| Title:
|
Construction of explicit and generalized Runge-Kutta formulas of arbitrary order with rational parameters (English) |
| Author:
|
Huťa, Anton |
| Author:
|
Strehmel, Karl |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
27 |
| Issue:
|
4 |
| Year:
|
1982 |
| Pages:
|
259-276 |
| Summary lang:
|
English |
| Summary lang:
|
Slovak |
| Summary lang:
|
Russian |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
explicit Runge-Kutta methods |
| Keyword:
|
ARK methods |
| Keyword:
|
S-stable |
| Keyword:
|
LS-stable |
| MSC:
|
65L05 |
| MSC:
|
65L20 |
| idZBL:
|
Zbl 0541.65047 |
| idMR:
|
MR0666905 |
| DOI:
|
10.21136/AM.1982.103971 |
| . |
| Date available:
|
2008-05-20T18:19:38Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103971 |
| . |
| Reference:
|
[1] J. C. Butcher: Implicit Runge-Kutta processes.Math. Соmр. 18, 50 (1964). Zbl 0123.11701, MR 0159424 |
| Reference:
|
[2] A. R. Curtis: An eight order Runge-Kutta process with eleven function evaluations per step.Numer. Math. 16, 268-277 (1970). MR 0270556, 10.1007/BF02219778 |
| Reference:
|
[3] G. Dahlquist: A special stability problem for linear multistep methods.BIT 3, 27-43 (1963). Zbl 0123.11703, MR 0170477, 10.1007/BF01963532 |
| Reference:
|
[4] B. L. Ehle, J. D. Lawson: Generalized Runge-Kutta processes for stiff initial-value problems.J. Inst. Math. Appl. 16, No. 1, 11-21 (1975). Zbl 0308.65046, MR 0391524, 10.1093/imamat/16.1.11 |
| Reference:
|
[5] A. Friedli: Verallgemeinertes Runge-Kutta-Verfahren zur Lösung steifer Differentialgleichungssysteme.Lect. Notes Math. 631, 35 - 50 (1978). MR 0494950, 10.1007/BFb0067462 |
| Reference:
|
[6] P. J. van der Houwen: Construction of integration formulas for initial value problems.Amsterdam: North Holland Publishing Company 1976. |
| Reference:
|
[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). |
| Reference:
|
[8] J. D. Lawson: Generalized Runge-Kutta processes for stable systems with large Lipschitz constants.SIAM J. Numer. Anal., Vol. 4, No. 3, 372-380 (1967). Zbl 0223.65030, MR 0221759, 10.1137/0704033 |
| Reference:
|
[9] K. Nickel, P. Rieder: Ein neues Runge-Kutta ähnliches Verfahren.In: ISNM 9, Numerische Mathematik, Differentialgleichungen, Approximationstheorie, 83 - 96, Basel: Birkhäuser 1968. Zbl 0174.47304, MR 0266436 |
| Reference:
|
[10] E. J. Nyström: Über die numerische Integration von Differentialgleichungen.Acta Soc. Sci. Fennicae, Tom 50, nr. 13, 1-55 (1925). |
| Reference:
|
[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). Zbl 0309.65034, MR 0331793 |
| Reference:
|
[12] K. Strehmel: Konstruktion von adaptiven Runge-Kutta-Methoden.ZAMM, to appear 1980. |
| Reference:
|
[13] J. G. Verwer: S-stability properties for generalized Runge-Kutta methods.Numer. Math. 27,359-370(1977). Zbl 0336.65036, MR 0438722 |
| Reference:
|
[14] J. G. Verwer: Internal S-stability for generalized Runge-Kutta methods.Report NW 21, Mathematisch Centrum, Amsterdam (1975). Zbl 0319.65044 |
| . |
