Study guide: Finite difference schemes for diffusion processes

$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\Aex}{{A_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\dfc}{\alpha} % diffusion coefficient $$

The computational algorithm for the Forward Euler scheme

compute $ u^0_i=I(x_i) $, $ i=0,\ldots,N_x $

for $ n=0,1,\ldots,N_t $:

compute $ u^{n+1}_i $ from (8) for all the internal spatial points $ i=1,\ldots,N_x-1 $

set the boundary values $ u^{n+1}_i=0 $ for $ i=0 $ and $ i=N_x $

Notice.

We visit one mesh point $ (x_i,t_{n+1}) $ at a time, and we have an explicit formula for computing the associated $ u^{n+1}_i $ value. The spatial points can be updated in any sequence, but the time levels $ t_n $ must be updated in cronological order: $ t_n $ before $ t_{n+1} $.