Celestial Mechanics and Dynamical Astronomy

, Volume 119, Issue 2, pp 143–168 | Cite as

Bandlimited implicit Runge–Kutta integration for Astrodynamics

  • Ben K. BradleyEmail author
  • Brandon A. Jones
  • Gregory Beylkin
  • Kristian Sandberg
  • Penina Axelrad
Original Article


We describe a new method for numerical integration, dubbed bandlimited collocation implicit Runge–Kutta (BLC-IRK), and compare its efficiency in propagating orbits to existing techniques commonly used in Astrodynamics. The BLC-IRK scheme uses generalized Gaussian quadratures for bandlimited functions. This new method allows us to use significantly fewer force function evaluations than explicit Runge–Kutta schemes. In particular, we use a low-fidelity force model for most of the iterations, thus minimizing the number of high-fidelity force model evaluations. We also investigate the dense output capability of the new scheme, quantifying its accuracy for Earth orbits. We demonstrate that this numerical integration technique is faster than explicit methods of Dormand and Prince 5(4) and 8(7), Runge–Kutta–Fehlberg 7(8), and approaches the efficiency of the 8th-order Gauss–Jackson multistep method. We anticipate a significant acceleration of the scheme in a multiprocessor environment.


Numerical integration Implicit Runge–Kutta Initial value problem Orbit propagation Symplectic property 



This research was made with Government support under and awarded by DoD, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a. B. Jones’ contribution to this work was funded by Air Force Research Laboratories contract FA9453-08-C-0165. The research of G. Beylkin was partially supported by AFOSR Grants FA9550-07-1-0135 and STTR Phase I Grant 1118-001-01.

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Ben K. Bradley
    • 1
    Email author
  • Brandon A. Jones
    • 1
  • Gregory Beylkin
    • 2
  • Kristian Sandberg
    • 3
  • Penina Axelrad
    • 1
  1. 1.Colorado Center for Astrodynamics ResearchUniversity of Colorado BoulderBoulderUSA
  2. 2.Department of Applied MathematicsUniversity of Colorado BoulderBoulderUSA
  3. 3.Computational Solutions, Inc.BoulderUSA

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