SEC.9.5 RUNGE-KUTTA METHODS 497
Runge-Kutta-Fehlberg Method (RKF45)
One way to guarantee accuracy in the solution of an I.V.P. is to solve the problem twice
using step sizes h and h/2 and compare answers at the mesh points corresponding to
the larger step size. But this requires a significant amount of computation for the
smaller step size and must be repeated if it is determined that the agreement is not
good enough.
The Runge-Kutta-Fehlberg method (denoted RKF45) is one way to try to resolve
this problem. It has a procedure to determine if the proper step size h is being used. At
each step, two different approximations for the solution are made and compared. If the
two answers are in close agreement, the approximation is accepted. If the two answers
do not agree to a specified accuracy, the step size is reduced. If the answers agree to
more significant digits than required, the step size is increased.
Each step requires the use of the following six values:
(28)
k
1
= hf(t
k
, y
k
),
k
2
= hf
t
k
+
1
4
h, y
k
+
1
4
k
1
,
k
3
= hf
t
k
+
3
8
h, y
k
+
3
32
k
1
+
9
32
k
2
,
k
4
= hf
t
k
+
12
13
h, y
k
+
1932
2197
k
1
−
7200
2197
k
2
+
7296
2197
k
3
,
k
5
= hf
t
k
+ h, y
k
+
439
216
k
1
− 8k
2
+
3680
513
k
3
−
845
4104
k
4
,
k
6
= hf
t
k
+
1
2
h, y
k
−
8
27
k
1
+ 2k
2
−
3544
2565
k
3
+
1859
4104
k
4
−
11
40
k
5
.
Then an approximation to the solution of the I.V.P. is made using a Runge-Kutta
method of order 4:
(29) y
k+1
= y
k
+
25
216
k
1
+
1408
2565
k
3
+
2197
4101
k
4
−
1
5
k
5
,
where the four function values f
1
, f
3
, f
4
, and f
5
are used. Notice that f
2
is not used
in formula (29). A better value for the solution is determined using a Runge-Kutta
method of order 5:
(30) z
k+1
= y
k
+
16
135
k
1
+
6656
12,825
k
3
+
28,561
56,430
k
4
−
9
50
k
5
+
2
55
k
6
.
The optimal step size sh can be determined by multiplying the scalar s times the
current step size h. The scalar s is
(31) s =
tol h
2|z
k+1
− y
k+1
|
1/4
≈ 0.84
tol h
|z
k+1
− y
k+1
|
1/4
