A robust trigonometrically fitted embedded pair for perturbed oscillators

Abstract

a b s t r a c tA new kind of trigonometrically fitted embedded pair of explicit ARKN methods for the numerical integration of perturbed oscillators is presented in this paper. This new pair is based on the trigonometrically fitted ARKN method of order five derived by Yang and Wu in [H.L. Yang, X.Y. Wu, Trigonometrically-fitted ARKN methods for perturbed oscillators, Appl. Numer. Math. 9 (2008) 1375-1395]. We analyze the stability properties, phase-lag (dispersion) and dissipation of the higher-order method of the new pair. Numerical experiments carried out show that our new embedded pair is very competitive in comparison with the embedded pairs proposed in the scientific literature.

An algorithm for second order initial and boundary value problems with an automatic error estimate based on a third derivative method

“…An error tolerance of 10 −11 was supplied to automatically generate the errors for specific values of N as given in Table 7. In Table 8, the results obtained in [10] for the ARKN5(3) are reproduced and compared to the TDM since their orders are very close. We remark that although the ARKN5(3) is expected to perform better because it was constructed using trigonometric basis functions and implemented as a variable-step method, the TDM is moderately competitive to ARKN5(3), especially as the step-size is decreased.…”

“…The problem was solved by Fang et al [10] using a variable step-size fifth-order trigonometrically fitted Runge-Kutta-Nyström method and a fifth-order Runge-Kutta-Nyström method (ARKN5(3)) which was constructed by Franco [11]. In Table 7, we show that the calculated ROC of the TDM is consistent with the theoretical order ( p = 6) behavior of the method, since on halving the step size, Err is reduced by a factor of about 2 6 .…”

“…We consider the nonlinear perturbed system which was also solved by Fang et al[10] on the range [0, 10], with ε = 10 −3 .…”

Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11

“…. , 12 are coefficients in R to be uniquely determined. To determine these coefficients, we interpolate at and +1 and then collocate at { + ℎ, = 0, 1, .…”

“…Special cases of (1) have also been extensively discussed in the literature ( [10][11][12][13][14], Hairer [15]). …”