On fitted modifications of Runge–Kutta–Nyström pairs

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

“…10 To obtain the adapted RKN pair, we take the coefficients of the lower order method in the RKN6(4)6 ER pair. Now we have to solve the system in ( 8)- (11) taking four of the coefficients as unknowns, specifically we take b1 , b2 , d1 , d2 as unknowns and obtain the following values:…”

“…When 𝜐 → 0, the obtained coefficients of the adapted fourth-order method become the constant coefficients of the counterpart method in the RKN6(4)6 ER approach. Similarly, for the sixth order method, if we consider as unknowns b 1 , b 3 , d 1 , d 2 in Equations ( 8)- (11), we obtain the following solution:…”

“…We compute the values of 𝑦 n , 𝑦 n+1 , 𝑦 ′ n , and 𝑦 ′ n+1 and introduce them in the equations of the method ( 5)- (7). If we use Euler's identity, e iv = cos(𝜐) + i sin(𝜐), and compare the real and imaginary parts of 𝑦 n+1 , 𝑦 ′ n+1 and 𝑦(x n+1 ), 𝑦 ′ (x n+1 ), respectively, we get the following system (see Tsitouras 11 ):…”

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

“…Alternatively, we chose to match all the entries of matrices $$$ R $$$ and $$$ \overline{R} $$$. We require then 9 $$$$ {\displaystyle \begin{array}{ccc}1+\zeta w{\left({I}_s-\zeta A\right)}^{-1}e& =& \cos \nu, \\ {}1+\zeta w{\left({I}_s-\zeta A\right)}^{-1}c& =& \frac{\sin \nu }{\nu },\\ {}\zeta {w}^{\prime }{\left({I}_s-\zeta A\right)}^{-1}e& =& -\nu \sin \nu, \\ {}1+\zeta {w}^{\prime}\left({I}_s-\zeta A\right)& =& \cos \nu \end{array}} $$$$ …”

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

“…Senu et al in [8] proposed an embedded explicit RKN method for solving oscillatory problems. Recently, 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.…”