Study guide: Truncation Error Analysis

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

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

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

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