= Discrete summation by parts
{title2=$\langle U,D_0V\rangle_h=-\langle D_0U,V\rangle_h$}
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 $\langle U,D_{xx}V\rangle_h=-\langle D_+U,D_+V\rangle_h$. 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>.
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