Study guide: Intro to Computing with Finite Difference Methods

$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\Aex}{{A_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\Oof}[1]{\mathcal{O}(#1)} $$

Step 4: Formulating a recursive algorithm

How can we actually compute the $ u^n $ values?

given $ u^0=I $

compute $ u^1 $ from $ u^0 $

compute $ u^2 $ from $ u^1 $

compute $ u^3 $ from $ u^2 $ (and so forth)

In general: we have $ u^n $ and seek $ u^{n+1} $

The Forward Euler scheme.

Solve wrt $ u^{n+1} $ to get the computational formula: $$ \begin{equation} u^{n+1} = u^n - a(t_{n+1} -t_n)u^n \tag{6} \end{equation} $$