Biomedical Engineering Reference
In-Depth Information
where
f
∼
results from the contribution of
B
(see Section
14.5
). This equation has to
be satisfied for all
∼
, hence
M
d
dt
+
K
∼
=
f
.
(15.27)
∼
This is a set of first-order differential equations having a similar structure as Eq.
(
15.11
). Therefore, application of the
θ
-scheme for temporal discretization yields
1
t
M
K
∼
n
+
1
=
1
t
M
)
K
∼
n
+
θ
,
+
θ
−
(1
−
θ
f
∼
(15.28)
with
θ
f
∼
=
θ
f
∼
n
+
1
+
(1
−
θ
)
f
∼
n
.
(15.29)
Clearly, in the steady case the set of equations, Eqs. (
15.27
), reduces to
K
∼
=
f
∼
.
(15.30)
Example 15.2
Consider the steady convection-diffusion problem, according to
v
du
c
du
dx
,
d
dx
dx
=
with the following parameter setting:
=
[0 1]
u
:
x
=
0 and
x
=
1
p
=∅
c
=
1
u
(
x
=
0)
=
0
u
(
x
=
1)
=
1
The convective velocity
v
will be varied. For
v
=
0, the diffusion limit, the solution
is obvious:
u
varies linearly in
x
from
u
=
0at
x
=
0to
u
=
1at
x
=
1.
Figs.
15.3
(a) to (d) show the solution for
v
1, 10, 25 and 100, respectively,
using a uniform element distribution with ten linear elements. For
v
=
=
10 the approximate solution
u
h
(solid line) closely (but not exactly) follows the
exact solution (dashed line). However, for
v
=
1 and
v
25 the numerical solution starts to
demonstrate an oscillatory behaviour that is more prominent for
v
=
100. Careful
analysis of the discrete set of equations shows that the so-called element Peclet
number governs this oscillatory behaviour. The element Peclet number is defined
as
=
vh
2
c
,
Pe
h
=
