Information Technology Reference
In-Depth Information
that finding the right scales is potentially very difficult, and that there are even some
non-trivial points in this simple introduction to diffusion PDEs.
Summary of the Scaled Problem
We can now summarize the equations in the scaled diffusion problem:
@
N
t
D
@
2
N
u
@
N
u
;
N
x
2
.0; 1/;
N
t>0;
(7.63)
@
N
x
2
N
u
.0;
N
t/
D
0;
N
t>0;
(7.64)
N
u
.1;
N
t/
D
1;
N
t>0;
(7.65)
N
u
.
N
x; 0/
D
0; 0
x
2
;
(7.66)
1;
2
<
N
x
1:
The most striking feature of (
7.63
)-(
7.66
)isthat
there are no physical parameters
,
no density, heat capacity, and so on. This means that
N
u
.
N
x;
N
t/ is in (
7.63
)-(
7.66
)
independent of %, c
v
, k, U
a
, U
b
, a,andb! More importantly, we can solve (
7.63
)-
(
7.66
) just once, and all information about the solution's dependence on the input
data such as k, %, and so on, must be inherited in
N
u
.
N
x;
N
t/ and the scaling. Suppose
we produce some curves
N
u
.
N
x;
N
t
`
/ for some time values
N
t
`
,say
N
t
1
D
0:1,
N
t
2
D
0:25,
and
N
t
3
D
1, as depicted in Fig.
7.10
. The physical temperatures
u
.x; t / are related to
the
N
u
.
N
x;
N
t
`
/ curves through the scaling; i.e., the real temperatures are
u
.x; t /
D
U
a
C
.U
b
U
a
/
N
u
x
a
:
tk
%c
v
.b
a/
2
b
a
;
(7.67)
This relation tells us everything about the influence of the original input data %, c
v
, k,
U
a
, U
b
, a,andb on the temperature. In other words, the 2,187 or 78,125 numerical
experiments we mentioned are reduced to the need for a single experiment! This
example should illustrate the usefulness of scaling.
Unfortunately, scaling away all the physical parameters is only possible in the
very simplest PDE problems, but frequently scaling helps reduce the number of
input data in the model, and hence the amount of numerical experiments.
It is tedious to write all the bars if we want to work further with the scaled prob-
lem. Therefore, simply dropping the bars is a common habit. We then write the
scaled PDE problem as
@t
D
@
2
u
@
u
;
x
2
.0; 1/; t > 0;
(7.68)
@x
2
u
.0; t /
D
0;
t > 0;
(7.69)
u
.1; t /
D
1;
t > 0;
(7.70)
u
.x; 0/
D
0; 0
x
c;
1;
N
c<x
1:
(7.71)
