Centered discrete advection is skew-adjoint ($\mathrm{Re}⟨u,D_{0}u{⟩}_h=0$)

With zero endpoints, the centered first-difference matrix has opposite off-diagonal entries and is a skew-adjoint operator in the mesh-weighted Euclidean inner product. Adding a real multiple of it to the negative symmetric second-difference matrix does not change the real energy form. The semidiscrete convection-diffusion solution therefore obeys a mesh-independent energy estimate, even when its drift is large compared with its diffusion coefficient.