$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Leading-order error terms in finite differences (2)
$$
\begin{align}
[\bar D_tu]^{n+\theta} &= \frac{u^{n+1} - u^{n}}{\Delta t} = u'(t_{n+\theta}) + R^{n+\theta}
\tag{11},\\
R^{n+\theta} &= \half(1-2\theta)u''(t_{n+\theta})\Delta t -
\frac{1}{6}((1 - \theta)^3 - \theta^3)u'''(t_{n+\theta})\Delta t^2 +
\Oof{\Delta t^3}
\tag{12}\\
\lbrack D_t^{2-}u \rbrack^n &= \frac{3u^{n} - 4u^{n-1} + u^{n-2}}{2\Delta t} = u'(t_n) + R^n
\tag{13},\\
R^n &= -\frac{1}{3}u'''(t_n)\Delta t^2 + \Oof{\Delta t^3}
\tag{14}\\
\lbrack D_tD_t u \rbrack^n &= \frac{u^{n+1} - 2u^{n} + u^{n-1}}{\Delta t^2} = u''(t_n) + R^n
\tag{15},\\
R^n &= \frac{1}{12}u''''(t_n)\Delta t^2 + \Oof{\Delta t^4}
\tag{16}
\end{align}
$$
