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3.2.1 One-Dimensional TAYLOR Expansion

For the sake of brevity, a one-dimensional investigation will be given. The results can be extended straightforwardly to higher dimensions [8, p.150]. Considering a set of grid points $ x_i$ the spacing is defined by

$\displaystyle h_i = x_{i + 1} - x_i \ , \qquad i = 1, \ldots, N - 1 \ .$ (3.1)

Fig. 3.2 shows the notation used.

Figure 3.2: Three adjacent grid points together with some notational abbreviations used in the derivation.
\includegraphics[width=.7\textwidth]{eps/FiniteDifferenceMethod.eps}

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 $ x_i + h_i / 2$ is considered

$\displaystyle f(x)$ $\displaystyle = \sum_{n = 0}^\infty \frac{f_{i + 1 / 2}^{(n)}}{n!} \, \Bigl(x - \Bigl(x_i + \frac{h_i}{2}\Bigr)\Bigr)^n =$ (3.2)
  $\displaystyle = f_{i + 1 / 2} + \frac{x - (x_i + \frac{h_i}{2})}{1!} \, f'_{i +... ...ac{(x - (x_i + \frac{h_i}{2}))^2}{2!} \, f (3.3)

To get an expression for the first order derivatives the series up to the order $ n = 2$ is evaluated at $ x_i$ and $ x_{i + 1}$

$\displaystyle f_i$ $\displaystyle = f_{i + 1 / 2}$   $\displaystyle - \frac{h_i}{2} \, f'_{i + 1 / 2}$   $\displaystyle + \frac{h_i^2}{8} \, f   $\displaystyle - \mathcal{O}(h^3) \ ,$ (3.4)
$\displaystyle f_{i + 1}$ $\displaystyle = f_{i + 1 / 2}$   $\displaystyle + \frac{h_i}{2} \, f'_{i + 1 / 2}$   $\displaystyle + \frac{h_i^2}{8} \, f   $\displaystyle + \mathcal{O}(h^3) \ .$ (3.5)

$ f_{i + 1 / 2}$ can be eliminated by subtracting the two equations and thus the first order derivative becomes

$\displaystyle f'_{i + 1 / 2} = \frac{f_{i + 1} - f_i}{h_i} + \mathcal{O}(h^2) \ .$ (3.6)

For the second order derivatives the TAYLOR series expansion around $ x_i$

$\displaystyle f(x) = \sum_{n = 0}^\infty \frac{f_i^{(n)}}{n!} \, (x - x_i)^n = ... ...c{x - x_i}{1!} \, f'_i + \frac{(x - x_i)^2}{2!} \, f (3.7)

is evaluated up to the order $ n = 2$ at $ x_{i - 1}$ and $ x_{i + 1}$

$\displaystyle f_{i - 1}$ $\displaystyle = f_i$   $\displaystyle - h_{i - 1} \, f'_i$   $\displaystyle + \frac{h_{i - 1}^2}{2} \, f   $\displaystyle - \mathcal{O}(h^3) \ ,$ (3.8)
$\displaystyle f_{i + 1}$ $\displaystyle = f_i$   $\displaystyle + h_i \, f'_i$   $\displaystyle + \frac{h_i^2}{2} \, f   $\displaystyle + \mathcal{O}(h^3) \ ,$ (3.9)

and by eliminating $ f'(x_i)$ the second order derivative is found to be

$\displaystyle f (3.10)

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 $ \mathcal{O}(h^3)$ in eqn. (3.6) and $ \mathcal{O}(h^2)$ in eqn. (3.10) [8, p.153].


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M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF