Centered Dirichlet drift-diffusion energy identity ($\mathrm{Re}⟨U,(D_2+\kappa D_1)U{⟩}_d=-d^{-1}{\sum }_{m=0}^M|u_{m+1}-u_m{|}^2$)

For the centered second difference and centered first difference on a finite interval with zero Dirichlet boundary conditions, the diffusion matrix is symmetric negative definite and the drift matrix is skew-symmetric. Discrete summation by parts gives the displayed mesh-weighted identity. It proves contraction in the discrete L2 norm without needing periodic modes, positive stencil coefficients or simultaneous diagonalization.