Construction of explicit and generalized Runge-Kutta formulas of arbitrary order with rational parameters

Anton Huťa; Karl Strehmel

Aplikace matematiky (1982)

  • Volume: 27, Issue: 4, page 259-276
  • ISSN: 0862-7940

Abstract

top
In the article containing the algorithm of explicit generalized Runge-Kutta formulas of arbitrary order with rational parameters two problems occuring in the solution of ordinary differential equaitions are investigated, namely the determination of rational coefficients and the derivation of the adaptive Runge-Kutta method. By introducing suitable substitutions into the nonlinear system of condition equations one obtains a system of linear equations, which has rational roots. The introduction of suitable symbols enables the authors to generalize the Runge-Kutta formulas. The starting point for the construction of adaptive R. K. method was the consistent s -stage R. K. formula. Finally, the S-stability of the ARK method is investigated.

How to cite

top

Huťa, Anton, and Strehmel, Karl. "Construction of explicit and generalized Runge-Kutta formulas of arbitrary order with rational parameters." Aplikace matematiky 27.4 (1982): 259-276. <http://eudml.org/doc/15247>.

@article{Huťa1982,
abstract = {In the article containing the algorithm of explicit generalized Runge-Kutta formulas of arbitrary order with rational parameters two problems occuring in the solution of ordinary differential equaitions are investigated, namely the determination of rational coefficients and the derivation of the adaptive Runge-Kutta method. By introducing suitable substitutions into the nonlinear system of condition equations one obtains a system of linear equations, which has rational roots. The introduction of suitable symbols enables the authors to generalize the Runge-Kutta formulas. The starting point for the construction of adaptive R. K. method was the consistent $s$-stage R. K. formula. Finally, the S-stability of the ARK method is investigated.},
author = {Huťa, Anton, Strehmel, Karl},
journal = {Aplikace matematiky},
keywords = {explicit Runge-Kutta methods; ARK methods; S-stable; LS-stable; explicit Runge-Kutta methods; ARK methods; S-stable; LS-stable},
language = {eng},
number = {4},
pages = {259-276},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Construction of explicit and generalized Runge-Kutta formulas of arbitrary order with rational parameters},
url = {http://eudml.org/doc/15247},
volume = {27},
year = {1982},
}

TY - JOUR
AU - Huťa, Anton
AU - Strehmel, Karl
TI - Construction of explicit and generalized Runge-Kutta formulas of arbitrary order with rational parameters
JO - Aplikace matematiky
PY - 1982
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 27
IS - 4
SP - 259
EP - 276
AB - In the article containing the algorithm of explicit generalized Runge-Kutta formulas of arbitrary order with rational parameters two problems occuring in the solution of ordinary differential equaitions are investigated, namely the determination of rational coefficients and the derivation of the adaptive Runge-Kutta method. By introducing suitable substitutions into the nonlinear system of condition equations one obtains a system of linear equations, which has rational roots. The introduction of suitable symbols enables the authors to generalize the Runge-Kutta formulas. The starting point for the construction of adaptive R. K. method was the consistent $s$-stage R. K. formula. Finally, the S-stability of the ARK method is investigated.
LA - eng
KW - explicit Runge-Kutta methods; ARK methods; S-stable; LS-stable; explicit Runge-Kutta methods; ARK methods; S-stable; LS-stable
UR - http://eudml.org/doc/15247
ER -

References

top
  1. J. C. Butcher, Implicit Runge-Kutta processes, Math. Соmр. 18, 50 (1964). (1964) Zbl0123.11701MR0159424
  2. A. R. Curtis, 10.1007/BF02219778, Numer. Math. 16, 268-277 (1970). (1970) Zbl0194.18902MR0270556DOI10.1007/BF02219778
  3. G. Dahlquist, 10.1007/BF01963532, BIT 3, 27-43 (1963). (1963) Zbl0123.11703MR0170477DOI10.1007/BF01963532
  4. B. L. Ehle, J. D. Lawson, 10.1093/imamat/16.1.11, J. Inst. Math. Appl. 16, No. 1, 11-21 (1975). (1975) Zbl0308.65046MR0391524DOI10.1093/imamat/16.1.11
  5. A. Friedli, 10.1007/BFb0067462, Lect. Notes Math. 631, 35 - 50 (1978). (1978) MR0494950DOI10.1007/BFb0067462
  6. P. J. van der Houwen, Construction of integration formulas for initial value problems, Amsterdam: North Holland Publishing Company 1976. (1976) Zbl0359.65057
  7. A. Huťa, The algorithm for computation of the n-th order formula for numerical solution of initial value problem of differential equations, 5th Symposium on Algorithms, 53 - 61, (І979). Zbl0449.65049
  8. J. D. Lawson, 10.1137/0704033, SIAM J. Numer. Anal., Vol. 4, No. 3, 372-380 (1967). (1967) Zbl0223.65030MR0221759DOI10.1137/0704033
  9. K. Nickel, P. Rieder, Ein neues Runge-Kutta ähnliches Verfahren, In: ISNM 9, Numerische Mathematik, Differentialgleichungen, Approximationstheorie, 83 - 96, Basel: Birkhäuser 1968. (1968) Zbl0174.47304MR0266436
  10. E. J. Nyström, Über die numerische Integration von Differentialgleichungen, Acta Soc. Sci. Fennicae, Tom 50, nr. 13, 1-55 (1925). (1925) Zbl51.0427.01
  11. A. Prothero, A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Соmр. 28, 145-162 (1974). (1974) Zbl0309.65034MR0331793
  12. K. Strehmel, Konstruktion von adaptiven Runge-Kutta-Methoden, ZAMM, to appear 1980. (1980) 
  13. J. G. Verwer, S-stability properties for generalized Runge-Kutta methods, Numer. Math. 27,359-370(1977). (1977) Zbl0336.65036MR0438722
  14. J. G. Verwer, Internal S-stability for generalized Runge-Kutta methods, Report NW 21, Mathematisch Centrum, Amsterdam (1975). (1975) Zbl0319.65044

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.