Graphics Programs Reference
In-Depth Information
=
.
With
h
0
5weencounteredinstability, asexpected:
>>
x
y1
y2
0.0000e+000 -9.0000e+000
0.0000e+000
1.0000e+001
2.7030e+020 -2.5678e+021
7.5
Adaptive Runge-Kutta Method
Determination of a suitable step size
h
can be a major headache in numerical inte-
gration. If
h
is too large, the truncation error may be unacceptable;if
h
is too small, we
aresquandering computational resources. Moreover, a constant step size may not be
appropriate for the entire rangeofintegration. For example, if the solution curvestarts
off with rapid changes before becoming smooth (as in a stiff problem), we shoulduse
a small
h
at the beginning and increase it as we reach the smooth region. This is where
adaptivemethods
come in. They estimate the truncation errorateach integration step
and automatically adjust the step size to keep the errorwithin prescribed limits.
The adaptive Runge-Kuttamethods use so-called
embedded integration formulas
.
These formulascome in pairs: oneformula has the integration order
m
, the other
one isoforder
m
+
1. The ideaistouse both formulastoadvance the solution from
+
+
+
x
to
x
h
.Denoting the results by
y
m
(
x
h
) and
y
m
+
1
(
x
h
), we may estimate the
truncation errorinthe formula of order
m
as
E
(
h
)
=
y
m
+
1
(
x
+
h
)
−
y
m
(
x
+
h
)
(7.17)
What makes the embedded formulas attractive isthat they share the points where
F
(
x
h
)has been computed, relatively
small additional effort is required to calculate
y
m
+
1
(
x
,
y
) isevaluated. This meansthatonce
y
m
(
x
+
h
).
Here are the Runge-Kutta embedded formulasoforders 5 and 4that wereorigi-
nallyderivedbyFehlberg; hence theyareknown as
Runge-Kutta-Fehlberg formulas
:
+
K
1
=
,
h
F
(
x
y
)
h
F
x
B
i j
K
j
,
i
−
1
(7.1)
K
i
=
+
A
i
h
,
y
+
i
=
2
,
3
,...,
6
j
=
0
6
y
5
(
x
+
h
)
=
y
(
x
)
+
C
i
K
i
(5th-order formula)
(7.19a)
i
=
1
6
y
4
(
x
+
h
)
=
y
(
x
)
+
D
i
K
i
(4th-order formula)
(7.19b)
i
=
1
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