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Exponentially fitted Runge-Kutta methods. (English) Zbl 0999.65065

Summary: Exponentially fitted Runge-Kutta methods with \(s\) stages are constructed, which exactly integrate differential initial-value problems whose solutions are linear combinations of functions of the form \(\{x^j\exp(\omega x),x^j\exp(-\omega x)\}\) \((\omega\in{\mathbb R}\) or \(i{\mathbb R}, j=0,1,\dots,j_{\max})\) where \(0\leq j_{\max}\leq\lfloor s/2-1\rfloor\), the lower bound being related to explicit methods and the upper bound applicable for collocation methods. Explicit methods with \(s\in\{2,3,4\}\) belonging to that class are constructed. For these methods, a study of the local truncation error is made, from which follows a simple heuristic to estimate the \(\omega\)-value. Error and step length control are introduced based on Richardson extrapolation ideas. Some numerical experiments show the efficiency of the methods introduced. It is shown that the same techniques can be applied to construct implicit exponentially fitted Runge-Kutta methods.

MSC:

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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