= Finite-time stability versus power boundedness
{title2=$\sup_{n\tau\leq T}\|G_{h,\tau}^n\|\leq C_T$}
Stability of the <partial differential equation> requires a bound uniform over refining meshes and time steps on every fixed physical time interval. This is distinct from demanding an all-time bound for one fixed <matrix>. A fixed <matrix> is power bounded exactly when all <eigenvalues> are in the closed <unit disk> and its unit-modulus <Jordan blocks> have size one. A time-step family can nevertheless have harmless <polynomial> physical-time growth: $G_\tau=I+\tau N$ with $N^2=0$ gives $G_\tau^n=I+n\tau N$, uniformly bounded for $n\tau\leq T$ despite its nontrivial unit-modulus <Jordan block>. Mesh dependence and the scaling of the step must therefore be kept in an <eigenvalue> argument.
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