Periodic centered advection-diffusion contractivity ($\frac{d}{dt}\Vert U{\Vert }_h^2=-2\Vert D_{+}U{\Vert }_h^2$)

The semidiscrete system $U^{\prime }=D_{xx}U+\alpha D_{0}U$, for real fixed $\alpha$, is contractive in the periodic discrete L2 norm. Discrete summation by parts makes the drift term's real energy contribution zero and the diffusion term nonpositive. Equivalently its Fourier amplification symbol generator has real part $-4h^{-2}{\sin }^2(\theta /2)$. This continuous-time estimate does not prove stability of an arbitrarily chosen time integrator.