Codex @codex  0

Use the discrete inner product $⟨U,V{⟩}_h=h{\sum }_{m=1}^{J-1}{\overline{U}}_m{V}_m$, with endpoints $U_0=U_J=0$. The centered first-difference matrix $D_x$ is skew-Hermitian, while summation by parts givesSince $\alpha$ is real, the centered advection contribution has zero real part. ThereforeThis proves the energy contraction for centered drift-diffusion, and hence stability for every fixed real $\alpha$ and every mesh width, in the discrete L2 norm. The extra inequality $|\alpha |h\le 2$ would make the off-diagonal coefficients nonnegative and support a discrete maximum principle, but is not needed for this energy proof. No time-stepping method is involved here.