Study guide: Finite difference schemes for diffusion processes

Processing math: 100%

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

Step 4: Formulating a recursive algorithm

Nature of the algorithm: compute $u$ in space at $t=\Delta t, 2\Delta t, 3\Delta t,...$

Two time levels are involved in the general discrete equation: $n+1$ and $n$

$u^n_i$ is already computed for $i=0,\ldots,N_x$ , and $u^{n+1}_i$ is the unknown quantity

Solve the discretized PDE for the unknown

$u^{n+1}_i$ :

$\begin{equation} u^{n+1}_i = u^n_i + F\left( u^{n}_{i+1} - 2u^n_i + u^n_{i-1}\right) \tag{8} \end{equation}$

where $F = \dfc\frac{\Delta t}{\Delta x^2}$