 Practical stability of a finite difference scheme

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 $C_T\Vert U^0\Vert$. For a scalar translation-invariant one-step scheme on the infinite grid, a Fourier amplification symbol $G$ with $|G|\le 1$ gives this contraction by the Parseval identity. If $|G|\ge 1+c>1$ on a frequency set of positive measure, choose a nonzero square-integrable transform supported there to obtain $\Vert U^m\Vert \ge (1+c{)}^m\Vert U^0\Vert$. 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 ${\sin }^2\theta \ge 1/2$. The distinction from finite-time stability is quantified by centered-advection refinement-dependent stability.