Biomedical Engineering Reference
In-Depth Information
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
u
0.5
u
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
x
x
(a)
(b)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
1.2
1
0.8
0.6
u
u
0.4
0.2
0
-0.2
0
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
x
x
(c)
(d)
Figure 15.3
Solution of the steady convection-diffusion equation for
v
= 1, 10, 25 and
v
= 100, respectively
using ten linear elements; solid line: approximate solution
u
h
, dashed line: exact solution
u
.
where
h
is the element length. Above a certain critical value of Pe
h
the solu-
tion behaves in an oscillatory fashion. To reduce possible oscillations the element
Peclet number should be reduced. For fixed
v
and
c
this can only be achieved by
reducing the element size
h
. For example, doubling the number of elements from
10 to 20 eliminates the oscillations at
v
25, see Fig.
15.4
.
The oscillations that appear in the numerical solution of the steady convection-
diffusion equation may be examined as follows. Consider a domain that is
subdivided in two linear elements, each having a length equal to
h
. At one end
of the domain the solution is fixed to
u
=
0, while at the other end the solution
is set to
u
=
1, or any other arbitrary non-zero value. For constant
v
and
c
the
governing differential equation may be rewritten as
=
d
2
u
dx
2
v
c
du
dx
−
=
0.
