$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
The discrete solution
- The numerical solution is a mesh function: \( u_i^n \approx \uex(x_i,t_n) \)
- Finite difference stencil (or scheme): equation for \( u^n_i \) involving
neighboring space-time points
