Abstract
We present a time-transformed leapfrog scheme combined with the extrapolation method to construct an integrator for orbits in N-body systems with large mass ratios. The basic idea can be used to transform any second-order differential equation into a form which may allow more efficient numerical integration. When applied to gravitating few-body systems this formulation permits extremely close two-body encounters to be considered without significant loss of accuracy. The new scheme has been implemented in a direct N-body code for simulations of super-massive binaries in galactic nuclei. In this context relativistic effects may also be included.
Similar content being viewed by others
References
Aarseth, S. J.: 1972, 'Binary evolution in stellar systems', In: M. Lecar (ed.), Gravitational N-body Problem, Reidel, pp. 88uu98.
Aarseth, S. J.: 1999a, 'From NBODY1 to NBODY6: The growth of an industry', PASP 111, 1333uu1346.
Aarseth, S. J.: 1999b, 'Star cluster simulations: The state of the art', In: J. Henrard and S. Ferraz-Mello (eds), Impact of Modern Dynamics in Astronomy, Kluwer, pp. 127uu137.
Aarseth, S. J. and Zare, K.: 1974, 'A regularization of the three-body problem', Cel. Mech. 10, 185uu205.
Ahmad, A. and Cohen, L.: 1973, 'A numerical integration scheme for the N-body gravitational problem', J. Comput. Phys. 12, 389uu402.
Bulirsch, R. and Stoer, J.: 1966, 'Numerical treatment of differential equations by extrapolation methods', Num. Math. 8, 1uu13.
Gragg, W. B.: 1964 'Repeated Extrapolation to the Limit in the Numerical Solution of Ordinary Differential Equations', PhD Thesis, University of California, Los Angeles.
Gragg, W. B.: 1965 'On extrapolation algorithms for ordinary initial value problems', SIAMJ. Numer. Anal. 2, 384uu403.
Heggie, D. C.: 1974, 'A global regularisation of the gravitational N-body problem', Celest Mech. 10, 217uu241.
Mikkola, S. and Aarseth, S. J.: 1993, 'An implementation of N-body chain regularization', Celest. Mech. & Dyn. Astr. 57, 439uu459.
Mikkola, S. and Tanikawa, K.: 1999a 'Explicit symplectic algorithms for time-transformed Hamiltonians', Celest. Mech. & Dyn. Astr. 74, 287uu295.
Mikkola, S. and Tanikawa, K.: 1999b 'Algorithmic regularization of the few-body problem', Mon. Not. R. Astr. Soc. 310, 745uu749.
Milosavljevi?, M. and Merritt, D.: 2001, 'Formation of galactic nuclei', Astro-ph/0103350.
Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T.: 1986, Numerical Recipes, Cambridge University Press.
Preto, M. and Tremaine, S.: 1999, 'A class of symplectic integrators with adaptive timestep for separable Hamiltonian systems', Astron J. 118, 2532uu2541.
Quinlan, G. D. and Hernquist, L.: 1997, 'The dynamical evolution of massive black hole binaries uu II. Self-consistent N-body integrations', New Astron. 2, 533uu554.
Soffel, M. H.: 1989, Relativity in Astrometry, Celestial Mechanics and Geodesy, Springer, Berlin, p. 141.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mikkola, S., Aarseth, S. A Time-Transformed Leapfrog Scheme. Celestial Mechanics and Dynamical Astronomy 84, 343–354 (2002). https://doi.org/10.1023/A:1021149313347
Issue Date:
DOI: https://doi.org/10.1023/A:1021149313347