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 linear system for a general $N_x$

$\begin{equation} - F_o u^n_{i-1} + \left(1+ 2F_o \right) u^{n}_i - F_o u^n_{i+1} = u_{i-1}^{n-1} \tag{11} \end{equation}$ for

$i=1,\ldots,Nx-1$ .

What are the unknowns in the linear system?

either $u^n_i$ for $i=1,\ldots,N_x-1$ (all internal spatial mesh points)

or $u^n_i$ , $i=0,\ldots,N_x$ (all spatial points)

The linear system in matrix notation:

$\begin{equation*} AU = b,\quad U=(u^n_0,\ldots,u^n_{N_x}) \end{equation*}$

The linear system for a general Nx N_x

The linear system for a general $N_x$