 Discrete summation by parts

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.