Summary
GeneralizedA(α)-stable Runge-Kutta methods of order four with stepsize control are studied. The equations of condition for this class of semiimplicit methods are solved taking the truncation error into consideration. For application anA-stable and anA(89.3°)-stable method with small truncation error are proposed and test results for 25 stiff initial value problems for different tolerances are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. New York: Dover Publ. Inc. 1970
Bedet, R.A., Enright, W.H., Hull, T.E.: STIFF DETEST: A program for comparing numerical methods for stiff ordinary differential equations. Tech. Rep. No. 81, University of Toronto 1975
Bulirsch, R., Stoer, J.: Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math.8, 1–13 (1966)
Burrage, K.: A special family of Runge-Kutta methods for solving stiff differential equations. BIT18, 22–41 (1978)
Dahlquist, G.: A special property for linear multistep methods. BIT3, 27–43 (1963)
Diekhoff, H.-J., Lory, P., Oberle, H.J., Pesch, H.-J., Rentrop, P., Seydel, R.: Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting. Numer. Math.27, 449–469 (1977)
Enright, W.H., Bedet, R., Farkas, I., Hull, T.F.: Test results on initial value methods for non-stiff ordinary differential equations. Techn. Rep. No. 68, University of Toronto 1974
Enright, W.H., Hull, T.E., Lindberg, B.: Comparing numerical methods for stiff systems of ordinary differential equations. Tech. Rep. No. 69, 1974 University of Toronto, see also BIT15, 10–48 (1975)
Gear, C.W.: Numerical initial value problems in ordinary differential equations. N.Y.: Prentice Hall 1970
Grigorieff, R.D.: Numerik gewöhnlicher Differentialgleichungen. Stuttgart: Teubner 1972
Hairer, E., Wanner, G.: On the Butcher group and general mutivalue methods. Computing13, 1–15 (1974)
Hindmarsh, A.C.: GEAR — ordinary differential equation system solver. UCID-30001, Rev. 2, University of California: Lawrence Livermore Laboratory 1972
Kaps, P.: Modifizierte Rosenbrockmethoden der Ordnung 4, 5 und 6 zur numerischen Integration steifer Differentialgleichungen. Dissertation, Universität Innsbruck, September 1977
Kaps, P., Wanner, G.: Rosenbrock-type methods of high order. In press (1979)
Nørsett, S.P.,C-polynomials for rational approximation to the exponential function. Numer. Math.25, 39–56 (1975)
Nørsett, S.P., Wanner, G.: The real-pole sandwich for rational approximations and oscillation equations. BIT19, 79–94 (1979)
Nørsett, S.P., Wolfbrandt, A.: Order conditions for Rosenbrock-type methods. Numer. Math.32, 1–15 (1979)
Rosenbrock, H.H.: Some general implicit processes for the numerical solution of differential equatons. Comp. J.5, 329–331 (1963)
Stoer, J., Bulirsch, R.: Einführung in die Numerische Mathematik II. Berlin, Heidelberg, New York: Springer Verlag, 2. Auflage, 1978
Wanner, G.: Letter to S.P. Nørsett and unpublished communications to P. Kaps and A. Wolfbrandt, 1973
Wolfbrandt, A.: A study of Rosenbrock processes with respect to order conditions and stiff stability. Dissertation, Research Rep. 77.01 R, University of Göteborg, March 1977
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kaps, P., Rentrop, P. Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations. Numer. Math. 33, 55–68 (1979). https://doi.org/10.1007/BF01396495
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01396495