We call a flux purely diffusive if the only driving force is a concentration gradient and neither other driving forces nor convection terms exist. Numerically, the treatment of such a term is straightforward.
We use the two-dimensional Taylor expansion of a function 
(3.4-20) to derive difference formulae for the partial
derivatives 
and 
at the fictive center point (
).
Now, we could match this polynomial to some of the points of the six point
stencil to get one possible discrete representation for the
derivatives. But we want to show that our discrete representations are
uniquely defined with respect to a polynomial ansatz which is quadratic in
and linear in 
.
The Taylor polynomial is matched the function values (
) at
the six points of the computational stencil. The six polynomial equations
can be combined in a matrix form 
,
using a spacing matrix 
, a coefficient vector 
containing
the polynomial coefficients which are the partial derivatives, and the
vector of the six function values.
The determinant of the spacing matrix (3.4-22) is nonzero for any
positive grid spacing and therefore the coefficient vector is
uniquely determined by the function values. We get (3.4-23)
for the inverse of using the shorthands
, 
, and 
. The discrete
representation of the partial derivatives are then calculated from
.
From the second and third row of (3.4-23) we get the
difference approximations (3.4-24) and (3.4-25) for the
partial derivatives. It is straightforward to verify that these expressions
are second order accurate. Analogous expressions hold for the points
.
The concentration values 
and the metric coefficients 
, 
, 
and 
are available at the grid points 1 to 6
(Figure 3.4-4, the Jacobian determinant 
is required at the
fictive center point and is gained by linear interpolation
(3.4-26). The (negative diffusion) coefficient 
is explicitly
available at the center point and may depend on the (interpolated)
concentration at the center point and all concentrations at the neighbor
points (3.4-27).
