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Zbl 0999.65065
Vanden Berghe, G.; De Meyer, H.; Van Daele, M.; Van Hecke, T.
Exponentially fitted Runge-Kutta methods. (English)
[J] J. Comput. Appl. Math. 125, No.1-2, 107-115 (2000). ISSN 0377-0427

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 $\{x^j\exp(\omega x),x^j\exp(-\omega x)\}$ $(\omega\in{\Bbb R}$ or $i{\Bbb R}, j=0,1,\dots,j_{\max})$ 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 2000:: *65L06 Multistep, Runge-Kutta, and extrapolation methods
65L20 Stability of numerical methods for ODE

Cited in: Zbl 1121.65332 Zbl 1014.65061

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