Our expressions derived for the transformed differential operators are the
result of a box integration over subdomains 
. These subdomains
partition the interior of the simulation domain without overlap or
exclusion. It is quite obvious to use subdomains (3.4-7)
that split the domain at the mid interval lines of the grid
(Figure 3.4-3), which are identical to the Voronoi cells
(cf. Figure 3.3-1).
We use the common notation for function values at mid interval points (3.4-8), (3.4-9) and the grid spacing in the computational domain (3.4-10).
In the following we repeatedly require discrete approximations for partial
derivatives of a function 
. If is sufficiently smooth (here three
times differentiable), we may use the difference expression
(3.4-11) for the partial derivatives. The local truncation error
is the residual which occurs when inserting the solution of the
continuous problem into the discrete scheme. The local truncation error in
our approximation (3.4-11) is of first order in the grid spacing
weighted with the second partial derivatives (3.4-12).
Therefore, we call the approximation (3.4-11) first order
accurate.
The most crucial and computationally expensive task is to find an
appropriate discrete representation for the flux components. To avoid
unnecessary complications we look closely at just one term of the sum in
(3.1-2), at the partial flux 
of quantity 
which
is related to (or driven by) quantity 
. The transformation of the
equations from physical space 
to computational space 
is linear and,
therefore, we can merely consider the term
and then discretize its transformed counterpart and accumulate the
discretized fluxes, without loss of generality. The term 
is not
explicitly indicated in the following derivations. It is treated as an
independent additional contribution to the flux 
.
The transformed flux is given by (3.4-14) and (3.4-15), which is obtained by applying (3.3-55) to (3.4-13).
For the transformed expression of the divergence (3.4-16), we
need both flux components 
and 
at all mid interval points
and 
. These points are marked by 
in
Figure 3.4-3.
For the discretization of these flux components and at a mid
interval point 
we use the local vertical six point stencil shown
in Figure 3.4-4. We denote values at the fictive center point
() with an index 
and the neighbor points (
) which are
actual grid points with indices 1 to 6. For the mid interval point
we use the analogous horizontal six point stencil.
For numerical reasons we distinguish between the cases where we have
