$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
2nd-order Adams-Bashforth scheme
$$
\begin{equation}
u^{n+1} = u^n + \half\Delta t\left( 3f(u^n, t_n) - f(u^{n-1}, t_{n-1})
\right)
\tag{44}
\end{equation}
$$
