Environmental Engineering Reference
In-Depth Information
Exercise 4.2: Instability of the Explicit Diffusion Simulator Check the simple
transport algorithm, introduced above, for Neumann-numbers
Neu ¼
0.1, 0.2, 0.3,
0.4, 0.5 ! Show that the algorithm becomes unstable, when
Neu
takes values greater
than ½!
The algorithm can be improved by using several
(M)
diffusion steps within one
time step simulation. In other words: a diffusion time step
D
t
diff
¼
D
t=M
can be
introduced in addition. Note that the time step of the splitting algorithm is not
arbitrary. It is determined by the velocity
v
and the cell extension
D
x
. The following
m-file, which is included in the accompanying software under
the name
'simpletrans.m'
, takes such a refinement into account.
After the specification of the input parameters the grid spacing
dx
is calculated in
several steps. First it is derived from the two input parameters time step and
velocity. The user may require smaller grid spacing, if the user-defined value for
dxmax
is smaller. In the latter case
K
is the integer factor, by which
dx
is reduced.
The time step, corresponding to the reduced
dx
is
dtadv
.
N
is the number of blocks,
