LECTURE 14
Numerical Integration
Numerical Integration proceeds from time
0
t
to the next time instant
h
t
t
+
=
0
where
h
is
the stepsize. For high accuracy, the stepsize must be kept small. In the RungeKutta
method of integrating
)
,
(
u
x
f
dt
x
d
=
for example, the right handside
)]
(
),
(
[
t
u
t
x
f
is
evaluated 4 times at each step. Usually a test is made by the software regarding the
accuracy of the predicted
)
(
0
h
t
x
+
. If unsatisfactory,
h
is halved and
)]
(
),
(
[
t
u
t
x
f
is
evaluated 4 times again and tested for accuracy again. The procedure is repeatedly until
the accuracy specified by the user is met. If the accuracy is too high,
h
is doubled. If the
satisfactory stepsize is
1
h
, the next step begins with time
1
0
h
t
t
+
=
and the numerical
integration predicts
)
(
2
1
0
h
h
t
x
+
+
. This continues until the endtime of the integration
specified by the user, at which point
)
....
(
2
1
0
N
h
h
h
t
x
+
+
+
is predicted.
A priori, it is not possible to know what stepsizes to use. Therefore, some exploratory
tests are required. This consists of carrying on the simulation with an educated guess of
h
.
Then
h
is reduced and if the second simulation shows that there are discrepancies,
h
must be reduced again and again until one is satisfied that
h
is fine enough.
Of course, one can carry on the reverse test by increasing
h
if there is reason to believe
that it is too fine and therefore wastefully time consuming.
Apart from accuracy, using too large a step size can meet numerical instability. Most of
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 Spring '09
 BonTekOoi
 Flux, Magnetic Field, Numerical Integration Numerical Integration

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