$$
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$$
The central difference for \( u'(t) \) (1)
For the central difference approximation,
$$ u'(t_n)\approx [ D_tu]^n, \quad [D_tu]^n =
\frac{u^{n+\half} - u^{n-\half}}{\Delta t},
$$
the truncation error is
$$ R^n = [ D_tu]^n - u'(t_n)\tp$$
Expand \( u(t_{n+\half}) \) and
\( u(t_{n-1/2}) \) in Taylor series around the point \( t_n \) where
the derivative is evaluated:
$$
\begin{align*}
u(t_{n+\half}) = &u(t_n) + u'(t_n)\half\Delta t +
{\half}u''(t_n)(\half\Delta t)^2 + \\
& \frac{1}{6}u'''(t_n) (\half\Delta t)^3
+ \frac{1}{24}u''''(t_n) (\half\Delta t)^4 + \Oof{\Delta t^5}\\
u(t_{n-1/2}) = &u(t_n) - u'(t_n)\half\Delta t +
{\half}u''(t_n)(\half\Delta t)^2 - \\
& \frac{1}{6}u'''(t_n) (\half\Delta t)^3
+ \frac{1}{24}u''''(t_n) (\half\Delta t)^4 + \Oof{\Delta t^5}
\tp
\end{align*}
$$