Flux-differencing SBP is conservative and semidiscretely entropy stable

For a fixed cell and direction write $F_{ij}^{ec}=F_{\mathrm{\ell }}^{ec}(U_i,U_j)$ and ${\psi }_i={\psi }_{\mathrm{\ell }}(U_i)$. Symmetry of $F_{\mathrm{\ell }}^{ec}$ from [F4] gives $2{\sum }_{i,j}{Q}_{ij}{v}_i^T{F}_{ij}^{ec}={\sum }_{i,j}{Q}_{ij}(v_i-v_j{)}^T{F}_{ij}^{ec}+{\sum }_{i,j}(Q_{ij}+Q_{ji})v_j^T{F}_{ij}^{ec}$. By [F4] the first sum is ${\sum }_{i,j}{Q}_{ij}({\psi }_i-{\psi }_j)=-{\sum }_j{\psi }_j(Q^T\mathbf{1}{)}_j=-{\sum }_F{\sum }_q{\omega }_{F,q}{n}_{K,F,\mathrm{\ell }}{\psi }_{\mathrm{\ell }}(U_{F,q})$, since $Q\mathbf{1}=0$ and $Q^T\mathbf{1}=(Q+Q^T)\mathbf{1}$ by [F2]. The second sum is ${\sum }_F{\sum }_q{\omega }_{F,q}{n}_{K,F,\mathrm{\ell }}v(U_{F,q}{)}^T{F}_{\mathrm{\ell }}(U_{F,q})$ by [F2], diagonal face selection, and consistency from [F4]. Summing directions and the potential identity in [F3] proves the volume entropy boundary identity.

The chain rule gives $\frac{d}{dt}{\sum }_i{H}_{ii}\eta (U_i)={\sum }_i{H}_{ii}v(U_i{)}^T{\dot{U}}_i$. Since the semidiscrete equation is $H\dot{U}+V+SAT=0$, the entropy rate is the negative of the volume and SAT contractions. Combining the volume contraction from step 1.1 with $-v^T{R}^{T}M({\widehat{F}}_n^{\ast }-F_n)$ cancels the physical flux and leaves, on each cell face, ${\sum }_q{\omega }_{F,q}({\psi }_{n_F}(U_{F,q})-v(U_{F,q}{)}^T{\widehat{F}}_{n_F,q}^{\ast })$.

Sum the cell entropy rates from step 2.1. By [F5] the two cell contributions at an ordinary interior face can be written in one orientation as $-[{\psi }_n]+[v{]}^T{\widehat{F}}_{n,F,q}^{\ast }$. For an identified periodic pair, [A2] supplies the same reversed orientation and matching weights, so the same calculation applies. Substitution of the flux definition in [F1] and the directional jump identity [F4] makes each interface contribution $-{\alpha }_{F,q}\parallel [v]{\parallel }^2\mathrm{/}2$. The remaining physical boundary-cell contributions are $-{\mathcal{B}}_h$ by definition. Moving these terms to the left gives the asserted identity, and [A1] gives nonincrease.

For each component, the constant-vector contraction of the volume term is $2{\sum }_{i,j}{Q}_{ij}{F}^{ec}(U_i,U_j)={\sum }_{i,j}(Q_{ij}+Q_{ji})F^{ec}(U_i,U_j)$. The SBP identity and diagonal face selection [F2] reduce this to the physical normal-flux quadrature on the cell boundary, since consistency of the directional fluxes in [F4] gives $F^{ec}(U_i,U_i)=F(U_i)$. The constant-vector contraction of the SAT [F1] subtracts that physical flux and leaves only the numerical normal-flux quadrature. Its contributions cancel on interior faces by normal reversal and flux conservation [F1, F5]; for the periodic conclusion, the paired contributions cancel by [A2]. Thus the quadrature-weighted total state is constant on a periodic mesh with the stated pairing.