A Zero-Dissipative Phase-Fitted Fourth Order Diagonally Implicit Runge-Kutta-Nyström Method for Solving 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
PFDIRKN4	∞	∞	—	—	—	—	(0, 4.02)
DIRKN4Senu	4	∞	2.05(−3)	1.48(−3)	2.23(−4)	—	(0, 8.20)
DIRKN4Senu2	6	5	1.88(−3)	1.70(−3)	4.71(−5)	1.19(−4)	(0, 8.10)
DIRKN4Franco	4	5	1.04(−3)	2.57(−3)	1.74(−3)	7.76(−3)	(0, ∞)
DIRKN4Raed	4	5	3.13(−2)	1.71(−2)	2.33(−2)	1.80(−2)	—
DIRKN4Sharp	6	5	1.85(−3)	6.26(−4)	2.82(−2)	1.03(−2)	(0, 12.00)
DIRKN4Som	4	5	6.35(−4)	1.59(−4)	3.18(−4)	1.43(−4)	(0, 9.51)

Note: ‖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.
1.23(−3)=1.23×10⁻³.