Information Technology Reference
In-Depth Information
Finally, by a straightforward induction argument we conclude that
u
`
C
1
u
0
D
max
i
j
f.x
i
/
j
;
which is the result we were seeking.
Assume that the discretization parameters t and x satisfy
t
x
2
1
2
:
˛
D
(8.87)
Then the approximations generated by the explicit scheme (7.91)
satisfy the bound
j
u
i
j
max
i
max
i
j
f.x
i
/
j
for `
D
0;:::;m;
(8.88)
where f is the initial condition in the model problem (
8.1
)-(
8.3
).
Thus, if (8.87) is satisfied then the numbers generated by (7.91) will not “blow
up”; they will always be bounded by the magnitude of the initial condition. Further-
more, it seems that the scheme produces accurate approximations of the solution
of the diffusion equation, provided that t and x are small. On the other hand,
if this condition is not satisfied, then the experiments presented above indicate that
this method does not work. Thus, the scheme is only conditionally stable!
8.2.10
Consequences of the Stability Criterion
Let us have a closer look at the implications of this stability condition. Assume
that we use 11 grid points in the space dimension, i.e., n
D
11 (corresponding to
x
D
0:1). Then t must satisfy t
0:005, provided that T
D
1. This means
that the number of time steps m must be at least 200. More generally, m and n must
satisfy
m
2T .n
1/
2
:
Hence, the number of time steps needed increases rapidly with the number of grid
points used in the space dimension.
Assume that we want to compute an approximation of the solution of the diffu-
sion equation in the time interval [0,1], i.e., T
D
1. Then, for n
D
101, m must
satisfy m
20;000, and in the case of n
D
1;001 at least 2
10
6
time steps
must be taken! For the present model problem this causes no problems on mod-
ern computers. However, in many realistic simulation processes, involving two or
three space dimensions and large solution domains, explicit schemes of this type
tend to “stall” due to the large number of time steps needed. To solve such problems
