= Practical stability of a finite difference scheme
{title2=$\|U^m\|\leq\|U^0\|$}
= Practical stability
{synonym}
= Practically stable
{synonym}
In the contraction convention for numerical evolution, practical stability means non-growth of the chosen <norm> for every datum and every permitted time level. This is stronger than finite-time mesh-uniform <stability> allowing a bound $C_T\|U^0\|$. For a scalar translation-invariant one-step scheme on the infinite grid, a <Fourier amplification symbol> $G$ with $|G|\leq1$ gives this contraction by the <Parseval identity>. If $|G|\geq1+c>1$ on a frequency set of positive measure, choose a nonzero square-integrable transform supported there to obtain $\|U^m\|\geq(1+c)^m\|U^0\|$. The constant $c$ may depend on the mesh. Centered pure advection with forward time has $c=\sqrt{1+(\Delta t/\Delta x)^2/2}-1$ on the set $\sin^2\theta\geq1/2$. The distinction from finite-time <stability> is quantified by <centered-advection refinement-dependent stability>.
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