For the discretization of the flux equations the derivatives in-between the grid points are important. Therefore a TAYLOR series expansion [47, p.415] around the mid point is considered
To get an expression for the first order derivatives the series up to the order is evaluated at and
For the second order derivatives the TAYLOR series expansion around
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 non-uniform 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 Silicon-on-Insulator MOSFETs PDF