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: $$\begin{align*} u(t_{n+\half}) = &u(t_n) + u'(t_n)\half\Delta t + {\half}u''(t_n)(\half\Delta t)^2 + \\ & \frac{1}{6}u'''(t_n) (\half\Delta t)^3 + \frac{1}{24}u''''(t_n) (\half\Delta t)^4 + \Oof{\Delta t^5}\\ u(t_{n-1/2}) = &u(t_n) - u'(t_n)\half\Delta t + {\half}u''(t_n)(\half\Delta t)^2 - \\ & \frac{1}{6}u'''(t_n) (\half\Delta t)^3 + \frac{1}{24}u''''(t_n) (\half\Delta t)^4 + \Oof{\Delta t^5} \tp \end{align*}$$