Energy contraction for centered drift-diffusion ($\frac{d}{dt}\Vert U{\Vert }_h^2=-\frac{2}{h}{\sum }_j|U_{j+1}-U_j{|}^2\le 0$)

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 $|\alpha |h\le 2$; 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.