For the conservative form of the transformed model equations (3.3-61) - (3.3-65), we can use finite difference expressions for the discretization. Discretizing the conservative form using finite differences is equivalent to returning to the integral form of the physical equations and to apply box integration methods for their discretization.
We find approximate representations of the differential operators which
appear in the transformed equations at a given point 
in terms of
the functions values at that point and at neighboring points. The finite
difference expressions will contain function values at nine points as shown
below (Figure 3.4-1). The governing physical equations do
not contain mixed derivatives and therefore a five point discretization
stencil were sufficient for finite differences discretization. The
additional dependencies on the diagonal points are caused by the cross
derivatives appearing in the transformed equations.
Since all the coefficients of the physical equations may be functions of the
dependent variables 
, the discrete representation of the PDEs
results in a system of nonlinear algebraic equations. This system is
solved with a modified Newton algorithm. The remainder of this section deals
with obtaining the discrete representation of the PDEs,
the modified Newton algorithm will be explained in Section 3.5.
First we show a useful method for calculating polynomial approximations from a set of given data points in Section 3.4.1. Then we derive the discrete representations of the different terms of the PDEs in the same order as they are processed in the program code. First we obtain the expressions for the fluxes (Section 3.4.2) and the divergence (Section 3.4.3), then the recombination terms (Section 3.4.4) and time derivatives (Section 3.4.5) and finally the boundary conditions (Section 3.4.6).
