A Phase-Fitted and Amplification-Fitted Explicit Runge–Kutta–Nyström Pair for Oscillating Systems

Abstract

An optimized embedded 5(3) pair of explicit Runge–Kutta–Nyström methods with four stages using phase-fitted and amplification-fitted techniques is developed in this paper. The new adapted pair can exactly integrate (except round-off errors) the common test: y″=−w2y. The local truncation error of the new method is derived, and we show that the order of convergence is maintained. The stability analysis is addressed, and we demonstrate that the developed method is absolutely stable, and thus appropriate for solving stiff problems. The numerical experiments show a better performance of the new embedded pair in comparison with other existing RKN pairs of similar characteristics.

Numerical Solution to Sixth Order Ordinary Differential Equation Using Three Stage Eighth Order Runge-Kutta Type Method

“…We assume  to be an Taylor series increment function for which the local truncation errors of , , , , , will be calculated after substituting the exact solution of [1] into [9][10][11][12][13][14][15].…”

“…Further, substituting equations [16][17][18][19] into equations [9][10][11][12][13][14][15], the increment functions…”

“…In the current year, 2021 A. Abdi et al (1), B. Huang et al (2) had contributed for finding error analysis for linear and non-linear differential equations. M. A. Demba et al (11) optimized sixth-order explicit RKN method to solve oscillating systems. Mechee et al (12) generalized Runge-Kutta integrators for fifth order ordinary differential equation.…”

A trigonometrically adapted 6(4) explicit Runge–Kutta–Nyström pair to solve oscillating systems

“…Recently, Demba et al 14 constructed a pair of explicit exponentially adapted RKN methods for solving the problem in (). Later, Demba et al developed a new phase‐ and amplification‐fitted explicit RKN pair for solving the problem in () 15 . In this paper, we obtain a new trigonometrically adapted embedded pair of explicit RKN methods based on the 6(4) pair of explicit type presented by El‐Mikkawy and Rahmo 3 for solving ().…”

“…Later, Demba et al developed a new phaseand amplification-fitted explicit RKN pair for solving the problem in (1). 15 In this paper, we obtain a new trigonometrically adapted embedded pair of explicit RKN methods based on the 6(4) pair of explicit type presented by El-Mikkawy and Rahmo 3 for solving (1). The obtained pair can solve exactly the common test oscillator: 𝑦 ′′ = −w 2 𝑦.…”