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Discrete Dynamics in Nature and Society
Volume 2008, Article ID 790619, 38 pages
http://dx.doi.org/10.1155/2008/790619
Research Article

Accelerated Runge-Kutta Methods

1Departments of Aerospace and Mechanical Engineering, Civil Engineering, Mathematics, and Information and Operations Management, 430K Olin Hall, University of Southern California, Los Angeles, CA 90089-1453, USA
2Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453, USA

Received 31 December 2007; Revised 8 March 2008; Accepted 27 April 2008

Academic Editor: Leonid Berezansky

Copyright © 2008 Firdaus E. Udwadia and Artin Farahani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. C. Butcher, “On Runge-Kutta processes of high order,” Journal of the Australian Mathematical Society, vol. 4, pp. 179–194, 1964. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Chichester, UK, 2003. View at Zentralblatt MATH · View at MathSciNet
  3. G. D. Byrne, “Parameters for pseudo Runge-Kutta methods,” Communications of the ACM, vol. 10, no. 2, pp. 102–104, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. G. D. Byrne and R. J. Lambert, “Pseudo-Runge-Kutta methods involving two points,” Journal of the Association for Computing Machinery, vol. 13, no. 1, pp. 114–123, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. F. Costabile, “Metodi pseudo Runge-Kutta di seconda specie,” Calcolo, vol. 7, no. 3-4, pp. 305–322, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. R. Dormand and P. J. Prince, “A family of embedded Runge-Kutta formulae,” Journal of Computational and Applied Mathematics, vol. 6, no. 1, pp. 19–26, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. W. B. Gruttke, “Pseudo-Runge-Kutta methods of the fifth order,” Journal of the Association for Computing Machinery, vol. 17, no. 4, pp. 613–628, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. T. E. Hull, W. H. Enright, B. M. Fellen, and A. E. Sedgwick, “Comparing numerical methods for ordinary differential equations,” SIAM Journal on Numerical Analysis, vol. 9, no. 4, pp. 603–637, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Z. Jackiewicz, R. Renaut, and A. Feldstein, “Two-step Runge-Kutta methods,” SIAM Journal on Numerical Analysis, vol. 28, no. 4, pp. 1165–1182, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Z. Jackiewicz and S. Tracogna, “A general class of two-step Runge-Kutta methods for ordinary differential equations,” SIAM Journal on Numerical Analysis, vol. 32, no. 5, pp. 1390–1427, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. W. Kutta, “Beitrag zur näherungsweisen integration totaler differentialgleichungen,” Zeitschrift für Mathematische Physik, vol. 46, pp. 435–453, 1901. View at Google Scholar
  12. The PARI Group, Bordeaux, France, version 2.3.2, 2007, http://pari.math.u-bordeaux.fr.
  13. P. Phohomsiri and F. E. Udwadia, “Acceleration of Runge-Kutta integration schemes,” Discrete Dynamics in Nature and Society, vol. 2004, no. 2, pp. 307–314, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. C. Runge, “Ueber die numerische auflösung von differentialgleichungen,” Mathematische Annalen, vol. 46, no. 2, pp. 167–178, 1895. View at Publisher · View at Google Scholar · View at MathSciNet
  15. L. F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, NY, USA, 1994. View at Zentralblatt MATH · View at MathSciNet
  16. S. Tracogna and B. Welfert, “Two-step Runge-Kutta: theory and practice,” BIT Numerical Mathematics, vol. 40, no. 4, pp. 775–799, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. X. Wu, “A class of Runge-Kutta formulae of order three and four with reduced evaluations of function,” Applied Mathematics and Computation, vol. 146, no. 2-3, pp. 417–432, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet