$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Correction terms in the Crank-Nicolson scheme (1)
$$ [D_t u = -a\overline{u}^t]^{n+\half},$$
Definition of the truncation error \( R \) and correction terms \( C \):
$$ [D_t \uex + a\overline{\uex}^{t} = C + R]^{n+\half}\tp$$
Must Taylor expand
- the derivative
- the arithmetic mean
$$
C^{n+\half} + R^{n+\half} =
\frac{1}{24}\uex'''(t_{n+\half})\Delta t^2
+ \frac{a}{8}\uex''(t_{n+\half})\Delta t^2 + \Oof{\Delta t^4}\tp$$
Let \( C^{n+\half} \) cancel the \( \Delta t^2 \) terms:
$$ C^{n+\half} =
\frac{1}{24}\uex'''(t_{n+\half})\Delta t^2
+ \frac{a}{8}\uex''(t_{n})\Delta t^2\tp$$
