$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
The truncation error for other schemes
Backward Euler:
$$ R^n \approx -\half\uex''(t_n)\Delta t $$
Crank-Nicolson:
$$ R^{n+\half} \approx \frac{1}{24}\uex'''(t_{n+\half})\Delta t^2$$
