= Energy contraction for centered drift-diffusion
{title2=$\frac{d}{dt}\|U\|_h^2=-\frac2h\sum_j|U_{j+1}-U_j|^2\leq0$}
For homogeneous Dirichlet endpoints, the centered first-difference <matrix> is skew-Hermitian and the centered second-difference <matrix> is negative definite. Thus $\dot U=(D_{xx}-\alpha D_x)U$ is contractive in the <discrete L2 norm> for every real $\alpha$ and every mesh width. The displayed identity follows by <summation by parts>. Nonnegative off-diagonal entries additionally require $|\alpha|h\leq2$; that maximum-principle condition is not necessary for the L2 energy estimate. <numerical consistency> and Duhamel's formula then give second-order <numerical convergence> for smooth solutions and compatible initial approximations.
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