Modeling Concepts - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 240

Implicit scheme, a = 1

120

100

analytical solution
time step 0.1
time step 0.5
time step 1.0
time step 1.5
time step 2.1

80

60

40

20

0

0 2 4 6 8 10 12

time

FIGURE 6.6 Solution of the decay equation (Equation (6.13)) with the implicit scheme with
different time steps.

As can be seen, this solution does not become unstable for any time step. The
concentrations will always remain positive and no undershoots will occur. This is
the desired property for the solution to the decay equation. Please note that the
implicit solutions all have higher values than the analytical solution, whereas the
stable and physical meaningful explicit solutions are all smaller than the analytical
one. The solutions for the implicit scheme can be seen in Figure 6.6.
If the growth equation is considered instead of the decay equation, all arguments
change. The growth equation reads

∂
∂

C
t

= β C

Clearly this equation can be reproduced by the decay equation just by setting α to

a negative value.
As can be seen easily, the growth equation remains stable if an explicit scheme
is used. However, if an implicit scheme is used, the solution will be stable only if
β

.
In summary, it seems clear that for the decay equation, we should always use
an implicit scheme in order to have a situation where solutions are stable for every
time step used. On the other hand, if the growth equation is to be solved, an explicit
scheme is better for the stability of the model.
The stability and accuracy associated with different options for temporal and
spatial discretizations of the advection and diffusion equations (Equation (6.2)) can
be examined by considering central explicit differences in the particular case of no

satisfies the stability criterion derived for α