Finite-time stability versus power boundedness (${\sup }_{n\tau \le T}\Vert G_{h,\tau }^n\Vert \le C_T$)

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 $n\tau \le T$ despite its nontrivial unit-modulus Jordan block. Mesh dependence and the scaling of the step must therefore be kept in an eigenvalue argument.