Study guide: Finite difference schemes for diffusion processes

Processing math: 100%

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

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 $t_{n+1}$ .