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Examples of transformations improving the numerical accuracy of the integration of differential equations

Part of the Lecture Notes in Mathematics book series (LNM,volume 362)

Abstract

Methods for transforming differential equations are developed in such a way that the transformed equation is better suited for numerical integration. In order to obtain theoretical insight into such mechanisms we discuss mostly differential equations which are also solvable in closed analytical form. The methods under consideration are aimed at application in cases where small perturbing terms are added to these equations.

Keywords

  • Fundamental Frequency
  • Celestial Mechanic
  • Control Term
  • Restricted Problem
  • Closed Analytical Form

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. V. Szebehely: Theory of orbits. The restricted problem of three bodies, Academic press 1967.

    Google Scholar 

  2. E.L. Stiefel, G. Scheifele: Linear and regular celestial mechanics, Springer 1971.

    Google Scholar 

  3. C.L. Siegel, J.K. Moser: Lectures on celestial mechanics, Springer 1971.

    Google Scholar 

  4. P. Kustaanheimo, E. Stiefel: Perturbation theory of Kepler motion based on spinor regularization, J.reine und angew. Math. 218, 1965, p. 204–219.

    MathSciNet  MATH  Google Scholar 

  5. E. Stiefel, M. Rössler, J. Waldvogel, C.A. Burdet: Methods of regularization for computing orbits in celestial mechanics, NASA contractor report CR-769, 1967.

    Google Scholar 

  6. H. Sperling: Computation of Keplerian conic sections, ARS journal, 1961, p. 660–661.

    Google Scholar 

  7. C.A. Burdet: Regularization of the two-body problem, ZAMP 18, 1967, p. 434–438.

    CrossRef  MATH  Google Scholar 

  8. : Theory of Kepler motion: The general perturbed two-body problem, ZAMP 19, 1968, p. 345–368.

    CrossRef  MATH  Google Scholar 

  9. K.F. Sundman: Mémoire sur le problème des trois corps, Acta math. 36, 1913, p. 105–179.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. G. Lemaître: Regularization dans le problème des trois corps, Acad. Roy. Belg. Bull.d.Sci. (5) 40, 1954, p. 759–767.

    MATH  Google Scholar 

  11. J. Waldvogel: A new regularization of the planar problem of three bodies, Cel. Mech., vol. 6, No. 2, 1972.

    Google Scholar 

  12. R. F. Arenstorf: New regularization of the restricted problem of three bodies, Astr. J. 68, No. 8, 1963., p. 548–555.

    CrossRef  MathSciNet  Google Scholar 

  13. J. Waldvogel: Die Verallgemeinerung der Birkhoff-Regularisierung auf das räumliche Dreikörperproblem, Bulletin astr. série 3, t. II, fasc. 2, 1967.

    Google Scholar 

  14. H. J. Sperling: Bibliography on the singularities of the equations of motion of celestial mechanics, Second edition, Internal note, Marshall Space Flight Center, Febr. 1970.

    Google Scholar 

  15. E. Stiefel, D.G. Bettis, Stabilization of Cowell's method. Numer. Math. 13, 1969, p. 154–175.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. D.G. Bettis, Stabilization of finite difference methods of numerical integration, Cel. Mech., vol. 2, 1970, p. 282–295.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. D.G. Bettis: Numerical integration of products of Fourier and ordinary polynomials, Numer. Math. 14, 1970, p. 421–434.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. G. Scheifele: On numerical integration of perturbed linear oscillating systems, ZAMP, vol. 22, fasc. 1, 1971, p. 186–210.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. P.E. Nacozy: The use of integrals in numerical integrations of the N-body problem, Astrophysics and space science 14, 1971, p. 40–51.

    CrossRef  Google Scholar 

  20. J. Baumgarte: Numerical stabilization of the differential equations of Keplerian motion, Cel. Mech., vol. 5, No. 4, p. 490–501.

    Google Scholar 

  21. J. Baumgarte: Stabilization of constraints and integrals of motion in dynamical systems, Comp. meth. in applied mech. and engineering 1, 1972, p. 1–16.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. G. Scheifele: On nonclassical canonical systems, Cel. Mech. 2, p. 296, 1970.

    CrossRef  MATH  Google Scholar 

  23. G. Scheifele, E. Stiefel: Canonical satellite theory based on independent variables different from time, Report to ESRO, ESOC contract 219/70/AR, 1972.

    Google Scholar 

  24. U. Kirchgraber: An analytical perturbation theory based on polar coordinates in the four-dimensional KS-space, Thesis ETH Zürich, 1972.

    Google Scholar 

  25. H.R. Schwarz: Stability of Kepler motion, Comp. meth. in appl. mech. and engineering 1. p. 279–299.

    Google Scholar 

  26. J. Baumgarte: Asymptotische Stabilisierung von Integralen bei gewöhnlichen Differentialgleichungen 1. Ordnung. To appear.

    Google Scholar 

  27. E. Hochfeld (ed): Stabilization of Computer Circuits, U. of Chicago, Nov. 1957. Available from US Govt., Printing Office, Washington D.C. as WADC TR 57-425 (AD 155740).

    Google Scholar 

  28. G.D. Birkhoff: The restricted problem of three bodies, Rendiconti Palermo 39, 1915, p. 314.

    Google Scholar 

  29. B. Gagliardi: Ueber die Dimensionserhöhung bei der Regularisierung des Keplerproblems, Thesis ETH Zürich, 1972

    Google Scholar 

  30. U. Kirchgraber: The transformational behaviour of perturbation theories. To appear in Cel. Mech., vol. 7, No. 4.

    Google Scholar 

  31. K. Kocher: Eine lineare Theorie der Optimierungsprobleme bei Raketen mit kleinem Schub (Low thrust), Thesis ETH Zürich, 1971.

    Google Scholar 

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© 1974 Springer-Verlag

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Baumgarte, J., Stiefel, E. (1974). Examples of transformations improving the numerical accuracy of the integration of differential equations. In: Bettis, D.G. (eds) Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations. Lecture Notes in Mathematics, vol 362. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066593

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  • DOI: https://doi.org/10.1007/BFb0066593

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06602-6

  • Online ISBN: 978-3-540-37911-9

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