Information Technology Reference
In-Depth Information
r
N
r
0
Tabl e 3. 2
The table shows
t , the number of time steps N , and the “
error”
r
0
r
N
r
0
t
N
r
0
10
1
10
2
10
16
2:6682
10
2
10
3
1:59986
10
17
10
3
10
4
10
17
3:97982
10
4
10
5
10
15
7:06021
for all n
0.InTable
3.2
we display the relative error defined by
r
N
r
0
r
0
:
(3.68)
We observe that the errors listed in Table
3.2
are much smaller than those in
Tab le
3.1
. Hence, we conclude that the Crank-Nicolson scheme generates better
solutions than that of the explicit scheme. In fact, it can be shown that the Crank-
Nicolson scheme produces the exact solution in this case. The errors in Table
3.2
are therefore due to round-off errors in the computations.
3.5
Exercises
Exercise 3.1.
Consider the system
F
0
D
.2
S/F; F.0/
D
F
0
;
(3.69)
S
0
D
.F
1/S; S.0/
D
S
0
;
and the numerical scheme
F
nC1
D
F
n
C
t.2
S
n
/F
n
;
(3.70)
S
nC1
D
S
n
C
t.F
n
1/S
n
:
(a) Write a computer program that implements the scheme (
3.70
). The program
should:
Accept S
0
, F
0
,andt as input.
Compute the numerical solution for t ranging from 0 to 10.
Plot the numerical solution both as a function of t and in the state space (the
F -S coordinate system).
(b) Use your program to determine a value of t such that if you reduce t , you
will not see any difference on your screen. In these computations you may use
F
0
D
1:9, S
0
D
0:1.
