A functional fitting Runge-Kutta method with variable coefficients

In this paper, we propose a functional fitting s-stage Runge-Kutta method w hich is based on the exact integration of the set of the linearly independe nt functions phi (i)(t), (i = 1,...,s). The method is exact when the soluti on of the ODE can be expressed as the linear combination of phi (i)(t), alt hough the method has an error for general ODE. In this work we investigate the order of accuracy of the method for general ODEs, and show that the ord er of accuracy of the method is at least s, if the functions phi (i)(t) are sufficiently smooth and the method is non-confluent. Furthermore, it is sh own that the attainable order of the method is 2s, like conventional Runge- Kutta methods. Two- and three-stage methods including embedded one of this type are developed.