Jovian Problem: Performance of Some High-Order Numerical Integrators

DOI: 10.4236/ajcm.2013.33028   PDF   HTML     3,791 Downloads   5,961 Views   Citations


N-body simulations of the Sun, the planets, and small celestial bodies are frequently used to model the evolution of the Solar System. Large numbers of numerical integrators for performing such simulations have been developed and used; see, for example, [1,2]. The primary objective of this paper is to analyse and compare the efficiency and the error growth for different numerical integrators. Throughout the paper, the error growth is examined in terms of the global errors in the positions and velocities, and the relative errors in the energy and angular momentum of the system. We performed numerical experiments for the different integrators applied to the Jovian problem over a long interval of duration, as long as one million years, with the local error tolerance ranging from 10-16 to 10-18.

Share and Cite:

S. Rehman, "Jovian Problem: Performance of Some High-Order Numerical Integrators," American Journal of Computational Mathematics, Vol. 3 No. 3, 2013, pp. 195-204. doi: 10.4236/ajcm.2013.33028.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] P. W. Sharp, “N-Body Simulations: The Performance of Some Integrators,” ACM Transactions on Mathematical Software, Vol. 32, No. 3, 2006, pp. 375-395. doi:10.1145/1163641.1163642
[2] K. R. Grazier, W. I. Newman, W. M. Kaula and J. M. Hyman, “Dynamical Evolution of Planetesimals in Outer Solar System,” Icarus, Vol. 140, No. 2, 1999, pp. 341-352. doi:10.1006/icar.1999.6146
[3] K. Heun, “Neue Methode zur Approximativen Integration der Differentialgleichungen einer Unabhangigen Veranderlichen,” Mathematical Physics, Vol. 45, 1900, pp. 23-38.
[4] M. W. Kutta, “Beitrag zur Naherungsweisen Integration totaler Differentialgleichungen,” Mathematical Physics, Vol. 46, 1901, pp. 435-453.
[5] F. T. Krogh, “A Variable Step Variable Order Multistep Methods for Ordinary Differential Equations,” Information Processing Letters, Vol. 68, 1969, pp. 194-199.
[6] E. J. Nystrom, “Uber die Numerische Integration von Differentialgleichungen,” Acta Societas Scientiarum Fennicae, Vol. 50, No. 13, 1925, pp. 1-54.
[7] C. Stormer, “Sur les Trajectoires des Corpuscles électrisés,” Acta Societas Scientiarum Fennicae, Vol. 24, 1907, pp. 221-247.
[8] D. Brouwer, “On the Accumulation of Errors in Numerical Integration,” Astronomical Journal, Vol. 46, No. 1072, 1937, pp. 149-153. doi:10.1086/105423
[9] K. R. Grazier, W. I. Newman, J. M. Hyman and P. W. Sharp, “Long Simulations of the Outer Solar System: Brouwer’s Law and Chaos,” In: R. May and A. J. Roberts, Eds., Proceedings of 12th Computational Techniques and Applications Conference CTAC-2004, ANZIAM Journal, Vol. 46, 2005, pp. C1086-C1103.
[10] E. Hairer, R. I. McLachlan and A. Razakarivony, “Achieving Brouwer’s Law with Implicit Runge-Kutta Methods,” BIT Numerical Mathematics, Vol. 48, No. 2, 2008, pp. 231-243. doi:10.1007/s10543-008-0170-3
[11] W. H. Enright, D. J. Higham, B. Owren and P. W. Sharp, “A Survey of the Explicit Runge-Kutta Method,” Technical Report, 291/94, Department of Computer Science, University of Toronto, Toronto, 1994.
[12] J. Dormand, M. E. A. El-Mikkawy and P. Prince, “Higher Order Embedded Runge-Kutta-Nystrom Formulae,” IMA Journal of Numerical Analysis, Vol. 7, No. 4, 1987, pp. 423-430. doi:10.1093/imanum/7.4.423
[13] L. F. Shampine and M. K. Gordon, “Computer Solution of Ordinary Differential Equations,” W. H. Freeman, San Francisco, 1975.
[14] E. Hairer, S. P. Norsett and G. Wanner, “Solving Ordinary Differential Equations I: Nonstiff Problems,” Springer-Verlag, Berlin, 1987.
[15] K. R. Grazier, “The Stability of Planetesimal Niches in the Outer Solar System: A Numerical Investigation,” PhD Thesis, University of California, 1997.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.