Information Technology Reference
In-Depth Information
1.2
1.2
t=0
t =0.01875
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
0
0.2
0.6
0.8
1
0.4
0
0.2
0.4
0.6
0.8
1
1.2
1.2
t=0.09875
t=0.49875
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Fig. 7.14
Visualization of the solution
u
.x; t
`
/ of (
7.102
)-(
7.105
) at the time levels `
D
D
D
x
2
=2
1; 16; 80; 400 (
top left
to
bottom right
); x
0:05, t
The analytical solution to this initial-boundary value problem can be shown to be
u
.x; t /
D
e
9
2
t
sin.3x/:
(7.106)
In Fig.
7.15
we have graphed this function and the numerical results generated by
Algorithm
7.1
at t
D
10 for various values of the discretization parameters in space
and time:
-
The solid line represents the analytical solution.
-
The dotted line represents results obtained with x
D
1=9, t
D
10=17,and
consequently ˛
D
0:4765.
-
The dash-dotted line represents numbers generated with x
D
1=19, t
D
10=82, implying that ˛
D
0:4402.
-
The dashed line represents approximations obtained by setting x
D
1=59,
t
D
10=706, and thus ˛
D
0:4931.
In these cases the scheme produces well-behaved approximations, and the approx-
imations seem to converge toward the solution of the problem as t and x tend
toward zero.
Note that, in each of these plots, ˛
0:5 and t is relatively much smaller
than x. Is this necessary, or can we increase t ? Unfortunately, if t is increased
such that ˛>0:5, then this scheme tends to produce oscillating numbers, and
