Information Technology Reference
In-Depth Information
As an example, consider the function
f.x/
D
A
C
B sin.!x/ :
Since f lies between A
C
B and A
B (the absolute value of the sine function is
at most one), a characteristic reference value f
r
is the average A. The characteristic
magnitude of f
A is B so we can introduce the scaling
f.x/
D
f.x/
A
B
D
sin !x :
We clearly see that max
x
j
f.x/
jD
1, as desired. In f.x/there is only one param-
eter left, !, i.e., A and B are in a sense “parameterized away”. No information is
lost, since we can easily construct f on the basis of f : f.x/
D
A
C
B f.x/.Inthis
example the reference value and the characteristic magnitude were obvious. Finding
the characteristic magnitude is occasionally quite demanding in PDE problems, and
it requires quite a bit of physical insight and approximate mathematical analysis of
the problem.
Let us apply a scaling procedure to (
7.52
)-(
7.55
). Our aim to is scale x, t ,
u
,and
I such that these variables have a magnitude between zero and order unity. The x
parameter varies between a and b,so
N
x
D
x
a
b
a
is a scaling where the scaled parameter
N
x varies between 0 and 1, as desired. Since
the initial time is 0, the reference value is 0. The characteristic magnitude of time,
call it t
c
, is more problematic. This t
c
can be the time it takes to experience signif-
icant changes in
u
. More analysis is needed to find candidate values for t
c
; thus we
just scale t as
N
t
D
t
t
c
;
and remember that t
c
must be determined later in the procedure. The function I.x/
is easy to scale since it has only two values. We can take U
a
as a reference value
and U
b
U
a
as a characteristic magnitude:
I.
N
x/
D
I.
N
xL/
U
a
U
b
U
a
:
Finally we need to scale
u
. Initially we could use the same scaling as for I ,but,
as time increases, the PDE governs the magnitude of
u
. Without any idea of how
the solution of the PDE behaves, it is difficult to determine an appropriate scaling.
Choosing a scaling based on the initial data,
N
u
D
u
U
a
U
b
U
a
;
