Frequency evaluation for exponentially fitted Runge–Kutta methods

This paper can be seen as a further investigation of the frequency evaluation techniques which are very recently proposed by Ixaru et al. for exponentially fitted multistep algorithms for first-order ordinary differential equations (ODEs). 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, 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 Ixaruʹs frequency evaluation algorithm for the exponentially fitted RadauIIA method of third order. It is an adaption of Ixaruʹs algorithm in the sense that instead being applied globally, it is applied stagewise. Numerical experiments illustrate the properties of the developed algorithm.