Advertisement

Numerical integration methods for orbital motion

  • O. Montenbruck
Article

Abstract

The present report compares Runge-Kutta, multistep and extrapolation methods for the numerical integration of ordinary differential equations and assesses their usefulness for orbit computations of solar system bodies or artificial satellites. The scope of earlier studies is extended by including various methods that have been developed only recently. Several performance tests reveal that modern single- and multistep methods can be similarly efficient over a wide range of eccentricities. Multistep methods are still preferable, however, for ephemeris predictions with a large number of dense output points.

Key words

Numerical Integration Runge-Kutta Methods Multistep Methods Extrapolation Methods Orbit Computation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berryman, K. W., Stanford, R. H., and Breckheimer, P. J.:1988, AIAA Conference Paper 88-4217-CP.Google Scholar
  2. Brankin, R. W., Dormand, J. R., Gladwell, I., Prince, P. J., and Seward, W. L.: 1987, Num. Anal. Rep. 136, University of Manchester.Google Scholar
  3. Brankin, R. W., Gladwell, I., Dormand, J. R., Prince, P. J., and Seward, W. L.: 1989, ACM Trans. Math. Soft., 15/1, 31–40.Google Scholar
  4. Bulirsch, R. and Stoer, J.: 1966, Num. Math. 8, 1–13.Google Scholar
  5. Cappellari, J. O., Velez, C. E., and Fuchs, A. J.: 1976, Mathematical Theory of the Goddard Trajectory Determination System: Goddard Space Flight Center, Greenbelt, Maryland.Google Scholar
  6. Corliss, G. and Chang Y. F.: 1982, ACM Trans. Math. Softw 8/2,114–144.Google Scholar
  7. Delva, M.: 1985, Astron. Astrophys. Suppl. Ser. 60, 277–284.Google Scholar
  8. Deuflhard, P.: 1983, Num. Math. 41, 399–422.Google Scholar
  9. Dormand, J. R. and Prince, P. J.: 1978, Cel. Mech. 18, 223–232.Google Scholar
  10. Dormand, J. R. and Prince, P. J.: 1980, J. Comp. Appl. Math. 6, No. 1, 19–26.Google Scholar
  11. Dormand, J. R. and Prince, P. J.: 1987, Comput. Math. Appl. 13, 937–949.Google Scholar
  12. Dormand, J. R. and Prince, P. J.: 1989, SIAM J. Sci. Stat. Comput. 5, 977–989.Google Scholar
  13. Dormand, J. R., El-Mikkawy, M. E. A., and Prince, P. J.: 1987, IMA J. Numer. Anal. 7, 423–430.Google Scholar
  14. Fehlberg, E.: 1968, Technical Report NASA TR R-287.Google Scholar
  15. Fehlberg, E.: 1975, Computing 14, 371–387.Google Scholar
  16. Filippi S. and Graf J.: 1986, J. Comp. Appl. Math 14, 361–370.Google Scholar
  17. Fox, K.: 1984, Cel. Mech. 33, 127–142.Google Scholar
  18. Gupta, G. K., Sacks-Davis, R., and Tischer, P. E.: 1985, Computing Surveys 17, 5.Google Scholar
  19. Hairer, E., Nørsett, S.P., and Wanner, G.: 1987, Solving Ordinary Differential Equations 1: SpringerVerlag, Berlin.Google Scholar
  20. Hairer, E. and Ostermann, A.: 1990, Num. Math. 58, 419–439.Google Scholar
  21. Herrick, S.: 1971, 1972, Astrodynamics I & II, Van Nostrand Reinhold, London.Google Scholar
  22. Hull, T. E., Enright, W. H., Fellen, B. M., and Sedgwick, A. E.: 1972, SIAM J. Numer. Anal. 9, 603–637.Google Scholar
  23. Hussels, H.-G.: 1973, Schrittweitensteuerung bei der Integration gewdhnlicher Differentialgleichungen mit Extrapolationsverfahren, Diplomarbeit, Köln.Google Scholar
  24. Kinoshita, H. and Nakai, H.: 1989, Cel. Mech. 45, 231–244.Google Scholar
  25. Lambert, J. D.: 1973, Computational Methods in Ordinary Differential Equations, John Wiley and Sons, London.Google Scholar
  26. Martin, T. V., Oh, I. H., Eddy, W. F., and Kogut, J. A.: 1976, GEODYN System Description, Vol. 1, EG&G/Washigton Analytical Science Center, Inc., Wolf Research & Development Company.Google Scholar
  27. Milani, A., and Nobili, A. M.: 1988, Cel. Mech. 43, 1–34.Google Scholar
  28. Montenbruck, O.: 1990, in Space Dynamics Conference, ONES Toulouse Nov. 1989, Cepadues Edition.Google Scholar
  29. Moore, H.: 1974, in Lecture Notes in Mathematics 362, 149, Springer-Verlag, New York.Google Scholar
  30. Prince, P. J.. and Dormand, J. R.: 1981, J. Comp. Appl. Math. 7, 67–75.Google Scholar
  31. Schastok, J., Gleixner, H.., Soffel, M., Ruder, H., and Schneider, M.: 1989, Comp. Phys. Commun. 54, 167–170.Google Scholar
  32. Schubart, J. and Stumpff P.: 1966, Veröffentlichungen des Astronomischen Rechen-Instituts Heidelberg Nr. 18.Google Scholar
  33. Schutz, B. E. and Tapley B. D.: 1980, Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, TR 80-1.Google Scholar
  34. Shampine, L. F. and Gordon, M. K.: 1975, Computer Solution of ordinary Differential Equations, Freeman and Comp., San Francisco.Google Scholar
  35. Shampine, L. F. and Watts H. A.: 1979, SAND79-2374, Sandia Laboratories (1979).Google Scholar
  36. Shampine, L. F., Baca, L. S., and Bauer, H.-J.: 1982, Tech. Rep. 196 SFB 123, Univ. Heidelberg.Google Scholar
  37. Sitarski, G.: 1979, Acta Astronomica 29, 401–411.Google Scholar
  38. Soop, E. M.: 1983, ESA SP-1053.Google Scholar
  39. Stoer, J. and Bulirsch, R.: 1983, Introduction to Numerical Analysis, Springer-Verlag, 2nd ed.Google Scholar
  40. Wanner, G.: 1969, Integration gewöhnlicher Differentialgleichungen Bibliographisches Institut, Mannheim.Google Scholar
  41. Watts, H. A. and Shampine, L. F. 1986, SIAM J. Sci. Stat. Comput. 7, 334–345.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • O. Montenbruck
    • 1
  1. 1.Deutsche Forschungsanstalt für Luft- and RaumfahrtGerman Space Operations Center (GSOC)Oberpfaffenhofen

Personalised recommendations