59
Views
2
CrossRef citations to date
0
Altmetric
Original Articles

Continuous extensions to high order runge-kutta methods

&
Pages 273-291 | Received 25 Jun 1996, Published online: 19 Mar 2007

References

  • Bogacki , P. and Shampine , L. F. 1990 . Interpolating High Order Runge-Kutta Formulas . Computers Math. Applic. , 20 : 489 – 504 .
  • Butcher , J. C. 1963 . Coefficients for the study of Runge-Kutta integration processes . J. Austral Math. Soc. , 3 : 185 – 201 .
  • Calvo , M. , Montijano , J. I. and Randez , L. 1990 . A fifth order interpolant for the Dormand & Prince Runge-Kutta method . J. Comput. Appl. Math. , 29 : 91 – 100 .
  • Dormand , J. R. and Prince , P. J. 1980 . A family of embedded formulae . J. Comp. Appl. Math. , 6 : 19 – 26 .
  • Dormand , J. R. and Prince , P. J. 1986 . Runge-Kutta Triples . Comput. Math. Appl. , 12A : 1007 – 1017 .
  • Enright , W. H. and Pryce , J. D. 1987 . Two FORTRAN packages for assessing initial value methods . ACM TOMS , 13 : 1 – 27 .
  • Enright , W. H. , Jackson , K. R. , Norsett , S. P. and Thomsen , P. G. 1986 . Interpolants for Runge-Kutta Formulas . ACM TOMS , 12 : 193 – 218 .
  • Gladwell , I. 1979 . Initial value routines in the NAG Library . ACM TOMS , 5 : 366 – 400 .
  • Gladwell , I. , Shampine , L. F. , Baca , L. S. and Brankin , R. W. 1987 . Practical aspects of interpolation in Runge-Kutta codes . SIAM J. Sci. Statist. Comput. , 8 : 322 – 341 .
  • Hall G. Enright W. H. Hull T. E. Sedgwick A. E. DETEST A program for comparing numerical methods for ordinary differential equations Dept. of Computer Science Tech. Rep., 60, University of Toronto 1973
  • Horn , M. K. 1983 . Fourth and fifth-order, scaled Runge-Kutta algorithms for treating dense output . SIAM J. Numer. Anal. , 20 : 558 – 568 .
  • Hull , T. E. , Enright , W. H. , Fellen , B. M. and Sedgwick , A. E. 1972 . Comparing numerical methods of ordinary differential equations . SIAM J. Numer. Anal. , 9 : 603 – 637 .
  • Owren B. Zennaro M. Derivation of efficient Continuous explicit Runge-Kutta methods Techn. Rep. No 240/90 Computer Science Department, University of Toronto, Canada 1990
  • Prince , P. J. and Dormand , J. R. 1981 . High order embedded Runge-Kutta formulae . J. Comput. Appl. Math. , 7 : 67 – 76 .
  • Shampine , L. F. 1985 . Interpolation for Runge-Kutta methods . SIAM J. Numer. Anal. , 22 : 1014 – 1027 .
  • Shampine , L. F. 1986 . Some practical Runge-Kutta formulas . Math. of Comput. , 46 : 135 – 150 .
  • Sharp , P. 1991 . Numerical Comparisons of some explicit R-K pairs of orders 4 through 8 . ACM TOMS , 17 : 387 – 409 .
  • Tsitouras Ch. Continuous extensions of Runge-Kutta type methods for intial value problems Doctoral Thesis, Deparatment of Mathematics, National Technical University of Athens Athens, Greece 1990 155
  • Tsitouras , Ch. and Papageorgiou , G. 1990 . Runge-Kutta interpolants based on values form two successive integration steps . Computing , 43 : 255 – 266 .
  • Verner , J. H. 1978 . Explicit Runge-Kutta methods with estimates of the local truncation error . SIAM J. Numer. Anal. , 15 : 772 – 790 .
  • Verner , J. H. 1993 . Differentiable interpolants for High-order Runge-Kutta methods . SIAM J. Numer. Anal. , 30 : 1446 – 1466 .
  • Zennaro , M. 1986 . Natural Continuous extensions of Runge-Kutta methods . Math. Comp. , 46 : 119 – 133 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.