Information Technology Reference
In-Depth Information
-
In the same way, we can show that if r
0
>R;then r
nC1
6
r
n
and
r
0
>
r
n
>
R:
(2.46)
-
Finally, we also note that if r
0
D
R; then
r
1
D
R
C
aRt
1
R
R
D
R;
and we can easily prove by induction that
r
n
D
R
(2.47)
for all n
>
0:
The Implicit Scheme
By proceeding as explained above, we can derive an implicit scheme
18
for the
logistic initial value problem. As usual, we replace the left hand side of
r
0
.t /
D
ar
.t /
1
r.t/
R
with a finite difference approximation, and we evaluate the right-hand side at time
t
nC1
: This gives the scheme
r
nC1
r
n
t
r
nC1
R
D
ar
nC1
.1
/;
which can be rewritten in the form
r
nC1
t
ar
nC1
.1
r
nC1
R
/
D
r
n
:
18
As discussed earlier, the prime reason for introducing implicit schemes is to get rid of a numer-
ical stability condition. We will see examples later on illustrating that this may be very important
and definitely worthwhile. However, for the logistic model this is not really a big issue, since the
condition
6
1=a
is not very strict for reasonable values of a. In fact, for practical computations we would probably
use such small time steps anyway. Since implicit schemes are harder to implement and analyze,
you may ask why we bother. The idea is this: By teaching both explicit and implicit schemes
for these simple models, it will be easier for you to understand implicit schemes for complicated
models when they are really needed. And when are they really needed? They are needed when
the restriction on the time step is so severe that we have to do millions, or perhaps even billions
or trillions, of time steps in order to compute an approximation of the solution. Then we have to
consider alternatives, and such alternatives almost always involve some sort of implicit procedure.
t
