Information Technology Reference
In-Depth Information
1.4
u(x,t=0)
u(x,t=0.1)
u(x,t=0.25)
u(x,t=1)
1.2
1
0.8
0.6
0.4
0.2
0
0
0.25
0.5
0.75
1
Fig. 7.10
Solution
u
.x; t / of (
7.63
)-(
7.66
)
When you meet a problem such as (
7.68
)-(
7.71
), where many (in this case all)
parameters have unit values, you should immediately recognize the problem as a
scaled problem. A common misinterpretation is to think that many physical param-
eters are set to the value one, but this can be completely unphysical. If you desire
to work with unit values, you should always scale the underlying physical problem
to arrive at the correct model, where “physical parameters” are removed. The infor-
mation provided here may be sufficient for scaling diffusion problems, but in other
PDE problems you probably have to gain more knowledge and experience about
the art of scaling in general in order to construct sound scalings. The technique of
scaling has been particularly important in applied mathematics, especially in pertur-
bation and similarity methods, so there is a vast literature on scaling. Unfortunately,
this literature does not discuss scaling in a modern scientific computing context, i.e.,
with the aim of reducing the amount of numerical experimentation to gain insight
into a model.
Extension: Variable Coefficients
A PDE with variable coefficients, e.g.,
k.x/
@
u
@x
C
f.x;t/;
%.x/c
v
.x/
@
u
@t
D
@
@x
