Construction of explicit and generalized Runge-Kutta formulas of arbitrary order with rational parameters. (English) Zbl 0541.65047

The construction of explicit Runge-Kutta methods is considered with the requirement that all coefficients should be rational. It is shown how to generalize this construction for the differential equation \(y'=Ay+g(x,y)\) in which the coefficients of the method are dependent on the matrix A. Stability questions for these so called ARK methods, where A is taken to be the Jacobian matrix, are investigated. In particular it is shown that all methods of this type are S-stable and conditions are derived under which they are also LS-stable.
Reviewer: J.C.Butcher


65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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[1] J. C. Butcher: Implicit Runge-Kutta processes. Math. Comp. 18, 50 (1964). · Zbl 0123.11701
[2] A. R. Curtis: An eight order Runge-Kutta process with eleven function evaluations per step. Numer. Math. 16, 268-277 (1970). · Zbl 0194.18902
[3] G. Dahlquist: A special stability problem for linear multistep methods. BIT 3, 27-43 (1963). · Zbl 0123.11703
[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
[5] A. Friedli: Verallgemeinertes Runge-Kutta-Verfahren zur Lösung steifer Differentialgleichungssysteme. Lect. Notes Math. 631, 35 - 50 (1978).
[6] P. J. van der Houwen: Construction of integration formulas for initial value problems. Amsterdam: North Holland Publishing Company 1976. · Zbl 0359.65057
[7] A. Hu\?a: 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)
[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
[9] K. Nickel, P. Rieder: Ein neues Runge-Kutta ähnliches Verfahren. ISNM 9, Numerische Mathematik, Differentialgleichungen, Approximationstheorie, 83 - 96, Basel: Birkhäuser 1968. · Zbl 0174.47304
[10] E. J. Nyström: Über die numerische Integration von Differentialgleichungen. Acta Soc. Sci. Fennicae, Tom 50, nr. 13, 1-55 (1925). · JFM 51.0427.01
[11] A. Prothero, A. Robinson: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comp. 28, 145-162 (1974). · Zbl 0309.65034
[12] K. Strehmel: Konstruktion von adaptiven Runge-Kutta-Methoden. ZAMM, to appear 1980.
[13] J. G. Verwer: S-stability properties for generalized Runge-Kutta methods. Numer. Math. 27,359-370(1977). · Zbl 0336.65036
[14] J. G. Verwer: Internal S-stability for generalized Runge-Kutta methods. Report NW 21, Mathematisch Centrum, Amsterdam (1975). · Zbl 0319.65044
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