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

Abstract

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 functionally fitted three-stage explicit singly diagonally implicit Runge-Kutta method

“…In [8] and [9], Ai = 0 and Fz = A for all i, that is, there exist s unknowns in each of the simultaneous equations, and all the functions Om (t) (m = 1, ..., s) are used to determine these coefficients. Therefore, the resulting method is necessarily a fully implicit one.…”

“…[2], [7], [10], [11], [13], [14]). To be able to fit Runge-Kutta (-Nystrom) methods to any desired functions, Ozawa [8], [9] has recently developed a technique to construct the RungeKutta (-Nystrom) method that is exact on the linear space of given functions. When the functions are polynomials, the method reduces to the collocation Runge-Kutta methods.…”

“…The purpose of this work is to develop a computationally cheap Runge-Kutta method which are exact for given functions, by using the technique used in Ozawa ([8] and [9]). …”

Functionally-fitted explicit pseudo two-step Runge–Kutta–Nyström methods

“…[16]). Thus, the coefficients b(t, h), d(t, h), A(t, h) converge to those of the conventional EPTRKN method when h → 0.…”

“…For example, trigonometric methods have been developed to solve periodic or nearly periodic problems (see, e.g., [7], [11], [17]). Numerical experiments have shown that trigonometrically-fitted methods are superior to classical Runge-Kutta methods for solving ODEs whose solutions are periodic or nearly periodic functions with known frequencies (see, e.g., [8], [11], [12], [13], [14], [16], [17]). …”