation. Discrete Contin. Dyn. Syst. 28 (2010) 1345-1367. | MR 2679714 | Zbl 1207.65093

[8] B. Engquist, A. Fokas, E. Hairer and A. Iserles Eds., Highly Oscillatory Problems. Cambridge University Press, Cambridge, UK (2009). | MR 2574216 | Zbl 1170.37002

[9] Y.N. Kyrychko and S.J. Hogan, On the use of delay equations in engineering applications. J. Vibr. Control 16 (2010) 943-960. | MR 2848163 | Zbl 1269.70002

[10] V.S. Udaltsov, J.P. Goedgebuer, L. Larger, J.B. Cuenot, P. Levy and W.T. Rhodes, Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations. Phys. Lett. A 308 (2003) 54-60. | MR 1972299 | Zbl 1008.94019

[11] G.D. Van Wiggeren and R. Roy, Communication with chaotic lasers. Science 279 (1998) 1198-1200.

[12] S. Wirkus and R. Rand, The dynamics of two coupled van der pol oscillators with delay coupling. Nonlinear Dyn. 30 (2002) 205-221. | MR 1935659 | Zbl 1021.70010