Abstract
The integrator SSS performs accurate N-body simulations of the Solar System when there is a mix of massive bodies and test particles. The orbital motion of all bodies at all times is integrated using a 12-10 explicit Runge-Kutta Nyström (RKN) pair. The test particles are divided into sets and each set integrated on a different processor. The explicit RKN pair uses an order 12 interpolant for the position and velocity when checking for collisions. We report on two significant improvements to SSS. The first improvement reduced the local round-off error in interpolated values by approximately four orders of magnitude, permitting more accurate modelling of collisions. The technique used to reduce the round-off error can be applied to other high-order interpolants. The second improvement is hand optimization of the implementation of SSS. This optimization increased the speed of SSS by approximately 60%, permitting more accurate modelling through the use of more test particles. We also present a summary of the numerical performance of SSS on a simulation of the Sun, the planets Earth to Neptune, and 500,000 test particles over 100 million years.
Similar content being viewed by others
References
Baker, T., Dormand, J., Gilmore, J., Prince, P.: Continuous approximation with embedded Runge-Kutta methods. Appl. Numer. Math. 22 (1–3), 51 – 62 (1996). https://doi.org/10.1016/S0168-9274(96)00025-6. http://www.sciencedirect.com/science/article/pii/S0168927496000256. Special Issue Celebrating the Centenary of Runge-Kutta Methods
Batygin, K., Brown, M.E.: Early dynamical evolution of the solar system: pinning down the initial conditions of the Nice model. Astrop. J. 716, 1323–1331 (2010). https://doi.org/10.1088/0004-637X/716/2/1323
Brasil, P.I.O., Nesvorný, D., Gomes, R.S.: Dynamical implantation of objects in the Kuiper Belt. Astron. J. 148, 56 (2014). https://doi.org/10.1088/0004-6256/148/3/56
Chambers, J.E.: A hybrid symplectic integrator that permits close encounters between massive bodies. MNRAS 304, 793–799 (1999). https://doi.org/10.1046/j.1365-8711.1999.02379.x
Dormand, J.R., El-Mikkaway, M.E.A., Prince, P.J.: High-order embedded Runge-Kutta-Nyström formuale. IMA J. Num. Anal. 7, 423–430 (1987)
Duncan, M.J., Levison, H.F., Lee, M.H.: A multiple time step symplectic algorithm for integrating close encounters. Astron. J. 116, 2067–2077 (1998). https://doi.org/10.1086/300541
Grazier, K.R., Castillo-Rogez, J.C., Horner, J.: It’s complicated: a big data approach to exploring planetesimal evolution in the presence of Jovian planets. Astron. J. 156, 232 (2018). https://doi.org/10.3847/1538-3881/aae095
Grazier, K.R., Newman, W.I., Sharp, P.W.: A multirate Störmer algorithm for close encounters. Astron. J. 145(4), 112 (2013). http://stacks.iop.org/1538-3881/145/i=4/a=112
Grimm, S.L., Stadel, J.G.: The GENGA code: gravitational encounters in n-body simulations with GPU acceleration. Astrop. J. 796, 23 (2014). https://doi.org/10.1088/0004-637X/796/1/23
Hairer, E., Wanner, G.: A theory for Nyström methods. Numer. Math. 25(4), 383–400 (1975). https://doi.org/10.1007/BF01396335
Horner, J., Jones, B.: Jupiter – friend or foe? I: the asteroids. Int. J. Astrobiol. 7, 251–261 (2008). https://doi.org/10.1017/S1473550408004187. http://journals.cambridge.org/article_S1473550408004187
Horner, J., Jones, B.: Jupiter – friend or foe? II: the centaurs. Int. J. Astrobiol. 8, 75–80 (2009). https://doi.org/10.1017/S1473550408004357. http://journals.cambridge.org/article_S1473550408004357
Horner, J., Jones, B., Chambers, J.: Jupiter – friend or foe? III: the Oort cloud comets. Int. J. Astrobiol. 9, 1–10 (2010). https://doi.org/10.1017/S1473550409990346. http://journals.cambridge.org/article_S1473550409990346
Horner, J., Jones, B.W.: Jupiter - friend or foe? IV: the influence of orbital eccentricity and inclination. Int. J. Astrobiol. 11, 147–156 (2012). https://doi.org/10.1017/S1473550412000043
Izidoro, A., Raymond, S.N., Morbidelli, A., Winter, O.C.: Terrestrial planet formation constrained by Mars and the structure of the asteroid belt. Mon. Not. Roy. Ast. Soc. 453, 3619–3634 (2015). https://doi.org/10.1093/mnras/stv1835
Kaufmann, D.: Swifter, http://www.boulder.swri.edu/swifter/ (2005)
Levison, H.F., Duncan, M.J.: Symplectically integrating close encounters with the Sun. Astron. J. 120, 2117–2123 (2000). https://doi.org/10.1086/301553
Levison, H.F., Morbidelli, A., Tsiganis, K., Nesvorný, D., Gomes, R.: Late orbital instabilities in the outer planets induced by interaction with a self-gravitating planetesimal disk. Astron. J. 142(5), 152 (2011). http://stacks.iop.org/1538-3881/142/i=5/a=152
Minton, D.A., Malhotra, R.: Dynamical erosion of the asteroid belt and implications for large impacts in the inner solar system. Icarus 207(2), 744–757 (2010). https://doi.org/10.1016/j.icarus.2009.12.008. http://www.sciencedirect.com/science/article/pii/S0019103509004953
Moore, A., Quillen, A.: QYMSYM: a GPU-accelerated hybrid symplectic integrator that permits close encounters. New Astron. 16(7), 445–455 (2011). https://doi.org/10.1016/j.newast.2011.03.009. http://www.sciencedirect.com/science/article/pii/S1384107611000303
Nesvorný, D., Vokrouhlický, D., Morbidelli, A.: Capture of Trojans by jumping Jupiter. Astrophys. J. 768, 45 (2013). https://doi.org/10.1088/0004-637X/768/1/45
Raymond, S.N., Armitage, P.J., Moro-MartÃn, A., Booth, M., Wyatt, M.C., Armstrong, J.C., Mandell, A.M., Selsis, F., West, A.A.: Debris disks as signposts of terrestrial planet formation. II. Dependence of exoplanet architectures on giant planet and disk properties. A & A 541, A11 (2012). https://doi.org/10.1051/0004-6361/201117049
Roy, A.: Orbital Motion, 3rd edn. Taylor & Francis, New York (1988). https://books.google.co.nz/books?id=yLGqnQAACAAJ
Sharp, P.W.: N-body simulations: the performance of some integrators. ACM Trans. Math. Softw. 32, 375–395 (2006). https://doi.org/10.1145/1163641.1163642
Sharp, P.W., Newman, W.I.: A multirate variable-timestep algorithm for n-body solar system simulations with collisions. Astron. J. 151(3), 64 (2016). http://stacks.iop.org/1538-3881/151/i=3/a= 64
Tsitouras, C.: A tenth order symplectic Runge-Kutta-Nyström method. Celestial Mech. Dyn. Astron. 74, 223–230 (1999). https://doi.org/10.1023/A:1008346516048
Wang, J.H., Brasser, R.: An Oort cloud origin of the Halley-type comets. Astron. Astrop. 563, A122 (2014). https://doi.org/10.1051/0004-6361/201322508
Wu, X., Wang, B.: Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations. https://doi.org/10.1007/978-981-10-9004-2 (2018)
Wu, X., Wang, B., Liu, K., Zhao, H.: ERKN methods for long-term integration of multidimensional orbital problems. Appl. Math. Model. 37, 2327–2336 (2013). https://doi.org/10.1016/j.apm.2012.05.021
Wu, X., You, X., Shi, W., Wang, B.: ERKN integrators for systems of oscillatory second-order differential equations. Comput. Phys. Commun. 181, 1873–1887 (2010)
Wu, X., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer Publishing Company, Incorporated (2015)
Acknowledgements
The author thanks the two referees for their careful reading of the paper and their suggestions on how to improve it. The author acknowledges the contribution of the NeSI high-performance computing facilities and the staff at the Centre for eResearch at the University of Auckland. New Zealand’s national facilities are provided by the New Zealand eScience Infrastructure (NeSI) and funded jointly by NeSI’s collaborator institutions and through the Ministry of Business, Innovation and Employment’s Infrastructure programme. http://www.nesi.org.nz. The author also acknowledges the use of the computational servers in the Department of Mathematics at the University of Auckland.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sharp, P.W. The performance of the N-body integrator SSS. Numer Algor 81, 1459–1472 (2019). https://doi.org/10.1007/s11075-019-00674-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-019-00674-1