Information Technology Reference
In-Depth Information
We note that in order to compute r
nC1
for a given value of r
n
; we have to solve a
second-order polynomial equation. In Exercise
2.6
below, see page
64
, we show that
this scheme is unconditionally stable and that it mimics the properties of the exact
solution in the same manner as we showed for the explicit scheme above.
Numerical Experiments
Since we have the exact solution of the logistic model, we are able to investigate
the accuracy of the numerical schemes by experiments. But let us first simplify the
equation a bit. Let s be the scaled number of rabbits defined as
r.t/
R
s.t/
D
:
Recall that
r
0
.t /
D
ar
.t /
1
:
r.t/
R
Since
r.t/
D
Rs
.t /;
it follows that
r
0
.t /
D
Rs
0
.t /;
and consequently
Rs
0
.t /
D
aRs
.t /
1
Rs
.t /
R
or
s
0
.t /
D
as
.t /.1
s.t //:
For simplicity,
19
we also choose a
D
1; and thus consider the model
s
0
.t /
D
s.t /.1
s.t //;
(2.48)
s.0/
D
s
0
:
For this model, we have the explicit scheme
y
nC1
D
y
n
C
ty
n
.1
y
n
/
(2.49)
19
We can also get rid of a by a scaling of time. Set
D
at
and
u
. /
D
s.t/: Then
du
. /
d
ds
.t /
dt
dt
d
D
1
a
ds
.t /
dt
1
a
.
as
.t /.1
D
D
s.t ///;
so we get the equation
u
0
. /
D
u
. /.1
u
. //:
