Study guide: Truncation Error Analysis

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

Correction terms in the Crank-Nicolson scheme (1)

$[D_t u = -a\overline{u}^t]^{n+\half},$ Definition of the truncation error

$R$ and correction terms

$C$ :

$[D_t \uex + a\overline{\uex}^{t} = C + R]^{n+\half}\tp$

Must Taylor expand

the derivative

the arithmetic mean

$C^{n+\half} + R^{n+\half} = \frac{1}{24}\uex'''(t_{n+\half})\Delta t^2 + \frac{a}{8}\uex''(t_{n+\half})\Delta t^2 + \Oof{\Delta t^4}\tp$ Let

$C^{n+\half}$ cancel the

$\Delta t^2$ terms:

$C^{n+\half} = \frac{1}{24}\uex'''(t_{n+\half})\Delta t^2 + \frac{a}{8}\uex''(t_{n})\Delta t^2\tp$