The accurate numerical solution of highly oscillatory ordinary differential equations
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- by Robert E. Scheid PDF
- Math. Comp. 41 (1983), 487-509 Request permission
Abstract:
An asymptotic theory for weakly nonlinear, highly oscillatory systems of ordinary differential equations leads to methods which are suitable for accurate computation with large time steps. The theory is developed for systems of the form \[ \begin {array}{*{20}{c}} {{\mathbf {Z}}’= (A(t)/\varepsilon ){\mathbf {Z}} + {\mathbf {H}}({\mathbf {Z}},t),} \hfill \\ {{\mathbf {Z}}(0,\varepsilon ) = {{\mathbf {Z}}_0},\quad 0 < t < T,0 < \varepsilon \ll 1,} \hfill \\ \end {array} \] where the diagonal matrix $A(t)$ has smooth, purely imaginary eigenvalues and the components of ${\mathbf {H}}({\mathbf {Z}},t)$ are polynomial in the components of Z with smooth t-dependent coefficients. Computational examples are presented.References
-
V. Amdursky & A. Ziv, On the Numerical Treatment of Stiff Highly-Oscillatory Systems, IBM Isreal Scientific Center Technical Report No. 15, Haifa, 1974.
V. Amdursky & A. Ziv, The Numerical Treatment of Linear Highly Oscillatory O.D.E. Systems by Reduction to Non-Oscillatory Type, IBM Israel Scientific Center Report No. 39, Haifa, 1976.
- Abraham Ziv and Vardy Amdursky, On the numerical solution of stiff linear systems of the oscillatory type, SIAM J. Appl. Math. 33 (1977), no. 4, 593–606. MR 455417, DOI 10.1137/0133042
- N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations, Translated from the second revised Russian edition, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, Inc., New York, 1961. MR 0141845
- G. Browning and H.-O. Kreiss, Problems with different time scales for nonlinear partial differential equations, SIAM J. Appl. Math. 42 (1982), no. 4, 704–718. MR 665380, DOI 10.1137/0142049
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
- Simeon Ola Fatunla, Numerical integrators for stiff and highly oscillatory differential equations, Math. Comp. 34 (1980), no. 150, 373–390. MR 559191, DOI 10.1090/S0025-5718-1980-0559191-X
- Walter Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math. 3 (1961), 381–397. MR 138200, DOI 10.1007/BF01386037
- C. W. Gear, Numerical solution of ordinary differential equations: is there anything left to do?, SIAM Rev. 23 (1981), no. 1, 10–24. MR 605438, DOI 10.1137/1023002
- O. F. Graf and D. G. Bettis, Modified multirevolution integration methods for satellite orbit computation, Celestial Mech. 11 (1975), 433–448. MR 373409, DOI 10.1007/BF01650283
- F. C. Hoppensteadt and Willard L. Miranker, Differential equations having rapidly changing solutions: analytic methods for weakly nonlinear systems, J. Differential Equations 22 (1976), no. 2, 237–249. MR 422791, DOI 10.1016/0022-0396(76)90026-7
- J. Kevorkian and Julian D. Cole, Perturbation methods in applied mathematics, Applied Mathematical Sciences, vol. 34, Springer-Verlag, New York-Berlin, 1981. MR 608029
- Heinz-Otto Kreiss, Difference methods for stiff ordinary differential equations, SIAM J. Numer. Anal. 15 (1978), no. 1, 21–58. MR 486570, DOI 10.1137/0715003
- Heinz-Otto Kreiss, Problems with different time scales for ordinary differential equations, SIAM J. Numer. Anal. 16 (1979), no. 6, 980–998. MR 551320, DOI 10.1137/0716072
- J. D. Lambert, Computational methods in ordinary differential equations, John Wiley & Sons, London-New York-Sydney, 1973. Introductory Mathematics for Scientists and Engineers. MR 0423815
- Bengt Lindberg, On smoothing and extrapolation for the trapezoidal rule, Nordisk Tidskr. Informationsbehandling (BIT) 11 (1971), 29–52. MR 281356, DOI 10.1007/bf01935326 G. Majda, "Filtering techniques for oscillatory stiff O.D.E.’s," SIAM J. Numer. Anal. (To appear.)
- Willard L. Miranker, Numerical methods for stiff equations and singular perturbation problems, Mathematics and its Applications, vol. 5, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. MR 603627
- Willard L. Miranker and Frank Hoppensteadt, Numerical methods for stiff systems of differential equations related with transistors, tunnel diodes, etc, Computing methods in applied sciences and engineering (Proc. Internat. Sympos., Versailles, 1973) Lecture Notes in Comput. Sci., Vol. 10, Springer, Berlin, 1974, pp. 416–432. MR 0436601
- W. L. Miranker and M. van Veldhuizen, The method of envelopes, Math. Comp. 32 (1978), no. 142, 453–496. MR 494952, DOI 10.1090/S0025-5718-1978-0494952-8
- W. L. Miranker and G. Wahba, An averaging method for the stiff highly oscillatory problem, Math. Comp. 30 (1976), no. 135, 383–399. MR 423817, DOI 10.1090/S0025-5718-1976-0423817-0
- A. Nadeau, J. Guyard, and M. R. Feix, Algebraic-numerical method for the slightly perturbed harmonic oscillator, Math. Comp. 28 (1974), 1057–1066. MR 349020, DOI 10.1090/S0025-5718-1974-0349020-9
- John C. Neu, The method of near-identity transformations and its applications, SIAM J. Appl. Math. 38 (1980), no. 2, 189–208. MR 564007, DOI 10.1137/0138017
- Ali Hasan Nayfeh, Perturbation methods, Pure and Applied Mathematics, John Wiley & Sons, New York-London-Sydney, 1973. MR 0404788
- Linda R. Petzold, An efficient numerical method for highly oscillatory ordinary differential equations, SIAM J. Numer. Anal. 18 (1981), no. 3, 455–479. MR 615526, DOI 10.1137/0718030 L. R. Petzold & C. W. Gear, Methods for Oscillating Problems, Dept. of Computer Science File #889, University of Illinois at Urbana-Champaign, 1977. R. E. Scheid, Jr., The Accurate Numerical Solution of Highly Oscillatory Ordinary Differential Equations, Ph.D. thesis, California Institute of Technology, 1982.
- Arthur David Snider and Gary Charles Fleming, Approximation by aliasing with application to “Certaine” stiff differential equations, Math. Comp. 28 (1974), 465–473. MR 343637, DOI 10.1090/S0025-5718-1974-0343637-3 C. E. Velez, Numerical Integration of Orbits in Multirevolution Steps, NASA Technical Note D-5915, Goddard Space Flight Center, Greenbelt, Maryland, 1970.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Math. Comp. 41 (1983), 487-509
- MSC: Primary 65L05; Secondary 34C29
- DOI: https://doi.org/10.1090/S0025-5718-1983-0717698-9
- MathSciNet review: 717698