Differential Equations

, Volume 55, Issue 7, pp 949–962 | Cite as

Adaptive Numerical Methods for Solving the Problem about Scattering on a Force Center

  • G. G. EleninEmail author
  • T. G. EleninaEmail author
Numerical Methods


We construct families of adaptive symplectic conservative numerical methods for solving problems about scattering on a force center. The methods preserve the global properties of the exact solution of the problem and approximate the dependences of the phase variables on time with the second, fourth, or sixth approximation order. The variable time step is selected automatically in two different ways depending on the properties of the solution.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Landau, L.D. and Lifshits, E.M., Mekhanika (Mechanics), Moscow: Nauka, 1973.Google Scholar
  2. 2.
    Hairer, E., Lubich, C., and Wanner, G., Geometric Numerical Integration, Berlin: Springer, 2006.zbMATHGoogle Scholar
  3. 3.
    Suris, Y.B., On the conservation of the symplectic structure in numerical solutions of Hamilton systems, in Numerical Solutions of Ordinary Differential Equations, Moscow: Keldysh Inst. Appl. Math, USSR Acad. Sci., 1988. pp. 148–160.Google Scholar
  4. 4.
    Sanz-Serna, J.M., Runge-Kutta schemes for Hamiltonian systems, BIT, 1988, vol. 28, no. 4, pp. 877–883.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Reich, S., Momentum conserving symplectic integrators, Physica D, 1994, vol. 76, pp. 375–383.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    McLachan, R.I., Quispel, G.R.W., and Robidoux, N., Geometric integration using discrete gradients, Phil. Trans. R. Soc. London Ser. A, 1999, vol. 357, pp. 1021–1045.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Marsden, J.E. and West, M., Discrete mechanics and variational integrators, Acta Numerica, 2001, vol. 10, pp. 1–158.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    LaBudde, R.A. and Greenspen, D., Discrete mechanics. A general treatment, J. Comput. Phys., 1974, vol. 15, pp. 134–167.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Minesaki, Y. and Nakamura, Y., A new discretization of the Kepler motion which conserves the Runge-Lenz vector, Phys. Lett. A, 2002, vol. 306, pp. 127–133.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kozlov, R., Conservative discretization of the Keplerian motions, J. Phys. A: Math. Theor., 2007, vol. 40, pp. 4529–4539.CrossRefzbMATHGoogle Scholar
  11. 11.
    Cieslinski, J.L., An orbit-preserving discretization of the classical Keplerian problem, Phys. Lett. A, 2007, vol. 370, pp. 8–12.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Elenin, G.G. and Elenina, T.G., A one-parameter family of difference schemes for the numerical solution of the Keplerian problem, Comput. Math. Math. Phys., 2015, vol. 55, no. 8, pp. 1292–1298.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Elenin, G.G. and Elenina, T.G., Adaptive symplectic conservative numerical methods for the Kepler problem, Differ. Equations, 2017, vol. 53, no. 7, pp. 923–934.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Elenin, G.G. and Elenina, T.G., Parametrization of the solution of the Kepler problem and new adaptive numerical methods based on this parametrization, Differ. Equations, 2018, vol. 54, no. 7, pp. 911–918.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ryzhik, I.S. and Grandshtein, I.M., Tablitsy integralov, summ, ryadov i proizvedenii (Tables of Integrals, Sums, Series, and Products), Moscow: Fizmatgiz, 1963.Google Scholar
  16. 16.
    Oewel, W. and Sofrouniou, M., Symplectic Runge-Kutta schemes II: Classification of symmetric methods, Preprint Univ. Paderborn, 1997.Google Scholar
  17. 17.
    Elenin, G.G. and Shlyakhov, P.I., The geometric structure of the parameter space of the three-stage symplectic Runge-Kutta methods, Math. Models Comput. Simul., 2011, vol. 23, no. 5, pp. 680–689.MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations