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Frequency determination and step-length control for exponentially-fitted Runge-Kutta methods. (English) Zbl 0991.65062

Summary: An exponentially fitted Runge-Kutta (EFRK) fifth-order method with six stages is constructed, which exactly integrates first-order differential initial-value problems whose solutions are linear combinations of functions of the form \(\{\exp(\omega x),\exp(-\omega x)\}\), (\(\omega\in\mathbb{R}\) or \(i\mathbb{R}\)). By combining this EFRK method with an equivalent classical embedded (4,5) Runge-Kutta method, a technique is developed for the estimation of the occurring \(\omega\)-values. Error and step-length control is carried out by using the Richardson extrapolation procedure. Some numerical experiments show the efficiency of the introduced methods.

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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