= Centered-advection refinement-dependent stability
{title2=$G=1-i(c\Delta t/h)\sin\theta$}
Forward time with centered advection has amplification magnitude greater than one at nonzero modes. Under a fixed nonzero Courant number, its powers blow up as the time step vanishes. If instead $\Delta t/h^2$ stays bounded, then $\log|G|\leq c^2\Delta t^2/(2h^2)$, so $|G|^n\leq\exp(c^2T\Delta t/(2h^2))$ is uniformly bounded on fixed time intervals. This severe parabolic refinement supplies stability without contraction; it does not validate the usual hyperbolic-step choice.
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