Inflow advection energy estimate from summation by parts ($HD+D^{T}H=e_N{e}_N^T-e_0{e}_0^T$)

For $U_t=-cDU$ with $c>0$, a positive discrete norm matrix $H$ satisfying the displayed identity gives $d(U^{T}HU)/dt=-c(U_N^2-U_0^2)$. Homogeneous inflow data $U_0=0$ therefore leave only the nonpositive outflow flux. Inhomogeneous inflow contributes a controlled boundary forcing term. This estimate includes the numerical boundary closure, unlike an interior-only Fourier stability analysis. For any dissipative semidiscrete operator in the $H$ norm, Backward Euler method steps preserve the contraction by the identity $\Vert U^{n+1}{\Vert }_H^2-\Vert U^n{\Vert }_H^2=2k⟨U^{n+1},A{U}^{n+1}{⟩}_H-\Vert U^{n+1}-U^n{\Vert }_H^2$.