Study guide: Truncation Error Analysis

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$\newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\vex}{{v_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)}$

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 $\Oof{\Delta t^4}$ .