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Numerical Integration Methods in Dynamical Astronomy

  • Hiroshi Kinoshita
  • Hiroshi Nakai
Conference paper

Abstract

We review what kinds of numerical integrators are used by astronomers in the field of dynamical astronomy and to what problems they are applied. This review is based on the questionaires distributed mainly to the members of IAU Commission 7(Celestial Mechanics). Because of the restriction to the Commission 7 members, the answers are mainly from astronomers in the solar system dynamics and problems mentioned in the answers are also related to celestial bodies in the solar system. Other than above, two questions, how to check the precision or accuracy of numerically integrated results and how to treat a close approach, are also surveyed. The problem of the suitable choice of a numerical integrator from various numerical integrators is out of the scope of this review, and it depends strongly on the dynamical nature of a particular dynamical system and the required accuracy.

Keywords

Multistep Method Numerical Integration Method Local Truncation Error Linear Multistep Method Dynamical Astronomy 
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. Aarseth, S. J., 1972, Numerical Experiments on the N-Body Problemen Gravitational N-Body Problem, ed. by Lecar, D. Reidel, 29–43.Google Scholar
  2. Aarseth, S. J., 1985, Direct Methods for N-Body Simulations, in Multiple Time Scales, ed. by Brockbill and Cohen, Academic Press, 377–418.Google Scholar
  3. Aarseth, S. J., 1988, Integration Methods for Small N-body System, in the Few Body Problem, ed. by Valtonen, Kluwer Academic Publisher, 287–307.Google Scholar
  4. Aarseth, S. J. and Zare, K., 1974, A Regularization of the Three-Body Problem, Celest. Mech., 1O, 185–205.ADSGoogle Scholar
  5. Aizu, K., 1985, Canonical Transformation Invariance and Linear Multistep Formula for Integration of Hamiltonian Systems, J. Comput. Phys. , 58, 270–274.MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. Baumgarte, J., 1972, Numerical Stabilization of the Differential Equations of Keplerian Motion, Celest. Mech., 5, 490–501.MathSciNetzbMATHGoogle Scholar
  7. Baumgarte, J., 1976, Stabilization by Making Use of a Generalized Hamiltonian Formalism, in Dynamics of Planets and Satellites and Theories of Their Motions, ed. by Szebehely, D. Reidel Pub. Co. , Dordrecht Hoiland, 149–156.Google Scholar
  8. Black, W., 1973, Practical Considerations in the Numerical Solution of the N-Body Problem by Taylor Series Methods, Celest. Mech., 8, 357–370.ADSzbMATHCrossRefGoogle Scholar
  9. Broucke, R., 1971, Solution of the N-Body Problem with Recurrent Power Series, Celest. Mech., 4, 110–115.MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. Brouwer, D., 1937, On the Accumulation of Errors in Numerical Integration, Astron. J., 46, 149–153.ADSCrossRefGoogle Scholar
  11. Burlirsch, R. and Stoer, J., 1966, Numerical Treatment of Ordinary Differential Equations by Extrapolation Methods, Num. Nath., 8, 1–13.CrossRefGoogle Scholar
  12. Channell, P. J., 1983, Symplectic Integration Algorithms, Los Alamos National Laboratory internal report AT-6:ATN-83–9.Google Scholar
  13. Cohen, C. T., Hubbard, E. C., and Oesterwinter, C., 1973, Elements of the Outer Planets for One Million Years, Astronomical Papaers of the American Ephemeris, 22, pt. 1.Google Scholar
  14. Dormand, J. R. and Prince, P. J., 1978, New Runge-Kutta Algorithms for Numerical Simulation in Dynamical Astronomy, Celest. Mech., 18, 223–232.MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. Dormand, J. R., El-Mikhawy, M. E. A., and Prince, P. J., 1987, IMAJ. Numerical Analysis(in press), (Teeside Polytechnic Mathematical Report, TPMR 86–2).Google Scholar
  16. Emslie, A. K. and Walker, I. W., 1973, Studies in the Application of Recurrence Relations to Special Perturbation Methods II, Celest. Mech., 19, 147–162.ADSGoogle Scholar
  17. Everhart, E., 1974, Implicit Single-Sequence Methods for Integrating Orbits, Celest. Mech., 10, 35–55. (Physics/Astronomy Tech. Rep., University of Denver:Listing and Exploration of Program Radau).Google Scholar
  18. Feagin, T., 1976, Multistep Methods of Numerical Integration Using BackCorrections, Celest. Mech., 13, 111–120.MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. Fehlberg, E., 1968, Classical Fifth-, Sixth-, Seventh-, and Eighth-order Runge-Kutta Formulas with Stepsize Control, NASA TR R-287.Google Scholar
  20. Fehlberg, E., 1974, Classical Fifth-, Sixth-, Seventh-, and Eighth-order Runge-Kutta-Nystrom Formulas with Stepsize Control for General Second-OrderDifferential Equations. NASA TR R-432.Google Scholar
  21. Fen, K. and Qin, M-Z., 1987, The Symplectic Methods for the Computation of Hamiltonian Equations, in Numerical Methods for Partial Differential Equations, ed. by Zhu and Guo.Google Scholar
  22. Fox, K., 1984, Numerical Integration of the Equations of Motion of Celestial Mechanics, Celest. Mech., 33, 127–142.zbMATHGoogle Scholar
  23. Fukushima, T., 1988, Tests of the Extrapolation Method for the Numerical Integration of the Keplerian Motion, in this issue.Google Scholar
  24. Gragg, W., 1963, Repeated Extrapolation to the Limit in the Numerical Solution of Ordinary Differential Equations, Doctoral Dissertation, University of California, Los Angels, pp. 103.Google Scholar
  25. Gragg, W., 1965, On Extrapolation Algorithms for Ordinary Initial Value Problems, SINUM, 2, 384–403.MathSciNetGoogle Scholar
  26. Grobner, W., 1967, Die Lie-Reihen und ihre Anwendungen, VEB Deutscher Verlar der Wissenschaften, Berlin.Google Scholar
  27. Hanslmeier, A. and Dvorak, R., 1984, Numerical Integration with Lie-Series, Astron. Astrophys., 132, 203–207.MathSciNetADSzbMATHGoogle Scholar
  28. Hanslmeier, A., 1984, Application of Lie-Series to Regularized Problem in Celestial Mechanics, Celest. Mech., 34, 135–143.ADSzbMATHCrossRefGoogle Scholar
  29. Heggie, D., 1974, A Global Regularization of the Gravitational N-Body Problem, Celest. Mech., 10, 217–241.ADSzbMATHCrossRefGoogle Scholar
  30. Hindmarsh, A. C. and Byrne, G. D., 1975, EPISODE, UCID-30112, Lawrence Livermore Laboratory.Google Scholar
  31. House, F., Weis, G., and Wiegant, R., 1978, Numerical Integration of Stellar orbits, Celest. Mech., 18, 311–318.zbMATHGoogle Scholar
  32. Huang, T-Y. and Innanen, K. A., 1983, The Accuracy Check in Numerical Integration of Dynamical Systems, Astron. J., 88, 870–876.ADSCrossRefGoogle Scholar
  33. Huang, T-Y. and Valtonen, M. J., 1988, Two Multiderivative Multistep (MDMS)Integrators, Celest. Mech., in press.Google Scholar
  34. Kalbouridis, T., Papageorgiou, G., and Simos, Th., 1988, On the Integration of the Magnetic-Binary Problem by Explicit-Runge-Kutta Methods, Dep. Mathematics, National Technical University of Athens, TR-1/1988.Google Scholar
  35. Kinoshita, H., 1976, Third-Order Solution of an Artificial-Satellite Theory, in Dynamics of Planets and Satellites and Theories of Their Motions, ed. by Szebehely, D. Reidel Pub. Co., Dordrecht-Holland, 241–257.Google Scholar
  36. Kinoshita, H., 1968, Propagation of the Discretization Error in the Two-Body Problem. Publ. Astron. soc. Japan, 20, 1–23.ADSGoogle Scholar
  37. Kinoshita, H. and Nakai, H., 1984, Motions of the Perihelions of Neptune and Pluto, Celest. Mech., 34, 203–217.ADSzbMATHCrossRefGoogle Scholar
  38. Krogh, F. T., 1970, Variable Order Integrators for the Numerical Solution of Ordinary Differential Equations, JPL Technical Utilization Document, No. CP-238.Google Scholar
  39. Lundberg, J. B., 1981, Multistep Integration Formulas for the Numerical Integration of the Satellite Problem, IASOM, TR 81–1.Google Scholar
  40. Menyuk, C. R., 1984, Some Properties of the Discrete Hamiltonian Methods, Physica, 11D, 109–129.MathSciNetADSGoogle Scholar
  41. Mikkola, S., 1985, A Practical and Regular Formulation of the N-Body Equations, Mon. Not. R. Astr. Soc., 215, 171–177.MathSciNetADSzbMATHGoogle Scholar
  42. Milani, A. and Nobili, A. M., 1988, Integration Error over Very Long Time Spans, Celest. Mech. , (in press).Google Scholar
  43. Moran, P. E., 1973, Studies in the Application of Recurrence Relations to Special Perturbation Methods II, Celest. Mech., 7, 122–135.ADSzbMATHCrossRefGoogle Scholar
  44. Moran, P. E., Roy, A. E., and Black, W., 1973, Studies in the Application of Recurrence Relations to Special Perturbation Methods II, Celest. Mech., 8, 405–428.ADSzbMATHCrossRefGoogle Scholar
  45. Neri, F. . 1987, Lie Algebras and Canonical Integration, University of Arizona, Dept. of Physics Technical Report, pp. 25.Google Scholar
  46. Parp, K. A., Innane, K. A., and Patrick, A. T., 1978, A Comparison of Five Algorithms for Numerical Orbit Computation in Galaxy Models, Celest. Mech., 18, 277–286.ADSCrossRefGoogle Scholar
  47. Parp, K. A., Innane, K. A., and Patrick, A. T., 1980, A Comparison of Five Algorithms for Numerical Orbit Computation in Galaxy Models, Celest. Mech., 21, 277–286.Google Scholar
  48. Rabe, E., 1961, Determination and Survey of Periodic Trojan Orbits in the Restricted Problem of Three Bodies, Astron. J., 66, 500–513.MathSciNetADSCrossRefGoogle Scholar
  49. Ruth, R., 1983, A Canonical Integration Technique. IEEE Trans. Nucl. Sci., 30(4), 2699–2671.ADSCrossRefGoogle Scholar
  50. Schubart, J. and Stumpf, P., 1966, On an N-Body Program of High Accuracy for the Computation of Ephemerides of Minor Planets and Comets, Verof. Astron. Rechen-Inst. , Heidelberg, Nr. 18.Google Scholar
  51. Steffensen, J. F., 1957, On the Problem of Three Bodies in the Plane, Math. Fys. Medd. Dansk. Vid. Selskap. 31, No. 3.MathSciNetGoogle Scholar
  52. Stetter, H. J., 1970, Symmetric Two-Step Algorithms for Ordinary Differential Equations, Computing, 5, 267–280.MathSciNetzbMATHCrossRefGoogle Scholar
  53. Stiefel, E. L. and Scheifele, G., 1971, Linear and Regular Celestial Mechanics, Springer, Berlin.zbMATHGoogle Scholar
  54. Sussman, G. J. and Wisdom, J., 1988, Numerical Evidence That the Motion of Pluto is Chaotic, Science, 241. 433–437.ADSCrossRefGoogle Scholar
  55. Szebehely, V. and Bettis, D. G., 1972, Recent Developments of Integrating the Gravitational Problem of N-Bodies, in Gravitational N-Body Problem, ed. by Lecar, D. Reidel 136–147.Google Scholar
  56. Wisdom, J., 1982, The Origin of the Kirkwood Gaps:A Mapping for Asteroidal Motion Near the 3/1 Commensurability, Astron. J., 87, 577–593.MathSciNetADSCrossRefGoogle Scholar

Textbooks

  1. Butcher, J. C., 1987, The Numerical Analysis of Ordinary Differential Equations:Runge-Kutta and General Linear Methods, John-Wiley & Sons. (C)Google Scholar
  2. Dekker, K. and Verwer, J. G., 1984, Stabitity of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holand. (C)Google Scholar
  3. Gear, C. W., 1971, Numerical Initial Value Problems in Ordinary Differentail Equations, Prenticehall, Englewood Cliffs. (A)Google Scholar
  4. Gekeler, E., 1984, Discretization Methods for Stable Initial Value Problems, Springer-Verlag. (C)Google Scholar
  5. Hairer, E., Norsett, S. P., and Wanner, G., 1984, Soluing Ordinary Differential Equations I:Nonstiff Problem, Springer-Verlag. (B)Google Scholar
  6. Henrici, P., 1961, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York. (B)Google Scholar
  7. Henrici, P., 1963, Error Propagation for Difference Methocls, John Wiley and Sons, New York. (B)Google Scholar
  8. Lambert, J. C., 1973, Computational Methods in Ordinary Differential Equations, Johm Wiley and Sons, London. (A)zbMATHGoogle Scholar
  9. Shampine, L. G. and Gordon, M. K., 1975, Computer Solutions of Ordianry Differential Equations, Freeman, San Francisco.Google Scholar
  10. Stetter, H. J., 1973, Analysis of Discretization Methods for Ordinary Differential Equations, Springer-Verlag, Berlin. (C)zbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • Hiroshi Kinoshita
    • 1
  • Hiroshi Nakai
    • 1
  1. 1.National Astronomical ObservatoryMitaka, TokyoJapan

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