Accuracy and Computational Cost of Interpolation Schemes While Performing N-Body Simulations

DOI: 10.4236/ajcm.2014.45037   PDF   HTML   XML   3,370 Downloads   3,899 Views  

Abstract

The continuous approximations play a vital role in N-body simulations. We constructed three different types, namely, one-step (cubic and quintic Hermite), two-step, and three-step Hermite interpolation schemes. The continuous approximations obtained by Hermite interpolation schemes and interpolants for ODEX2 and ERKN integrators are discussed in this paper. The primary focus of this paper is to measure the accuracy and computational cost of different types of interpolation schemes for a variety of gravitational problems. The gravitational problems consist of Kepler’s two-body problem and the more realistic problem involving the Sun and four gas-giants—Jupiter, Saturn, Uranus, and Neptune. The numerical experiments are performed for the different integrators together with one-step, two-step, and three-step Hermite interpolation schemes, as well as the interpolants.

Share and Cite:

Rehman, S. (2014) Accuracy and Computational Cost of Interpolation Schemes While Performing N-Body Simulations. American Journal of Computational Mathematics, 4, 446-454. doi: 10.4236/ajcm.2014.45037.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Nyström, E.J. (1925) Über die numerische Integration von Differentialgleichungen. Acta Societatis Scientiarum Fennicae, 50, 1-54.
[2] Dormand, J., El-Mikkawy, M.E.A. and Prince, P. (1987) Higher Order Embedded Runge-Kutta-Nyström Formulae. IMA Journal of Numerical Analysis, 7, 423-430.
http://dx.doi.org/10.1093/imanum/7.4.423
[3] Dormand, J.R. and Prince, P.J. (1987) New Runge-Kutta Algorithms for Numerical Simulation in Dynamical Astronomy. Celestial Mechanics, 18, 223-232.
http://dx.doi.org/10.1007/BF01230162
[4] Baker, T.S., Dormand, J.R. and Prince, P.J. (1999) Continuous Approximation with Embedded Runge-Kutta-Nyström Methods. Applied Numerical Mathematics, 29, 171-188.
http://dx.doi.org/10.1016/S0168-9274(98)00065-8
[5] Hairer, E., Nørsett, S.P. and Wanner, G. (1987) Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag, Berlin.
[6] Störmer, C. (1907) Sur les trajectoires des corpuscles électrisés. Acta Societatis Scientiarum Fennicae, 24, 221-247.
[7] Grazier, K.R., Newman, W.I., Kaula, W.M. and Hyman, J.M. (1999) Dynamical Evolution of Planetesimals in Outer Solar System. ICARUS, 140, 341-352.
http://dx.doi.org/10.1006/icar.1999.6146
[8] Grazier, K.R. (1997) The Stability of Planetesimal Niches in the Outer Solar System: A Numerical Investigation. Ph.D. Thesis, University of California, Berkeley.
[9] Grazier, K.R., Newman, W.I. and Sharp, P.W. (2013) A Multirate Störmer Algorithm for Close Encounters. The Astronomical Journal, 145, 112-119.
http://dx.doi.org/10.1088/0004-6256/145/4/112
[10] Rehman, S. (2013) Jovian Problem: Performance of Some High-Order Numerical Integrators. American Journal of Computational Mathematics, 3, 195-204.
http://dx.doi.org/10.4236/ajcm.2013.33028

  
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.