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Discretization

The Taylor series expansion of a function $ f(x)$ can be written as

\begin{equation*}\left\{ \begin{split}f(x+h)=f(x)+\frac{\partial f(x)}{\partial ...
...frac{\partial^3 f(x)}{\partial x^3}h^3+\ldots \end{split} \right.\end{equation*} (29)

It leads to

\begin{equation*}\left\{ \begin{split}\frac{f(x+h)+f(x-h)}{2} &=f(x)+\frac{1}{2!...
...frac{\partial^5 f(x)}{\partial x^5}h^5+\ldots \end{split} \right.\end{equation*} (30)

Let $ h=\Delta x/2$ . This implies

\begin{equation*}\left\{ \begin{split}\frac{\partial f(x)}{\partial x}=\frac{f(x...
...\Delta x/2)+f(x-\Delta x/2)}{2}+O(\Delta x^2) \end{split} \right.\end{equation*} (31)



Subsections


2021-08-31