$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Computing correction terms
- Can we add terms to the ODE such that the truncation error
is improved?
$$ [D_tD_t \uex + \omega^2\uex =C + R]^n,$$
- Idea: choose \( C^n \) such that it absorbs the \( \Delta t^2 \) term in \( R^n \),
$$ C^n = \frac{1}{12}\uex''''(t_n)\Delta t^2\tp$$
- Downside: got a \( u'''' \) term
- Remedy: use the ODE \( u''=-\omega^2u \) to see that \( u'''' = \omega^4 u \).
- Just apply the standard scheme to a modified ODE:
$$ [D_tD_t u + \omega^2(1 - \frac{1}{12}\omega^2\Delta t^2)u=0]^n,$$
- Accuracy is \( \Oof{\Delta t^4} \).
