A 5(4) Embedded Pair of Explicit Trigonometrically-Fitted Runge–Kutta–Nyström Methods for the Numerical Solution of Oscillatory Initial Value Problems

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.

Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems

“…Senu [8] proposed an embedded explicit RKN method for solving oscillatory problems, Fawzi et al [9] derived an embedded 6(5) pair of explicit Runge-Kutta methods for periodic ivps, Franco [10] developed two new embedded pairs of explicit Runge-Kutta methods adapted to the numerical solution of oscillatory problems, and Anastassi [11] constructed a 6(4) optimized embedded Runge-Kutta-Nyström pair for the numerical solution of periodic problems. Recently, Demba et al [12,13] constructed two new embedded explicit trigonometrically-fitted RKN methods for solving the problem in Equation (1).…”

Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

“…Among these time steppers, very wide-spread are the ones with variable coefficients, such as phase-fitted and/or amplification-fitted, trigonometrically-fitted etc. (e.g., [19][20][21][22][23][24][25]). This approach requires a well-defined dominant frequency of the oscillations along the propagation coordinate and, when this is not the case, they under-perform.…”

“…Another methodology that is used along with single-step methods is the step size control, which allows the method to automatically adjust the step size and thus reduce the computation effort [19][20][21][22][23]26,29,[31][32][33][34][35][36]. For some types of methods, the local error estimation can be performed using an embedded estimator, which has many advantages over other techniques, such as extrapolation [32].…”

Trigonometrically-fitted Higher Order Two Derivative Runge-Kutta Method for Solving Orbital and Related Periodical IVPs

“…Meanwhile, in [7] and [8], Demba et al derived four-stage fourth order explicit trigonometrically -fitted Runge-Kutta-Nyström (RKN) method and fifth-order four-stage explicit trigonometrically-fitted RKN method respectively for the numerical solution of secondorder initial value problems with oscillatory solutions based on Simos' RKN method. Other than that, Demba et al in [11] and [9] constructed embedded explicit 4(3) and 5(4) pairs trigonometrically-fitted RKN methods for solving oscillatory problems respectively. In [10], again Demba et al derived a symplectic third-order three-stage explicit trigonometrically-fitted RKN method for the numerical solution of second order initial value problems with periodic solutions.…”