(3.1) 

For the discretization of the flux equations the derivatives inbetween the grid points are important. Therefore a TAYLOR series expansion [47, p.415] around the mid point is considered
(3.2)  
(3.3) 
To get an expression for the first order derivatives the series up to the order is evaluated at and
(3.4)  
(3.5) 
For the second order derivatives the TAYLOR series expansion around
(3.7) 
(3.8)  
(3.9) 
No assumption about the uniformity of the grid has been made during the derivation of eqn. (3.6) and eqn. (3.10), so the estimated truncation errors are valid for a nonuniform grid. If a uniform grid spacing is assumed, the truncation error will be of order in eqn. (3.6) and in eqn. (3.10) [8, p.153].
M. Gritsch: Numerical Modeling of SilicononInsulator MOSFETs PDF