Transport Solutions - Environmental Modeling - page 83

An alternative is the contourf - command

by which the transition zone can be visualized nicely. The transition zone is the
multi-colored region between the plateaus of initial concentration (blue) and the
inflow concentration (red) -see Fig. 4.8 . For more details concerning 2D graphics
see Chap. 14.

Sidebar 4.2: Derivation of the diffusion algorithm
As noted, diffusion is described by Fick's Law , which in 1D is stated as:

j ¼D @c
@x

with diffusivity or dispersivity parameter D (see (3.5)). It was shown that the
application of the principle of mass conservation together with Fick's Law
leads to the transport equation, which is a differential equation for the
concentration c . In the derivation the mass balance was set up for a control
volume (Eq. 3.3):

c ð x ; t þ D t Þ c ð x ; t Þ
D t

j x þ ð x ; t Þ j x ð x ; t Þ
D x

¼

In order to describe the change of concentration in the finite system of
cells, the same equation can be used for each cell. We choose an arbitrary cell
at position i of the series with neighbour cells at positions i + 1 and i 1. The

concentrations are designated as c i , c i+1 and c i 1 . The fluxes in x -direction
across the corresponding faces of the cell can be approximated by a finite
version of Fick's Law:

j xþ ¼D c i þ 1 c i
D x

j x ¼D c i c i 1
D x

and

Note that these formulae are not valid exactly, but may serve well as
approximations. It is the idea that small errors in the approximation may lead
to small deviations between analytical and numerical solutions after
performing the algorithm. Mathematicians speak of stability at this point, as
not all algorithms tend out to be stable. Replacing the finite difference terms
in the mass conservation equation above, yields:

c i ; new c i
D t ¼ D x D c i þ 1 c i

1

D x þ D c i c i 1

D x

and

c i ; new c i
D t

¼ D c iþ 1

2 c i þ c i 1
D x 2

(continued)