The accurate numerical solution of highly oscillatory ordinary differential equations
Author:
Robert E. Scheid
Journal:
Math. Comp. 41 (1983), 487509
MSC:
Primary 65L05; Secondary 34C29
MathSciNet review:
717698
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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 where the diagonal matrix has smooth, purely imaginary eigenvalues and the components of are polynomial in the components of Z with smooth tdependent coefficients. Computational examples are presented.
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 V. Amdursky & A. Ziv, On the Numerical Treatment of Stiff HighlyOscillatory Systems, IBM Isreal Scientific Center Technical Report No. 15, Haifa, 1974.
 [2]
 V. Amdursky & A. Ziv, The Numerical Treatment of Linear Highly Oscillatory O.D.E. Systems by Reduction to NonOscillatory Type, IBM Israel Scientific Center Report No. 39, Haifa, 1976.
 [3]
 V. Amdursky & A. Ziv, "On the numerical solution of stiff linear systems of the oscillatory type," SIAM J. Appl. Math., v. 33, 1977, pp. 593606. MR 0455417 (56:13655)
 [4]
 N. N. Bogoliubov & Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961. MR 0141845 (25:5242)
 [5]
 G. Browning & H.O. Kreiss, "Problems with different time scales for nonlinear partial differential equations," SIAM J. Appl. Math., v. 42, 1982, pp. 704718. MR 665380 (84g:35014)
 [6]
 E. A. Coddington & N. Levinson, Theory of Ordinary Differential Equations, McGrawHill, New York, 1955. MR 0069338 (16:1022b)
 [7]
 S. O. Fatunla, "Numerical integrators for stiff and highly oscillatory differential equations," Math. Comp., v. 34, 1980, pp. 373390. MR 559191 (81k:65080)
 [8]
 W. Gautschi, "Numerical integration of ordinary differential equations based on trigonometric polynomials," Numer. Math., v. 3, 1961, pp. 381397. MR 0138200 (25:1647)
 [9]
 C. W. Gear, "Numerical solution of ordinary differential equations: Is there anything left to do?," SIAM Rev., v. 23, 1981, pp. 1024. MR 605438 (82c:65043)
 [10]
 O. F. Graff & D. G. Bettis, "Modified multirevolution integration methods for satellite orbit computation," Celestial Mech., v. 11, 1975, pp. 433448. MR 0373409 (51:9609)
 [11]
 F. C. Hoppensteadt & W. L. Miranker, "Differential equations having rapidly changing solutions: Analytic methods for weakly nonlinear systems," J. Differential Equations, v. 22, 1976, pp. 237249. MR 0422791 (54:10777)
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 J. Kevorkian & J. D. Cole, Perturbation Methods in Applied Mathematics, SpringerVerlag, New York, 1981. MR 608029 (82g:34082)
 [13]
 H.O. Kreiss, "Difference methods for stiff ordinary differential equations," SIAM J. Numer. Anal., v. 15, 1978, pp. 2158. MR 486570 (80a:65149)
 [14]
 H.O. Kreiss, "Problems with different time scales for ordinary differential equations." SIAM J. Numer. Anal., v. 16, 1979, pp. 980998. MR 551320 (81a:65087)
 [15]
 J. D. Lambert, Computational Methods in Ordinary Differential Equations, Wiley, Chichester, England, 1973. MR 0423815 (54:11789)
 [16]
 B. Lindberg, "On smoothing and extrapolation for the trapezoidal rule," BIT, v. 11, 1971, pp. 2952. MR 0281356 (43:7074)
 [17]
 G. Majda, "Filtering techniques for oscillatory stiff O.D.E.'s," SIAM J. Numer. Anal. (To appear.)
 [18]
 W. L. Miranker, Numerical Methods for Stiff Equations and Singular Perturbation Problems, Reidel, Dordrecht, Holland, 1981. MR 603627 (82h:65058)
 [19]
 W. L. Miranker & F. Hoppensteadt, Numerical Methods for Stiff Systems of Differential Equations Related With Transistors, Tunnel Diodes, etc., Lecture Notes in Comput. Sci., Vol. 10. SpringerVerlag, New York, 1974. MR 0436601 (55:9544)
 [20]
 W. L. Miranker & M. van Veldhuizen, "The method of envelopes," Math. Comp., v. 32, 1978, pp. 453498. MR 0494952 (58:13727)
 [21]
 W. L. Miranker & G. Wahba, "An averaging method for the stiff highly oscillatory problem," Math. Comp., v. 30, 1976, pp. 383399. MR 0423817 (54:11791)
 [22]
 A. Nadeau, J. Guyard & M. R. Feix, "Algebraicnumerical method for the slightly perturbed harmonic oscillator," Math. Comp., v. 28, 1974, pp. 10571066. MR 0349020 (50:1514)
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 J. C. Neu, "The method of nearidentity transforms and its applications," SIAM J. Appl. Math., v. 38, 1980, pp. 189200. MR 564007 (81m:70031)
 [24]
 A. H. Nayfeh, Perturbation Methods, Wiley, New York, 1973. MR 0404788 (53:8588)
 [25]
 L. R. Petzold, "An efficient numerical method for highly oscillatory ordinary differential equations," SIAM J. Numer. Anal., v. 18, 1981, pp. 455479. MR 615526 (82h:65059)
 [26]
 L. R. Petzold & C. W. Gear, Methods for Oscillating Problems, Dept. of Computer Science File #889, University of Illinois at UrbanaChampaign, 1977.
 [27]
 R. E. Scheid, Jr., The Accurate Numerical Solution of Highly Oscillatory Ordinary Differential Equations, Ph.D. thesis, California Institute of Technology, 1982.
 [28]
 A. D. Snyder & G. C. Fleming, "Approximation by aliasing with applications to "Certaine" stiff differential equations," Math. Comp., v. 28, 1974, pp. 465473. MR 0343637 (49:8377)
 [29]
 C. E. Velez, Numerical Integration of Orbits in Multirevolution Steps, NASA Technical Note D5915, Goddard Space Flight Center, Greenbelt, Maryland, 1970.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198307176989
PII:
S 00255718(1983)07176989
Keywords:
Oscillatory,
numerical solution of ordinary differential equations,
stiff equations
Article copyright:
© Copyright 1983
American Mathematical Society
