Study guide: Finite difference schemes for diffusion processes

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

Results for stability

We always have $ A\leq 1 $. The condition $ A\geq -1 $ implies $$ 4F\sin^2p\leq 2 $$ The worst case is when $ \sin^2 p=1 $, so a sufficient criterion for stability is $$ F\leq {\half} $$ or: $$ \Delta t\leq \frac{\Delta x^2}{2\dfc} $$

Implications of the stability result.

Less favorable criterion than for $ u_{tt}=c^2u_{xx} $: halving $ \Delta x $ implies time step $ \frac{1}{4}\Delta t $ (not just $ \half\Delta t $ as in a wave equation). Need very small time steps for fine spatial meshes!