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Practical Runge-Kutta processes. (English) Zbl 0704.65053

An embedded Runge-Kutta process is a combination of two Runge-Kutta formulae of different order. The paper contains a review of results on such processes, especially developed by the authors [J. Comput. Appl. Math. 6, 19-26 (1980; Zbl 0448.65045)]. An estimation of the local error and a control of the step size is possible using the different approximations given by the two formulae.
Also, the global error can be estimated using an idea of P. E. Zadunaisky [Proc. Int. Astronaut. Union 25, 281-287, Acad. Press (1966)] to solve a neighbouring problem with known solution by the same process. Moreover, the global error can be estimated by solving the error equation [cf. R. D. Skeal, Numer. Math. 48, 1-20 (1986; Zbl 0562.65050)].
By a third Runge-Kutta formula values of the solution at intermediate points can be gained. By using this dense output the construction of an interpolating polynomial as a base for the error estimation is possible. The paper contains also test results and an extension to Runge-Kutta- Nyström processes for differential equations of second order.
Reviewer: K.-H.Bachmann

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
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