An Efficient Integrator that Uses Gauss-Radau Spacings

  • Edgar Everhart
Part of the Astrophysics and Space Science Library book series (ASSL, volume 115)

Abstract

This describes our integrator RADAU, which has been used by several groups in the U.S.A., in Italy, and in the U.S.S.R, over the past 10 years in the numerical integration of orbits and other problems involving numerical solution of systems of ordinary differential equations. First- and second-order equations are solved directly, including the general second-order case. A self-starting integrator, RADAU proceeds by sequences within which the substeps are taken at Gauss-Radau spacings. This allows rather high orders of accuracy with relatively few function evaluations. After the first sequence the information from previous sequences is used to improve the accuracy. The integrator itself chooses the next sequence size. When a 64-bit double word is available in double precision, a 15th-order version is often appropriate, and the FORTRAN code for this case is included here. RADAU is at least comparable with the best of other integrators in speed and accuracy, and it is often superior, particularly at high accuracies.

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References

  1. Batrakov, Yu. N .:1982, ‘Methods of Computation of the Perturbed Motion of Small Bodies in the Solar System’ inSun and Planetary System, (Proceedings of the VI European Meeting in Astronomy), W. Fricke and G. Teleki, editors, D. Reidel Publishing Co., Dordrecht. pp415–419Google Scholar
  2. Bettis, D. G. and Szebehely, V.: 1972, ‘Treatment of Close Approaches in the Numerical Integration of the Gravitational Problem of N-bodies’ Gravitational N-Body Problem, M. Lecar, editor, D. Reidel Publishing Co., Dordrecht, pp388–405. See p395Google Scholar
  3. Bulirsch, R. and Stoer, J.: 1966. ‘Numerical Treatment of Ordinary Differential Equations by Extrapolation Methods’Num. Math.,8, pp1–13MathSciNetMATHCrossRefGoogle Scholar
  4. Butcher, JC.: 1964, ‘Integration Processes Based on Radau Quadrature Formulas’,Math. Comp.18, pp233–344MathSciNetMATHCrossRefGoogle Scholar
  5. Carusi, A., Kresak, L., Perozzi, E., and Valsecchi, G. B.: 1985 ‘The Long-Term Evolution Project’ inDynamics of Comets: Their Origin and Evolution, A. Carusi and G.B. Valsecchi, editors, D. Reidel Publishing Co., Dordrecht. (IAU Colloq. 83, Rome, June 1984 ) This volumeGoogle Scholar
  6. Eckert, W. J., Brouwer, D., and Clemence, G. M. : 1951 ‘Coordinates of the Five Outer Planets, 1653–2060’Astron. Papers Am. Ephemeris,12 Google Scholar
  7. Everhart, E.: 1974a, ‘Implicit Single-Sequence Methods for Integrating Orbits’, Celest. Mech. 10, pp35–55MathSciNetADSMATHCrossRefGoogle Scholar
  8. Everhart, E.: 1974b ‘An Efficient Integrator of Very High Order and Accuracy’ Denver Res. Inst. Tech. Report, 1 July 1974 (unpublished)Google Scholar
  9. Everhart E. and Marsden, B. G.: 1983, ‘New Original and Future Cometary Orbits’Astron. J.,88, pp135–137ADSCrossRefGoogle Scholar
  10. Fehlberg, E.: 1972, ‘Classical Eighth- and Lower Order Runge-Kutta-Nystrom Formulas with Stepsize Control for Special Second-Order Differential Equations’,NASA Tech. Rept., NASA TR R-381Google Scholar
  11. Gallaher, L. J. and Perlin, I. E.: 1966. E.: 1966, ‘A comparison of Several Methods of Numerical Integration of Nonlinear Differential Equations’. Presented at the SIAM meeting, Univ. of Iowa, March. 1966 (unpublished). See Krogh, 1973.Google Scholar
  12. House, F., Weiss, G., and Weigandt, R. : 1978, ‘Numerical Integration of Stellar Orbits’,Celest. Mech.,18, pp311–318ADSMATHCrossRefGoogle Scholar
  13. Krogh, F. T. : 1973, ‘On Testing a Subroutine for Numerical Integration of Ordinary Differential Equations’, J. Assoc. Comput. Mach., 20, pp545–562MATHGoogle Scholar
  14. Levi-Civita, T.: 1903, ‘Traiettorie singulars ed arts nel problems ristretto del tre corp’,Annali di Mat.9, pp1–32Google Scholar
  15. Marsden, B. G., Sekanina, Z., and Everhart, E. : 1978, ‘New Osculating Orbits for 110 Comets and Analysis of Original Orbits for 200 Comets’.Astron. J.,83, pp64–71ADSCrossRefGoogle Scholar
  16. Newton, R. R . : 1959, ‘Periodic Orbits of a Planetoid’,Smithson. Contrib. Astrophys.3, No. 7, pp69–78ADSGoogle Scholar
  17. Papp, K. A., Innanen, K. A., and Patrick, A. T. : 1977, 1980, ‘A Comparison of Five Algolrithms for Numerical Orbit Computation in Galaxy Models’,Celest. Mech.,18, pp277–286, and 21, 337–349ADSGoogle Scholar
  18. Papp, K. A., Innanen, K. A., and Patrick, A. T. : 1977, 1980, ‘A Comparison of Five Algolrithms for Numerical Orbit Computation in Galaxy Models’,Celest. Mech.,18, pp277–286, and 21, 337–349ADSCrossRefGoogle Scholar
  19. Shefer, V.A.: 1982, ‘Variational Equations of the Perturbed Two Body Problem in Regularized Form’, Institute of Theoretical Astronomy, Leningrad. Publication Na.37. An 11th-order version of RADAR is usedGoogle Scholar
  20. Stroud, A. H. and Secrest, D. : 1966,Gaussian Quadrature Formulas Prentice Hall, Inc. Englewood Cliffs, N.J., See Table 12 (Radau spacings).MATHGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1985

Authors and Affiliations

  • Edgar Everhart
    • 1
  1. 1.Physics Department and Chamberlin ObservatoryUniversity of DenverDenverUSA

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