Study guide: Truncation Error Analysis

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

Increasing the accuracy by adding correction terms

Question.

Can we add terms in the differential equation that can help increase the order of the truncation error?

To be precise for the Forward Euler scheme, can we find $ C $ to make $ R $ $\Oof{\Delta t^2}$? $$ \begin{equation} \lbrack D_t^+ \uex + a\uex = C + R \rbrack^n\tp \tag{30} \end{equation} $$

$$ \half\uex''(t_n)\Delta t - \frac{1}{6}\uex'''(t_n)\Delta t^2 + \Oof{\Delta t^3} = C^n + R^n\tp$$ Choosing $$ C^n = \half\uex''(t_n)\Delta t,$$ makes $$ R^n = \frac{1}{6}\uex'''(t_n)\Delta t^2 + \Oof{\Delta t^3}\tp$$