Information Technology Reference
In-Depth Information
and the implicit scheme
z
nC1
t
z
nC1
.1
z
nC1
/
D
z
n
:
(2.50)
Both schemes are started using the initial condition
y
0
D
z
0
D
s
0
where s
0
is given. By solving the algebraic equation (2.50), we find that the implicit
scheme can be written as
1
C
t
C
p
.1
t/
2
C
4t
z
n
:
1
2t
z
nC1
D
(2.51)
In Fig.
2.5
we show the numerical solutions as time ranges from 0 to 10: We h ave
used N
D
100; so t
D
1=10: We have also plotted the analytical solution given
by
s
0
s
0
C
e
t
.1
s
0
/
;
s.t/
D
(2.52)
with s
0
D
0:2: Note that the three solutions are more or less indistinguishable.
We also recognize the properties of the logistic model discussed above. For a short
Explicit, Implicit and Exact solution, N =100, T = 10.0, s0 = 0.2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
1
2
3
4
5
6
7
8
9
10
t
Fig. 2.5
The figure shows the analytical solution and two numerical approximations using
t
D
1=20. Both numerical solutions provide excellent approximations to the analytical solution
