Explicit algorithmic regularization in the few-body problem for velocity-dependent perturbations

Abstract

A new algorithm is presented for the numerical integration of second-order ordinary differential equations with perturbations that depend on the first derivative of the dependent variables with respect to the independent variable; it is especially designed for few-body problems with velocity-dependent perturbations. The algorithm can be used within extrapolation methods for enhanced accuracy, and it is fully explicit, which makes it a competitive alternative to standard discretization methods.

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References

  1. Bulirsch R., Stoer J.: Numerical treatment of ordinary differential equations by extrapolation methods. Numerische Mathematik 8, 1–13 (1966)

    MATH  Article  MathSciNet  Google Scholar 

  2. Chambers J.: Symplectic integrators with complex time steps. Astron. J. 126, 1119–1126 (2003)

    Article  ADS  Google Scholar 

  3. Dunning, R.: The orbital mechanics of flight mechanics. Tech. rep. (1973)

  4. Gragg W.: On extrapolation algorithms for ordinary initial value problems. SIAM J. Numer. Anal. 2, 384–403 (1965)

    MathSciNet  Google Scholar 

  5. Hairer E., Stoffer D.: Reverisble long-term integration with variable step sizes. SIAM J. Sci. Comput. 18, 257–269 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  6. Hairer E., Lubich C., Wanner G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2006)

    Google Scholar 

  7. Harfst S., Gualandris A., Merritt D., Mikkola S.: A hybrid N-body code incorporating algorithmic regularization and post-newtonian forces. Mon. Not. R. Astron. Soc. 389, 2–12 (2008)

    Article  ADS  Google Scholar 

  8. Kane C., Marsden J., Ortiz M., West M.: Variational integrators and the newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49, 1295–1325 (2000)

    MATH  Article  MathSciNet  Google Scholar 

  9. Kirk D.: Optimal Control Theory. An Introduction. Dover Publications, Mineola, NY (2004)

    Google Scholar 

  10. Marsden J., West M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  11. Meditch J.: On the problem of optimal thrust for a lunar soft landing. IEEE Trans. Automat. Contr. 4, 477–484 (1964)

    Article  MathSciNet  Google Scholar 

  12. Mikkola S., Aarseth S.: A time-transformed leapfrog scheme. Celest. Mech. Dyn. Astron. 84, 343–354 (2002)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  13. Mikkola S., Merritt D.: Algorithmic regularization with velocity-dependent forces. Mon. Not. R. Astron. Soc. 372, 219–223 (2006)

    Article  ADS  Google Scholar 

  14. Mikkola S., Merritt D.: Implementing few-body algorithmic regularization with post-newtonian terms. Astron. J. 135, 2398–2405 (2008)

    Article  ADS  Google Scholar 

  15. Mikkola S., Tanikawa K.: Algorithmic regularization of the few-body problem. Mon. Not. R. Astron. Soc. 310, 745–749 (1999a)

    Article  ADS  Google Scholar 

  16. Mikkola S., Tanikawa K.: Explicit symplectic algorithms for time-transformed hamiltonians. Celest. Mech. Dyn. Astron. 74, 287–295 (1999b)

    MATH  Article  ADS  Google Scholar 

  17. Mititelu G.: Analytical solutions for the equations of motion of a space vehicle during the atmospheric re-entry phase on a 2-D trajectory. Celest. Mech. Dyn. Astron. 103, 327–342 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  18. Pástor P., Klačka J., Kómar L.: Motion of dust in mean motion resonances with planets. Celest. Mech. Dyn. Astron. 103, 343–364 (2009)

    Article  Google Scholar 

  19. Pavlis, N., Holmes, S., Kenyon, S., Factor, J.: An earth gravitational model to degree 2160: EGM2008. presented at the 2008 general assembly of the European geosciences union, Vienna, Austria, 13–18 April (2008)

  20. Preto M., Tremaine S.: A class of symplectic integrators with adaptive timestep for separable hamiltonian systems. Astron. J. 118, 2532–2541 (1999)

    Article  ADS  Google Scholar 

  21. Soffel M.: Relativity in Astrometry. Celestial Mechanics and Geodesy. Springer, Berlin (1989)

    Google Scholar 

  22. Stengel R.: Optimal Control and Estimation. Dover Publications, Mineola, NY (1994)

    Google Scholar 

  23. Vallado D.: Fundamentals of Astrodynamics and Applications, Space Technology Library, vol. 21. Springer, Hawthorne, CA (2007)

    Google Scholar 

  24. Yoshida H.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262–268 (1990)

    Article  MathSciNet  ADS  Google Scholar 

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Correspondence to Christian Hellström.

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Hellström, C., Mikkola, S. Explicit algorithmic regularization in the few-body problem for velocity-dependent perturbations. Celest Mech Dyn Astr 106, 143–156 (2010). https://doi.org/10.1007/s10569-009-9248-8

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Keywords

  • Algorithmic regularization
  • Extrapolation methods
  • Auxiliary velocity algorithm (AVA)
  • Bulirsch-Stoer