New Phase-Fitted and Amplification-Fitted Fourth-Order and Fifth-Order Runge-Kutta-Nyström Methods for Oscillatory Problems

Table 1

Summary of the properties of the methods.

Method	𝑞	𝑟	‖𝜏^(𝑝+1)‖2	‖𝜏^′(𝑝+1)‖2	DPC	DSC	S.I/P.I
SRKN3V	6	∞	2.750(−3)	9.007(−4)	2.126(−4)	—	(0, 7.57)
PFERKN4P	∞	5	—	—	—	—	—
OPTRKN4P	∞	∞	1/2160	—	—	—	—
RKN4G	4	5	1/1440	3.130(−3)	1/640	1/1440	(0, 8.77)
PFAFRKN4	∞	∞	1/1440	—	—	—	(0, 8.94)
OPTRKN5K1	∞	∞	—	1.598(−4)	—	—	—
OPTRKN5K2	∞	∞	—	1.598(−4)	—	—	—
RKN5H	6	5	1/21600	1.449(−4)	1/6720	1/3600	—
PFAFRKN5	∞	∞	—	—	—	—	(0, 8.01)

Note: 2.750(−3)=2.750×10⁻³.
‖𝜏^(𝑝+1)‖₂ is principal local truncation error constant for 𝑦.
‖𝜏^′(𝑝+1)‖₂ is principal local truncation error constant for 𝑦^′.
DPC is dispersion constant.
DSC is dissipation constant.
P.I is periodicity interval.
S.I is stability interval.