Summary
In this paper we develop a general theory of implicit extrapolation methods for ordinary differential equations. We demonstrate how to implement such methods in practice. We first consider the common principles of constructing extrapolation methods. After that we discuss implicit extrapolation methods. Then we derive quadratic extrapolation for implicit one-step methods possessing a quadratic asymptotic expansion of the global error Finally, we present a brief outline of a theory of minimally implicit methods that extends the concept of linearly implicit methods to arbitiary iterative methods. In addition, the paper gives numerical examples which clearly confirm all theoretical results.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Hairer, E., Nørsett, S.R., Wanner, G. (1987): Solving ordinary differential equations. I. Nonstiff problems. Springer, Berlin
Hairer, E., Wanner, G. (1996): Solving ordinary differential equations. II Stiff and differential-algebraic problems. Springer, Berlin
Kulikov, G.Yu. (1998): Numerical solution of the Cauchy problem for a system of differential-algebraic equations with the use of implicit Runge-Kutta methods with a nontrivial predictor (Russian) Zh. Vychisl. Mat. Mat. Fiz. 38, 68–84; translation in Comput. Math. Math. Phys. 38, 64–80
Kulikov, G.Yu. (1998) Asymptotic error estimates for the method of simple iterations and for the modified and generalized Newton methods. (Russian) Mat. Zametki 63, 562–571; transladon in Math. Notes 63, 494–502
Kulikov, G.Yu. (1998): Revision of the theory of symmetric one-step methods for ordinary differential equations. Korean J. Comput. Appl. Math. 5, 579–600
Kulikov, G.Yu. (2002): On implicit extrapolation methods for ordinary differential equations. Russian J. Numer. Anal. Math. Modelling 17, 41–69
Kulikov, G.Yu. (2003): Symmetric Runge-Kutta methods and their stability. Russian J. Numer. Anal. Math. Modelling, to appear
Kulikov, G.Yu. (2003): On stability of symmetric Runge-Kutta formulas. (Russian) Dokl. Akad. Nauk, to appear
Ortega, J.M., Rheinboldt, W.C. (1970): Iterative solution of nonlinear equations in several variables. Academic Press, New York
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Italia
About this paper
Cite this paper
Kulikov, G.Y. (2003). Theory of implicit extrapolation methods for ordinary differential equations. In: Brezzi, F., Buffa, A., Corsaro, S., Murli, A. (eds) Numerical Mathematics and Advanced Applications. Springer, Milano. https://doi.org/10.1007/978-88-470-2089-4_82
Download citation
DOI: https://doi.org/10.1007/978-88-470-2089-4_82
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-2167-9
Online ISBN: 978-88-470-2089-4
eBook Packages: Springer Book Archive