A functional fitting Runge-Kutta-Nyström method with variable coefficients

A new type of variable coefficient Runge-Kutta-Nyström methods is proposed for solving the initial value problems of the special formy″(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 variablet. It is shown that the method has the same algebraic order as those of the direct collocation Runge-Kutta-Nyström method by Van der Houwenet al., when the collocation points are identical with that method. Experimental results which demonstrate the validity of the theoretical analysis are presented.