Study guide: Truncation Error Analysis

Processing math: 100%

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

Taylor series inserted in the backward difference approximation

$\begin{align*} [D_t^-u]^n - u'(t_n) &= \frac{u(t_n) - u(t_{n-1})}{\Delta t} - u'(t_n)\\ &= \frac{u(t_n) - (u(t_n) - u'(t_n)\Delta t + {\half}u''(t_n)\Delta t^2 + \Oof{\Delta t^3} )}{\Delta t}\\ &\quad -u'(t_n)\\ &= -{\half}u''(t_n)\Delta t + \Oof{\Delta t^2} ) \end{align*}$

Result: $\begin{equation} R^n = -{\half}u''(t_n)\Delta t + \Oof{\Delta t^2} ) \tp \end{equation}$ The difference approximation is of first order in $\Delta t$ . It is exact for linear $\uex$ .