= Stability of a numerical method
= Stability
{synonym}
= Numerical stability
{synonym}
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 is
$$
\max_{nh\leq T}\|\delta_n\|\leq C_T\left(\|\delta_0\|+\sum_{nh<T}\|d_n\|\right),
$$
where $C_T$ 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 $y'=-2y$ 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 $h\to0$ 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.https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/a9d2bd9be098f0ada172af40379a17cc_MIT18_330S12_Chapter5.pdf[Laurent Demanet, MIT 18.330, Chapter 5, Example 14, 2012]{ref}
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