$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
The unifying \( \theta \)-rule
The Forward Euler, Backward Euler, and Crank-Nicolson schemes can be
formulated as one scheme with a varying parameter \( \theta \):
$$
\begin{equation}
\frac{u^{n+1}-u^{n}}{t_{n+1}-t_n} = -a (\theta u^{n+1} + (1-\theta) u^{n})
\tag{13}
\end{equation}
$$
- \( \theta =0 \): Forward Euler
- \( \theta =1 \): Backward Euler
- \( \theta =1/2 \): Crank-Nicolson
- We may alternatively choose any \( \theta\in [0,1] \).
\( u^n \) is known, solve for \( u^{n+1} \):
$$
\begin{equation}
u^{n+1} = \frac{1 - (1-\theta) a(t_{n+1}-t_n)}{1 + \theta a(t_{n+1}-t_n)}
\tag{14}
\end{equation}
$$
