Discrete summation by parts ($⟨U,D_{0}V{⟩}_h=-⟨D_{0}U,V{⟩}_h$)

On a periodic uniform grid, shifting an index in the discrete inner product gives the displayed skew-adjoint identity for the centered first difference. It also gives $⟨U,D_{xx}V{⟩}_h=-⟨D_{+}U,D_{+}V{⟩}_h$. With finite boundaries there are additional boundary terms unless the prescribed endpoint traces eliminate them. These identities are discrete counterparts of integration by parts and prove energy estimates for finite difference methods.

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 $\Vert U^{n+1}{\Vert }_H^2-\Vert U^n{\Vert }_H^2=2k⟨U^{n+1},A{U}^{n+1}{⟩}_H-\Vert U^{n+1}-U^n{\Vert }_H^2$.