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

Abstract

In this study, a trigonometrically adapted 6(4) explicit Runge-Kutta-Nyström (RKN) pair with six stages is formulated, considering a previous method developed by El-Mikkawy and Rahmo. The obtained adapted pair integrates exactly the usual test equation: 𝑦 ′′ = −w 2 𝑦. The local truncation error of the new method is presented, showing that the algebraic order of the original method is maintained. The periodicity interval of the new method is computed, showing that the developed method is "almost" P-stable. The numerical examples considered clearly show the superiority of the new developed embedded pair over other RKN methods of algebraic orders 6(4) with six stages appeared in the literature.

A new block method with variable stepsize implementation for solving third‐order differential systems

“…(15) is solved using three different RETOLs; as reported in Table 3, the proposed OHBM performs better than THBM and VSMM methods. Furthermore, Figure 2 shows the agreement between the exact solution and the discrete one obtained with OHBM using RETOL = 10 −10 and h 0 = 10 (16) with h 0 = 10 −1 .…”

“…Additional numerical techniques for solving () and related problems can be found in the literature. Examples of these methods include the harmonic balance method, spline method, energy conservation method, finite element method, boundary value method, multi‐derivative method, Trigonometric fitted method, embedded explicit method, direct integration Obrechkoff method, mesh‐free technique, Nyström method, and block method; see [13–33] for further details.…”

Fitted modifications of Runge–Kutta–Nyström pairs of orders 7(5) for addressing oscillatory problems

“…We tested pairs that use effectively six stages per step, namely, The six stages pair ER6(4) given in El‐Mikkawy and Rahmo 12 The six stages trigonometric fitted pair ER6(4)f given in Demba et al 13 The effectively six stages pair RKNST7(5) given in Simos and Tsitouras 14 The effectively six stages trigonometric fitted pair RKNST7(5)f presented here …”

Derivation of Embedded Diagonally Implicit Methods for Directly Solving Fourth-order ODEs

“…However, it would be more efficient if numerical methods could be used to solve the problem accurately and quickly. In [4][5][6][7][8][9][10][11][12] contain such works. Multistep strategies for solving ODEs require initial values.…”