Study guide: Finite difference schemes for diffusion processes

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

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!