
Fourth-order RK formulas are the most common of all the RK numerical integrators for reasons
we shall discuss shortly. The derivation is patterned after the approach used for the lower-order RK
methods. Flexibili ty in the choice of several parameters results in a family of RK-4 integrators.
A popular RK-4 integrator is illustrated in Figure 6.5.
The derivative function evaluations are computed according to
k
1
¼ f [t
i
, x
A
(i)], x
A
i þ
1
2
¼ x
A
(i) þ
T
2
k
1
(6:60)
k
2
¼ ft
iþ1=2
, x
A
i þ
1
2
,
^
x
A
i þ
1
2
¼ x
A
(i) þ
T
2
k
2
(6:61)
k
3
¼ ft
iþ1=2
,
^
x
A
i þ
1
2
,
^
x
A
(i þ 1) ¼ x
A
(i) þ Tk
3
(6:62)
k
4
¼ f [t
iþ1
,
^
x
A
(i þ 1)] (6:63)
and the updated state x
A
(i þ1) is obtained from
x
A
(i þ 1