in the iteration (3.5-2) needs not to be the
exact Jacobian in order to get a convergent scheme, nevertheless an exact
Jacobian is desirable since it yields quadratic convergence.
As far as possible we use exact values for the Jacobian. The Jacobian matrix
is assembled in exactly the same order as the right hand side 
.
Each entry 
in consists of a sum of contributions
from the nine points of the discretization
stencil. The subscript 
indicates the quantity, the subscripts 
identify the grid point. The coefficient 
consists of the physical
parameters which may depend on the concentrations 
, and 
is a
concentration independent function of grid spacing or time step size.
Corresponding to the nine points of the discretization stencil the
Jacobian has nine block-diagonals (Figure 3.5-1),
each block 
(
in Figure 3.5-1)
consisting of a possibly full 
matrix (3.5-8).
Following the chain rule we have 
. For the fluxes the coefficient
may depend on concentrations at the grid points and on the
concentration 
at a mid interval point (3.4-27),
(3.4-31). The derivative 
is divided up among the grid points consistently to the interpolation used for
.
