Abstract
A 5(4) pair of embedded explicit trigonometrically-fitted Runge–Kutta–Nyström (EETFRKN) methods especially designed for the numerical integration of second order initial value problems with oscillatory solutions is presented in this paper. Algebraic order analysis and the interval of absolute stability for the new method are also discussed. The new method is capable of integrating the test equation y ″ = − w 2 y . The new method is much more efficient than the other existing Runge–Kutta and Runge–Kutta–Nyström methods.
Highlights
Our focus is on the numerical solution of the initial value problems (IVPs) of second-order differential equations; whose first derivative does not appear explicitly of the form : y = f (x, y), y(x0) = y0, x ∈ [x0, X], y (x0) = y0, (1)
ERKN4(3): The embedded Runge–Kutta–Nyström method obtained by Van de Vyver in [6]
ARKN5(3)S: A 5(3) pair of explicit adapted Runge–Kutta–Nyström method derived by Franco in [5]
Summary
Our focus is on the numerical solution of the initial value problems (IVPs) of second-order differential equations; whose first derivative does not appear explicitly of the form :. Franco in [5] proposed a 5(3) pair of explicit adapted Runge–Kutta–Nyström methods for the numerical integration of perturbed oscillators, Van de Vyver in [6,7] proposed a Runge–Kutta–Nyström pair for the numerical integration of perturbed oscillators, a 5(3) pair of explicit RKN methods for oscillatory problems. Tsitouras in [9] proposed fitted modifications of RKN pairs, Franco et al in [10] proposed two new embedded pair of explicit Runge-Kutta methods adapted to the numerical solution of oscillatory problems, and Anastassi and Kosti in [11] proposed a 6(4) optimized embedded Runge–Kutta–Nyström pair for the numerical solution of periodic problems. We develop a new efficient embedded explicit trigonometrically-fitted RKN method based on the technique proposed by Simos in [12] for Runge–Kutta (RK) methods.
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