$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Correction terms in the Crank-Nicolson scheme (2)
- Must replace \( u''' \) and \( u'' \) in correction term
- Using \( u'=-au \): \( u''=a^2u \) and \( u'''=-a^3u \)
Result: solve the perturbed ODE by a Crank-Nicolson method,
$$ u' = -\hat a u,\quad \hat a = a(1 - \frac{1}{12}a^2\Delta t^2)\tp$$
and experience an error \( \Oof{\Delta t^4} \).
