Stability of a numerical method

For a numerical evolution, numerical stability means that perturbations remain controlled over a fixed finite interval by a bound independent of sufficiently small discretization parameters. With perturbations ${\delta }_n$ to starting data and added step defects $d_n$, a typical estimate iswhere $C_T$ is independent of $h$. For a linear homogeneous recurrence, a uniform bound on powers of its update matrix supplies this control. von Neumann stability analysis and the energy method provide complementary tools.

Finite-interval numerical stability differs from absolute stability at a fixed step size. Applying the Euler method to $y^{\prime }=-2y$ multiplies a perturbation by $1-2h$ each step. At $h=0.2$ this is $0.6$, so perturbations shrink; at $h=1.2$ it is $-1.4$, so they alternate and grow. Nevertheless the Euler method converges as $h\to 0$ for this problem: failure at a coarse step does not contradict the convergence theorem. Absolute stability asks whether a selected step preserves boundedness of the Dahlquist test equation over arbitrarily many steps. MIT's analysis uses strict decay, $|1+h\lambda |<1$, rather than the boundary-inclusive boundedness convention.^[ref]

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.