$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Implicit 2-step backward scheme
$$ u'(t_{n+1}) \approx \frac{3u^{n+1} - 4u^{n} + u^{n-1}}{2\Delta t}$$
Scheme:
$$ u^{n+1} = \frac{4}{3}u^n - \frac{1}{3}u^{n-1} +
\frac{2}{3}\Delta t f(u^{n+1}, t_{n+1})
\tag{39}
$$
Nonlinear equation for \( u^{n+1} \).
