$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Example: The backward difference for \( u'(t) \)
Backward difference approximation to \( u' \):
$$
\begin{equation}
\lbrack D_t^- u\rbrack^n = \frac{u^{n} - u^{n-1}}{\Delta t} \approx u'(t_n)
\tag{1}
\tp
\end{equation}
$$
Define the truncation error of this approximation as
$$
\begin{equation}
R^n = [D^-_tu]^n - u'(t_n)\tp
\tag{2}
\end{equation}
$$
The common way of calculating \( R^n \) is to
- expand \( u(t) \) in a Taylor series around the point where the
derivative is evaluated, here \( t_n \),
- insert this Taylor series in (2),
and
- collect terms that cancel and simplify the expression.
