Abstract
N-body simulations of the Solar System form a challenging set of initial value problems for numerical integrators. The challenge comes from the variety of problems and their size – one recent simulation had 300,015 second order equations and required 9×1010 integration steps. A number of packages for specific types of simulations are available. I discuss what is required of a package intended to efficiently perform a wide range of N-body simulations.
Similar content being viewed by others
References
M.P. Calvo and J.M. Sanz-Serna, The development of variable-step symplectic integrators, with applications to the two-body problem, SIAM J. Sci. Statist. Comput. 14 (1993) 936–952.
M.P. Calvo and J.M. Sanz-Serna, High-order symplectic Runge-Kutta-Nyström methods, SIAM J. Sci. Comput. 14(5) (1993) 1237–1252.
J.E. Chambers, A hybrid symplectic integrator that permits close encounters between massive bodies, Monthly Notices Royal Astronom. Soc. 304 (1999) 793–799.
J.E. Chambers and M.A. Murison, Pseudo-high-order symplectic integrators, Astron. J. 199 (2000) 425–433.
L-Y. Chou, On order 5 and 6 symplectic explicit Runge-Kutta-Nyström methods, M.Sc. thesis, Department of Mathematics, University of Auckland (2000).
L-Y. Chou and P.W. Sharp, On order 5 symplectic explicit Runge-Kutta-Nyström methods, J. Appl. Math. Decision Sci. 4(2) (2000) 143–150.
J.R. Dormand, M.E.A. El-Mikkawy and P.J. Prince, High-order embedded RKN formulae, IMA J. Numer. Anal. 7 (1987) 423–430.
M. Duncan and H. Levison, SWIFT, http://k2.boulder.swri.edu/?hal/swift.html.
J.M. Fine, Low order practical Runge-Kutta-Nyström methods, Computing 38 (1987) 281–297.
K.R. Grazier, The stability of planetesimal niches in the outer solar system: A numerical study, Ph.D. thesis, University of California, Los Angeles (1997).
K. Grazier, W.I. Newman, W.M. Kaula and J.M. Hyman, Dynamical evolution of planetesimals in the outer solar system: I. The Jupiter/Saturn zone, Icarus 140 (1999) 341–352.
E. Hairer, S.P. N¸rsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics, Vol. 8 (Springer, New York, 1987).
T.-Y. Huang and K. Innanen, A survey of multiderivative multistep integrators, Astron. J. 112(1) (1996) 1254–1262.
M.A. López-Marcos, J.M. Sanz-Serna and R.D. Skeel, Explicit symplectic integrators using Hessianvector products, SIAM J. Sci. Comput. 18(1) (1997) 233–238.
R. McLachlan, On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM J. Sci. Comput. 16 (1995) 151–168.
S. Moshier, DE118i, http://people.ne.mediaone.net/moshier/index.html.
X.X. Newhall, E.M. Standish, Jr. and J.G. Williams, DE102: A numerically integrated ephemeris of the Moon and the planets spanning forty-four centuries, Astron. Astrophys. 125 (1983) 150–167.
D. Okunbor and R.D. Skeel, Explicit canonical methods for Hamiltonian systems, Math. Comp. 59 (1992) 439–455.
T. Quinn, N. Katz, J. Stadel and G. Lake, Timing stepping N-body simulations, submitted to Astron. J.
G. Rowlands, A numerical algorithm for Hamiltonian systems, J. Comput. Phys. 97 (1991) 235–239.
P. Saha and S. Tremaine, Long-term planetary integration with individual time steps, Astron. J. 108 (1994) 1962–1969.
L.F. Shampine and M.K. Gordon, Computer Solution of Ordinary Differential Equations; The Initial Value Problem (Freeman, San Francisco, CA, 1975).
P.W. Sharp, Order five explicit second-derivative Runge-Kutta pairs with interpolants, Report Series No. 312, Department of Mathematics, University of Auckland (1994).
P.W. Sharp, Comparisons of high order Stormer and explicit Runge-Kutta methods for N-body simulations of the Solar system, Report Series 449, Department of Mathematics, University of Auckland (2000).
P.W. Sharp and J.M. Fine, Some Nyström pairs for the general second order initial value problem, J. Comput. Appl. Math. 42 (1992) 279–291.
F. Varadi, NBI, http://www.astrobiology.ucla.edu/?varadi/NBI/NBI.html.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sharp, P. Requirements of a Package for N-Body Simulations of the Solar System. Numerical Algorithms 31, 271–279 (2002). https://doi.org/10.1023/A:1021151220482
Issue Date:
DOI: https://doi.org/10.1023/A:1021151220482