$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Lowering the order of the derivative in the correction term
- \( C^n \) contains \( u'' \)
- Can discretize \( u'' \) (requires \( u^{n+1} \), \( u^n \), and \( u^{n-1} \))
- Can also express \( u'' \) in terms of \( u' \) or \( u \)
$$ u' = -au,\quad\Rightarrow\quad u''=-au'=a^2u\tp$$
Result for \( u''=a^2u \): apply Forward Euler to a perturbed ODE,
$$
\begin{equation}
u' = -\hat au ,\quad \hat a = a(1 - {\half}a\Delta t),
\tag{31}
\end{equation}
$$
to make a second-order scheme!
