Loading [MathJax]/extensions/TeX/boldsymbol.js
\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} .
