In this paper, we propose a functional fitting s-stage Runge-Kutta method which is based on the exact integration of the set of the linearly independent functions pt(t), (i = 1, ... , s). The method is exact when the solution of the ODE can be expressed as the linear combination of cpt(t), although 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 order of accuracy of the method is at least s, if the functions w(t) are sufficiently smooth and the method is non-confluent. Furthermore, it is shown 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.

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A special class of Runge-Kutta(-Nystrom) methods called functionally fitted (or functional fitting) Runge-Kutta (FRK) methods has recently been proposed by the author. This class of methods is designed to be exact, if the solution of the equation to be solved is an element of the linear space of given functions, which is called the basis functions. The purpose of this article is to develop a functionally fitted Runge-Kutta method that is cheap to implement. The method proposed in this paper is a three-stage explicit singly diagonally implicit Runge-Kutta (ESDIRK) method, which requires one LU decomposition per step. This method is exact if the basis functions are properly chosen, and is moderately accurate even if the choice of the functions is inappropriate, since the method is shown to be of order 4 for general cases. An embedded pair of this type is also developed. Several numerical experiments show the superiority of the methods to conventional ones for the particular case that a suitable set of the basis functions can be found.

A new type of variable coefficient Runge-Kutta-Nyström methods is proposed for solving the initial value problems of the special form y"(t) = f (t, y(t)). The method is based on the exact integration of some given functions. If the second derivative of the solution is a linear combination of these functions, then the method is exact, and if this is not the case, the algebraic order (order of accuracy) of the method is very important. The algebraic order of the method is investigated by using the power series expansions of the coefficients, which are functions of the stepsize and independent variable t. It is shown that the method has the same algebraic order as those of the direct collocation Runge-Kutta-Nyström method by Van der Houwen et al., when the collocation points are identical with that method. Experimental results which demonstrate the validity of the theoretical analysis are presented.

A new implicit Runge-Kutta-Nyström method with variable coefficients is developed for solving the periodic initial value problem of the differential equation y" = f (t, y). The proposed method, whose coefficients are functions of the frequency and the stepsize, integrates exactly the equation, if the solution is a periodic function with a single Fourier component and the frequency is known. On the other hand, the order of accuracy of the method is shown to be 4 for the case that an estimated frequency, instead of the exact one, is applied to evaluate the coefficients, as well as for that the solution is non-periodic.