Information Technology Reference
In-Depth Information
Tabl e 2. 1
The table shows the analytical, explicit and implicit solutions at time t
D
1 for problem
(
2.38
). Note that the implicit scheme provides reasonable approximations for any values of t ,
whereas the e
xplicit scheme requires t to be small in order to give solutions less th
an zero
N
t
y.1/
Explicit at t
D
1
Implicit at t
D
1
10
11
1
1;000
0:9091
1;000
0:9071
0:9111
10
11
1
100
0:9091
100
0:8891
0:9288
10
11
1
25
0:9091
25
0:9871
0:8256
10
11
1
12
0:9091
12
0:6239
1:0703
10
11
1
11
0:9091
11
0:4835
1:0848
10
11
1
10
0:9091
10
0:0
1:1022
10
11
1
9
0:9091
9
5:7500
1:1235
10
11
1
8
0:9091
10
3
8
6:4
1:1501
10
11
1
7
0:9091
10
7
7
1:8014
1:1843
10
11
1
5
0:9091
10
7
5
1:6317
1:2936
10
11
1
2
0:9091
2
840
1:8575
and consequently we must require that
1
C
ty
n
>0:
If we set n
D
0; we have y
0
D
10; and hence we require
1
10t > 0
which implies that
1
10
;
t <
or
N > 10:
The computations presented in the table shows that breaking this criterion for the
explicit scheme leads to completely erroneous numerical solutions.
The Implicit Scheme
For completeness, we also look a bit closer at the implicit scheme. We observed
above that the implicit scheme can be formulated as follows: Suppose
z
n
is given.
Then solve the equation
x
tx
2
D
z
n
(2.39)
