= Periodic centered advection-diffusion contractivity
{title2=$\frac d{dt}\|U\|_h^2=-2\|D_+U\|_h^2$}
The semidiscrete system $U'=D_{xx}U+\alpha D_0U$, 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.
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