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Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary differential equations: part I

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Abstract

This work presents methods of efficient numerical approximation for linear and nonlinear systems of highly oscillatory ordinary differential equations. We show how an appropriate choice of quadrature rule improves the accuracy of approximation as the frequency of oscillation grows. We present asymptotic and Filon-type methods to solve highly oscillatory linear systems of ODEs, and WRF method, representing a special combination of Filon-type methods and waveform relaxation methods, for nonlinear systems. Numerical examples support this paper.

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References

  1. U. M. Ascher and S. Y. P. Chan, On parallel methods for boundary value ODEs, Computing, 46 (1991), pp. 1–17.

    Article  MATH  MathSciNet  Google Scholar 

  2. U. M. Ascher, H. Huang, and K. Van Den Doel, Artificial time integration, BIT, 47 (2007), pp. 3–25.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. G. De Bruijn, Asymptotic Methods in Analysis, Dover Publications Inc., New York, 1981.

    MATH  Google Scholar 

  4. E. A. Flinn, A modification of Filon’s method of numerical integration, J. Assoc. Comput. Mach., 7 (1960), pp. 181–184.

    MATH  MathSciNet  Google Scholar 

  5. M. J. Gander and A. M. Stuart, Space-time continuous analysis of waveform relaxation for the heat equation, SIAM J. Sci. Comput., 19 (1998), pp. 2014–2031 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  6. M. J. Gander and S. Vandewalle, Analysis of the parareal time-parallel time-integration method, SIAM J. Sci. Comput., 29 (2007), pp. 556–578 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  7. V. Grimm and M. Hochbruck, Error analysis of exponential integrators for oscillatory second-order differential equations, J. Phys. A, 39 (2006), pp. 5495–5507.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Huybrechs and S. Olver, Highly oscillatory quadrature, in Highly Oscillatory Problems: Computation, Theory and Applications, Cambridge University Press, to appear, 2008.

  9. D. Huybrechs and S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal., 44 (2006), pp. 1026–1048 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 461 (2005), pp. 1383–1399.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Iserles, S. P. Nørsett, and S. Olver, Highly oscillatory quadrature: the story so far, in Numerical Mathematics and Advanced Applications, Springer, Berlin, 2006, pp. 97–118.

    Chapter  Google Scholar 

  12. Z. Jackiewicz, B. Owren, and B. Welfert, Pseudospectra of waveform relaxation operators, Comput. Math. Appl., 36 (1998), pp. 67–85.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Janssen and S. Vandewalle, On SOR waveform relaxation methods, SIAM J. Numer. Anal., 34 (1997), pp. 2456–2481.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Khanamiryan, Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary differential equations: Part II, to be submitted to BIT, 2008.

  15. M. Khanamiryan, Levin-type method for highly oscillatory systems of ordinary differential equations, preprint, 2008.

  16. D. Levin, Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations, Math. Comp., 38 (1982), pp. 531–538.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. Levin, Analysis of a collocation method for integrating rapidly oscillatory functions, J. Comput. Appl. Math., 78 (1997), pp. 131–138.

    Article  MATH  MathSciNet  Google Scholar 

  18. C. Lubich and A. Ostermann, Multigrid dynamic iteration for parabolic equations, BIT, 27 (1987), pp. 216–234.

    Article  MATH  MathSciNet  Google Scholar 

  19. U. Miekkala and O. Nevanlinna, Convergence of dynamic iteration methods for initial value problem, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 459–482.

    Article  MATH  MathSciNet  Google Scholar 

  20. O. Nevanlinna, Remarks on Picard–Lindelöf iteration. I, BIT, 29 (1989), pp. 328–346.

    Article  MATH  MathSciNet  Google Scholar 

  21. O. Nevanlinna, Remarks on Picard–Lindelöf iteration. II, BIT, 29 (1989), pp. 535–562.

    Article  MATH  MathSciNet  Google Scholar 

  22. F. W. J. Olver, Asymptotics and special functions, in Computer Science and Applied Mathematics, Academic Press, New York, London, 1974.

    Google Scholar 

  23. S. Olver, Numerical approximation of vector-valued highly oscillatory integrals, BIT, 47 (2007), pp. 637–655.

    Article  MATH  MathSciNet  Google Scholar 

  24. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  25. S. Vandewalle, Parallel multigrid waveform relaxation for parabolic problems, in Teubner Scripts on Numerical Mathematics, B. G. Teubner, Stuttgart, 1993.

    Google Scholar 

  26. J. White, F. Odeh, A. L. Sangiovanni-Vincentelli, and A. Ruehli, Waveform Relaxation: Theory and Practice, EECS Department, University of California, Berkeley, 1985.

    Google Scholar 

  27. R. Wong, Asymptotic Approximations of Integrals, Classics in Applied Mathematics, vol. 34, SIAM, Philadelphia, 2001.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom

    M. Khanamiryan

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  1. M. Khanamiryan
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Correspondence to M. Khanamiryan.

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Dedicated to the memory of Rudolf Khanamiryan.

AMS subject classification (2000)

65L05, 34E05, 34C15

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Khanamiryan, M. Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary differential equations: part I . Bit Numer Math 48, 743–761 (2008). https://doi.org/10.1007/s10543-008-0201-0

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  • DOI: https://doi.org/10.1007/s10543-008-0201-0

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