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Frequency evaluation for exponentially fitted Runge-Kutta methods. (English) Zbl 1077.65082

Summary: This paper can be seen as a further investigation of the frequency evaluation techniques which are very recently proposed by L. Gr. Ixaru, M. Rizea, G. Vanden Berghe, and H. De Meyer [ibid. 132, No. 1, 83–93 (2001; Zbl 0991.65061)], and by L. Gr. Ixaru, G. Vanden Berghe, and H. De Meyer [ibid. 140, No. 1–2, 423–434 (2002; Zbl 0996.65075; Comput. Phys. Commun. 150, No. 2, 116–128 (2003)] for exponentially fitted multistep algorithms for first-order ordinary differential equations. The question answered was how the frequencies should be tuned in order to have a maximal benefit from exponentially fitted methods.

In a previous paper by G. Vanden Berghe, L. Gr. Ixaru, and M. Van Daele [Comput. Phys. Commun. 140, No.3, 346–357 (2001; Zbl 0990.65080)] this frequency evaluation algorithm was successfully applied in a direct way to a second-order exponentially fitted Runge-Kutta (EFRK) method of collocation type but such a direct application is impossible for higher-order EFRK methods. To overcome this difficulty we develop an efficient extension of Ixaru’s frequency evaluation algorithm for the exponentially fitted RadauIIA method of third order. It is an adaption of Ixaru’s algorithm in the sense that instead being applied globally, it is applied stagewise. Numerical experiments illustrate the properties of the developed algorithm.

MSC:
65L06Multistep, Runge-Kutta, and extrapolation methods
65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general