Study guide: Finite difference schemes for diffusion processes

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

Two classes of discretization methods: explicit and implicit

Implicit.

Discretization methods that lead linear systems are known as implicit methods.

Explicit.

Discretization methods that avoid linear systems and have an explicit formula for each new value of the unknown are called explicit methods.