With the flux components 
and 
at all mid interval points
and 
(see 
in Figure 3.4-3)
we are ready to discretize the divergence expression
(3.4-39). We apply first order difference approximations
(3.4-11) for the partial derivatives and get (3.4-40)
for the divergence at point 
.
We have already mentioned that the truncation error 
in
(3.4-40) grows linearly with the grid spacing and the second
derivative of the flux components (3.4-41) [Sel84].
A small implementation detail deserves attention. Of course, we do not
evaluate (3.4-40) for each grid point for the calculation of
the divergence, this would waste a lot of computational expenses. Actually,
we calculate for each mid interval point 
the contribution
, which is identical to the contribution 
for the grid point 
.
Then, this contribution , weighted by 
, is added to the divergence at point , and the same
weighted by 
is added to the
divergence at point 
. After that, for each mid interval
point 
the contributions 
are added accordingly.
