Centered-advection refinement-dependent stability ($G=1-i(c\Delta t/h)\sin \theta$)

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 $\log |G|\le c^2\Delta t^2/(2h^2)$, so $|G{|}^n\le \exp (c^{2}T\Delta t/(2h^2))$ is uniformly bounded on fixed time intervals. This severe parabolic refinement supplies stability without contraction; it does not validate the usual hyperbolic-step choice.