$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Truncation error
- How good is the discrete equation?
- Possible answer: see how well \( \uex \) fits the discrete equation
$$ \lbrack D_t u = -au\rbrack^n$$
i.e.,
$$ \frac{u^{n+1}-u^n}{\Delta t} = -au^n$$
Insert \( \uex \) (which does not in general fulfill this equation):
$$
\begin{equation}
\frac{\uex(t_{n+1})-\uex(t_n)}{\Delta t} + a\uex(t_n) = R^n \neq 0
\tag{31}
\end{equation}
$$
