Biomedical Engineering Reference
In-Depth Information
t
=
0.01 : 0.05 : 0.26
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Figure 15.5
Solution of the unsteady convection-diffusion problem using 20 elements at
v
= 10.
very fine meshes, an alternative, stabilized formulation has been developed: the
so-called SUPG (Streamline-Upwind/Petrov-Galerkin) formulation. A discussion
of this method however, is beyond the scope of the present topic.
Example 15.3
Let us consider the instationary convection-diffusion problem. For this prob-
lem the same outline as in the previous example for
v
=
10 is chosen and
we use a uniform distribution of 20 linear elements. The initial condition is
u
(
x
,
t
=
0)
=
0 throughout the domain. At the first time step the boundary
condition
u
(
x
=
1,
t
)
=
1 is imposed. The unsteady solution is obtained using a
time step of
t
=
0.01, while
θ
=
0.5 is selected for the
θ
-scheme. Fig.
15.5
shows the time-evolution of the solution towards the steady state value (denoted
by the dashed line) for
v
=
10.
Exercises
15.1 Consider the domain
=
[ 0 1]. On the domain the one-dimensional steady
convection-diffusion equation:
dx
=
c
d
2
u
v
du
dx
2
