Hyperbolic DG and summation-by-parts methods

Let $a<b$ and choose an integer $N\ge 1$ and points $a=x_0<x_1<\cdots <x_N=b\mathrm{.}$ The oriented interval mesh is the ordered family of cells $I_j=(x_{j-1},x_j)$, $1\le j\le N$, each oriented from left to right. Write $h_j=x_j-x_{j-1}>0$. For an integer $p\ge 0$, the broken polynomial space is $V_h^{\mathrm{b}\mathrm{r}}={\prod }_{j=1}^N{\mathbb{P}}_p(I_j),$ with functions interpreted cell by cell. At an interior node $x_j$, write $w_j^{-}=w{\mathrm{\mid }}_{I_j}(x_j)$ for the trace from the left cell and $w_j^{+}=w{\mathrm{\mid }}_{I_{j+1}}(x_j)$ for the trace from the right cell; the chosen face normal points from left to right, and the jump is $[w{]}_j=w_j^{+}-w_j^{-}$. At $a$ and $b$, the outward normals are $-1$ and $+1$, respectively.

The published matching-affine-simplicial-mesh convention used elsewhere on the A page has dimension $d\ge 2$. This definition supplies the distinct one-dimensional interval and trace convention; it makes no assertion about multidimensional cells.

On an affine element in dimension $d\ge 1$, diagonal-norm SBP data consist of a nodal state vector with $N_K$ entries, a positive diagonal matrix $H_K\in {\mathbb{R}}^{N_K\times N_K}$, and derivative matrices $D_{K,\mathrm{\ell }}$, $1\le \mathrm{\ell }\le d$. Put $Q_{K,\mathrm{\ell }}=H_K{D}_{K,\mathrm{\ell }}$. For each face $F$ let $R_{K,F}$ select its face nodes from the volume nodes, let $M_{K,F}$ be a positive diagonal face quadrature matrix, and let $n_{K,F}$ be its constant outward unit normal. The data are SBP compatible when $D_{K,\mathrm{\ell }}\mathbf{1}=0,R_{K,F}\mathbf{1}={\mathbf{1}}_F,Q_{K,\mathrm{\ell }}+Q_{K,\mathrm{\ell }}^T={\sum }_{F\subset \mathrm{\partial }K}{n}_{K,F,\mathrm{\ell }}{R}_{K,F}^T{M}_{K,F}{R}_{K,F}$ for each coordinate $\mathrm{\ell }$. The rows of $R_{K,F}$ select the nodes used by the face quadrature; on an interior face, the two cells have the same face nodes and weights under the face identification and opposite normals. In dimension one, $R_L$ and $R_R$ select the left and right endpoint nodes with normals $-1$ and $+1$.

This is conditional algebraic data: the definition does not assert that every mesh has such a rule. The identities include exact affine metric factors and no curved-element metric approximation.

Let $d\ge 1$, let $\beta \in {\mathbb{R}}^d$ be constant, and consider $u_t+\mathrm{\nabla }\cdot (\beta u)=0$ on a fixed domain. For $d=1$, use Oriented interval meshes and broken polynomial spaces. For $d\ge 2$, use a finite matching affine simplicial mesh ${\mathcal{T}}_h$ and its broken space $V_h^{\mathrm{b}\mathrm{r}}={\prod }_{K\in {\mathcal{T}}_h}{\mathbb{P}}_p(K)$ from Matching affine simplicial meshes and shape regularity and Broken and conforming polynomial spaces on a matching mesh. On each cell $K$, let $u_h{\mathrm{\mid }}_K\in {\mathbb{P}}_p(K)$, with $p\ge 0$.

In this discretization, a cell face means a facet of dimension $d-1$ when $d\ge 2$, and an endpoint when $d=1$; all face sums below use only these faces, with counting measure at endpoints. For each cell face $F\subset \mathrm{\partial }K$, let $n_{K,F}$ be the outward unit normal, $b_{K,F}=\beta \cdot n_{K,F}$, and $u_K$ the trace of $u_h{\mathrm{\mid }}_K$. On an interior face with neighbor $K^{\mathrm{\prime }}$, define the upwind trace by ${\widehat{u}}_{K,F}=u_K$ when $b_{K,F}>0$ and ${\widehat{u}}_{K,F}=u_{K^{\mathrm{\prime }}}$ when $b_{K,F}<0$; either trace may be used when $b_{K,F}=0$. On a physical boundary face, use the prescribed inflow datum $g$ when $b_{K,F}<0$ and $u_K$ when $b_{K,F}\ge 0$. The numerical flux is ${\widehat{f}}_{K,F}=b_{K,F}{\widehat{u}}_{K,F}$.

The upwind DG semidiscretization seeks a differentiable cellwise polynomial $u_h(t)$ such that for every $K\in {\mathcal{T}}_h$ and $v_h\in {\mathbb{P}}_p(K)$, ${\int }_K({\mathrm{\partial }}_t{u}_h)v_{h}dx-{\int }_K\beta u_h\cdot \mathrm{\nabla }{v}_{h}dx+{\sum }_{F\subset \mathrm{\partial }K}{\int }_F{\widehat{f}}_{K,F}{v}_{h}dS=0.$ This definition is spatial only. It asserts neither a fully discrete bound nor a shock-convergence result.

For the face orientation, in the diagram $F$ is the shared vertical edge, $n_F=(1,0)$ points from $K^{-}$ to $K^{+}$, and $b=\beta \cdot n_F$. The flux takes the trace on the side from which the sign of $b$ transports data. The coordinates and orientation specify the diagram completely; the proofs use the displayed flux formula, not the picture.

Input a dimension $d\ge 2$, a finite matching affine simplicial mesh ${\mathcal{T}}_h$ whose boundary faces are paired by translations to form a periodic mesh, a constant velocity $\beta \in {\mathbb{R}}^d$, one cell mean $U_K\in \mathbb{R}$ per cell, and a step $\mathrm{\Delta }t>0$. The output is either CFL-inadmissible or an updated vector of cell means $U^{+}=(U_K^{+}{)}_{K\in {\mathcal{T}}_h}$. Let $V_K={\int }_{K}1dx$ be the cell volume and $A_F={\int }_{F}1dS$ the face area. Here a cell face is a codimension-one facet, as in Upwind discontinuous Galerkin discretization of constant transport. For each face $F\subset \mathrm{\partial }K$, let $n_{K,F}$ be the outward unit normal, $b_{K,F}=\beta \cdot n_{K,F}$, and $K_F$ the neighboring cell across that periodic face. With $b^{+}=\max (b,0)$ and $b^{-}=\min (b,0)$, the upwind flux out of $K$ is ${\widehat{f}}_{K,F}=b_{K,F}^{+}{U}_K+b_{K,F}^{-}{U}_{K_F}\mathrm{.}$

Compute ${\lambda }_K=\frac{\mathrm{\Delta }t}{V_K}{\sum }_{F\subset \mathrm{\partial }K}{A}_F{b}_{K,F}^{+}\mathrm{.}$ If ${\lambda }_K>1$ for any cell, return CFL-inadmissible without updating the state. Otherwise, the degree-zero upwind forward-Euler update is admissible and its output is $U_K^{+}=U_K-\frac{\mathrm{\Delta }t}{V_K}{\sum }_{F\subset \mathrm{\partial }K}{A}_F{\widehat{f}}_{K,F}\mathrm{.}$ This is the degree-zero specialization of Upwind discontinuous Galerkin discretization of constant transport.

Let $K$ be a cell in the one-dimensional interval or $d\ge 2$ matching affine-simplex branch of Upwind discontinuous Galerkin discretization of constant transport, and fix an integer degree $p\ge 0$. Let $(K,\mathcal{A},\mu )$ be a finite measure space as in Measure spaces, let ${\mathcal{L}}^2(\mu )$ have the raw measurable-function meaning of The function space ${\mathcal{L}}^p(\mu )$ for $0<p<\mathrm{\infty }$, and let $W={\mathbb{P}}_p(K)\subseteq {\mathcal{L}}^2(\mu )$. For $u\in {\mathcal{L}}^2(\mu )$ define its cellwise L2 projection relation to be the set ${\mathcal{P}}_K(u)=\{p\in {\mathbb{P}}_p(K):{\int }_K(u-p)wd\mu =0\text{ for every }w\in {\mathbb{P}}_p(K)\}\mathrm{.}$ For a mesh, the broken relation is the product of these cell relations. The measure $\mu$ is required to give positive measure to every nonempty relatively open subset of $K$ when existence and uniqueness are claimed. This definition does not itself assert that the set is a singleton. Whenever a later result proves that the relation has a unique member, write ${\mathrm{\Pi }}_{K}u$ for that member.

Fix constant transport velocity $\beta \in {\mathbb{R}}^d$ and SBP data from Diagonal-norm summation-by-parts element data. For each cell $K$, let $U_K(t)$ be the vector of nodal values, $Q_{K,\mathrm{\ell }}=H_K{D}_{K,\mathrm{\ell }}$, and $U_{K,F}=R_{K,F}{U}_K$. Set $b_{K,F}=\beta \cdot n_{K,F}$. At each interior face, use the trace from the cell with positive outward speed when $b_{K,F}>0$, and the neighboring trace when $b_{K,F}<0$; at boundary faces use the inflow data on $b_{K,F}<0$ and the interior trace on $b_{K,F}\ge 0$. Denote this upwind scalar trace by ${\widehat{U}}_{K,F}$.

The upwind SBP-SAT semidiscretization is $H_K{\dot{U}}_K+{\sum }_{\mathrm{\ell }=1}^d{\beta }_{\mathrm{\ell }}{Q}_{K,\mathrm{\ell }}{U}_K+{\sum }_{F\subset \mathrm{\partial }K}{R}_{K,F}^T{M}_{K,F}{b}_{K,F}({\widehat{U}}_{K,F}-U_{K,F})=0.$ The sign of the SAT correction is fixed by this equation. The definition is spatial only and makes no fully discrete claim.

Let $d\ge 1$. If $d=1$, let ${\mathcal{T}}_h$ be a finite oriented interval mesh of $\mathrm{\Omega }=(a,b)$, $a<b$, from Oriented interval meshes and broken polynomial spaces; if $d\ge 2$, let ${\mathcal{T}}_h$ be a finite matching affine simplicial mesh of $\mathrm{\Omega }$. Let $\beta \in {\mathbb{R}}^d$ be constant, and let $u_h$ solve Upwind discontinuous Galerkin discretization of constant transport with inflow data $g$. Assume Countable Choice when $d\ge 2$, as required by the affine-simplex Green supplier; no choice assumption is needed for the interval branch. Write ${\mathrm{\Gamma }}_{+}=\{x\in \mathrm{\partial }\mathrm{\Omega }:\beta \cdot n>0\}$ and ${\mathrm{\Gamma }}_{-}=\{x\in \mathrm{\partial }\mathrm{\Omega }:\beta \cdot n<0\}$. For an interior facet $F=K^{-}\cap K^{+}$ choose a unit normal $n_F$ from $K^{-}$ to $K^{+}$ and write $[u_h]=u_h^{+}-u_h^{-}$. In dimension one, integrals over facets and boundary points mean sums with respect to zero-dimensional surface measure. Then $\frac{1}{2}\frac{d}{dt}{\sum }_{K\in {\mathcal{T}}_h}\mathrm{\parallel }{u}_h{\mathrm{\parallel }}_{L^2(K)}^2+\frac{1}{2}{\sum }_{F\text{ interior}}{\int }_F\mathrm{\mid }\beta \cdot n_F\mathrm{\mid }[u_h{]}^{2}dS+\frac{1}{2}{\int }_{{\mathrm{\Gamma }}_{+}}(\beta \cdot n)u_h^{2}dS+\frac{1}{2}{\int }_{{\mathrm{\Gamma }}_{-}}\mathrm{\mid }\beta \cdot n\mathrm{\mid }(u_h-g{)}^{2}dS=\frac{1}{2}{\int }_{{\mathrm{\Gamma }}_{-}}\mathrm{\mid }\beta \cdot n\mathrm{\mid }{g}^{2}dS\mathrm{.}$ The equality is understood in $\mathbb{R}\cup \{+\mathrm{\infty }\}$. It is a finite energy balance whenever ${\int }_{{\mathrm{\Gamma }}_{-}}\mathrm{\mid }\beta \cdot n\mathrm{\mid }{g}^{2}dS<\mathrm{\infty }$ at the time in question; otherwise both sides are $+\mathrm{\infty }$ and give no finite inflow bound. In particular, zero inflow data give a nonincreasing semidiscrete $L^2$ norm. This result is semidiscrete and makes no claim about a time integrator.

Let $d\ge 2$, let ${\mathcal{T}}_h$ be a finite matching affine simplicial mesh, and consider $U_t+{\sum }_{\mathrm{\ell }=1}^d{\mathrm{\partial }}_{x_{\mathrm{\ell }}}{F}_{\mathrm{\ell }}(U)=0$ with state domain $\mathcal{A}\subseteq {\mathbb{R}}^m$. Assume a common convex entropy $\eta$ and, for each direction $\mathrm{\ell }$, an entropy flux $q_{\mathrm{\ell }}$ so that $(\eta ,q_{\mathrm{\ell }})$ satisfies Convex entropy pairs, entropy variables, and entropy potentials for systems for the directional flux $F_{\mathrm{\ell }}$. Write $v=\mathrm{\nabla }\eta$ and ${\psi }_{\mathrm{\ell }}=v^T{F}_{\mathrm{\ell }}-q_{\mathrm{\ell }}$. For every direction, supply a symmetric consistent two-point flux $F_{\mathrm{\ell }}^{ec}$ satisfying Tadmor entropy-conservative two-point flux, namely $(v(U_R)-v(U_L){)}^T{F}_{\mathrm{\ell }}^{ec}(U_L,U_R)={\psi }_{\mathrm{\ell }}(U_R)-{\psi }_{\mathrm{\ell }}(U_L)\mathrm{.}$

On each cell use diagonal-norm SBP data from Diagonal-norm summation-by-parts element data. Put $Q_{K,\mathrm{\ell }}=H_K{D}_{K,\mathrm{\ell }}$; let $U_i^K\in \mathcal{A}$ be the nodal states; and let the face restrictions select paired face nodes with matching positive weights on the two incident cells. For an oriented interior face node with normal $n$ from the minus trace to the plus trace, set $F_n(U)={\sum }_{\mathrm{\ell }}{n}_{\mathrm{\ell }}{F}_{\mathrm{\ell }}(U)$ and ${\psi }_n(U)={\sum }_{\mathrm{\ell }}{n}_{\mathrm{\ell }}{\psi }_{\mathrm{\ell }}(U)$. Given any fixed coefficient ${\alpha }_{F,q}\ge 0$, define the normal interface flux ${\widehat{F}}_{n,F,q}^{\ast }(U^{-},U^{+})={\sum }_{\mathrm{\ell }=1}^d{n}_{\mathrm{\ell }}{F}_{\mathrm{\ell }}^{ec}(U^{-},U^{+})-\frac{{\alpha }_{F,q}}{2}(v(U^{+})-v(U^{-}))\mathrm{.}$ The coefficient is the same when the face orientation is reversed. Thus the flux is conservative and consistent, and its entropy contraction is $[v{]}^T{\widehat{F}}_{n,F,q}^{\ast }=[{\psi }_n]-\frac{{\alpha }_{F,q}}{2}\parallel [v]{\parallel }^2\le [{\psi }_n],$ by the directional Tadmor identities in Tadmor entropy-conservative two-point flux. On physical boundary faces supply a consistent normal flux and require the boundary entropy contribution below to be nonnegative.

Write ${\omega }_{F,q}$ for the $q$th diagonal entry of $M_F$. Define the boundary entropy contribution ${\mathcal{B}}_h(t)={\sum }_{F\subset \mathrm{\partial }\mathrm{\Omega }}{\sum }_q{\omega }_{F,q}(v(U_{F,q}{)}^T{\widehat{F}}_{n_F}^{\ast }-{\psi }_{n_F}(U_{F,q}))\mathrm{.}$ The selected boundary flux is entropy admissible when ${\mathcal{B}}_h(t)\ge 0$. The flux-differencing SBP semidiscretization on each element is $H_K{\dot{U}}^K+2{\sum }_{\mathrm{\ell }=1}^d(Q_{K,\mathrm{\ell }}\circ F_{\mathrm{\ell }}^{ec}(U^K,U^K))\mathbf{1}+{\sum }_{F\subset \mathrm{\partial }K}{R}_{K,F}^T{M}_{K,F}({\widehat{F}}_{n_{K,F}}^{\ast }-F_{n_{K,F}}(U_{K,F}))=0,$ where $\circ$ is entrywise matrix multiplication and $F_{\mathrm{\ell }}^{ec}(U^K,U^K)$ is the matrix with $(i,j)$ entry $F_{\mathrm{\ell }}^{ec}(U_i^K,U_j^K)$. These are conditional method data: compatible SBP data and directional entropy-conservative fluxes are hypotheses, not asserted to exist for every mesh or entropy pair. The interior numerical flux is explicitly defined once those data and ${\alpha }_{F,q}$ are supplied. Curved elements are excluded.

For every finite measure $\mu$ on a bounded nondegenerate interval or nondegenerate affine simplex $K$ that assigns positive measure to every nonempty relatively open subset, every integer degree $p\ge 0$ with ${\mathbb{P}}_p(K)\subseteq {\mathcal{L}}^2(\mu )$ as in Cellwise L2 projection relation onto broken polynomials, and every $u\in {\mathcal{L}}^2(\mu )$, the set ${\mathcal{P}}_K(u)$ contains exactly one polynomial ${\mathrm{\Pi }}_{K}u$. For each $w\in {\mathbb{P}}_p(K)$, $\mathrm{\parallel }u-{\mathrm{\Pi }}_{K}u{\mathrm{\parallel }}_{L^2(K)}\le \mathrm{\parallel }u-w{\mathrm{\parallel }}_{L^2(K)}\mathrm{.}$

Let ${\mathcal{T}}_h$ be a finite periodic matching affine simplicial mesh in $d\ge 2$, let $\beta \in {\mathbb{R}}^d$ be constant, and apply Forward-Euler step for degree-zero upwind DG transport with ${\lambda }_K\le 1$ for every cell. Assume Countable Choice for the imported affine-simplex Green formula. Then the cell-volume-weighted mean is conserved: ${\sum }_K{V}_K{U}_K^{+}={\sum }_K{V}_K{U}_K\mathrm{.}$ For every convex function $\eta :\mathbb{R}\to \mathbb{R}$, the weighted discrete entropy does not increase: ${\sum }_K{V}_K\eta (U_K^{+})\le {\sum }_K{V}_K\eta (U_K)\mathrm{.}$

Let $d\ge 1$. If $d=1$, let the cells be a finite oriented interval mesh from Oriented interval meshes and broken polynomial spaces; if $d\ge 2$, let them form a finite matching affine simplicial mesh. On every cell assume diagonal-norm SBP data of Diagonal-norm summation-by-parts element data, with paired interior face nodes, equal paired weights, and opposite outward normals. Let $\beta \in {\mathbb{R}}^d$ be constant, and let the differentiable nodal state $U_K(t)$ solve the upwind SBP-SAT method of Upwind SBP-SAT discretization of constant transport with prescribed inflow values $g_{F,q}$ where $b_{K,F}:=\beta \cdot n_{K,F}<0$. Write ${\omega }_{F,q}$ for the diagonal entries of the face norm. On an interior face oriented from $K^{-}$ to $K^{+}$, set $b_F=\beta \cdot n_F$ and $[U{]}_{F,q}=U_{F,q}^{+}-U_{F,q}^{-}$. Let ${\mathrm{\Gamma }}_{+}$ and ${\mathrm{\Gamma }}_{-}$ denote the boundary faces with positive and negative outward speed, respectively. For $d=1$, each face has one endpoint node and its face sum is a one-point sum. The discrete energy $E_h=\frac{1}{2}{\sum }_K{U}_K^T{H}_K{U}_K$ satisfies $\frac{d{E}_h}{dt}+\frac{1}{2}{\sum }_{F\text{ interior}}{\sum }_q{\omega }_{F,q}\mathrm{\mid }{b}_F\mathrm{\mid }[U{]}_{F,q}^2+\frac{1}{2}{\sum }_{F\subset {\mathrm{\Gamma }}_{+}}{\sum }_q{\omega }_{F,q}{b}_{K,F}{U}_{F,q}^2+\frac{1}{2}{\sum }_{F\subset {\mathrm{\Gamma }}_{-}}{\sum }_q{\omega }_{F,q}\mathrm{\mid }{b}_{K,F}\mathrm{\mid }(U_{F,q}-g_{F,q}{)}^2=\frac{1}{2}{\sum }_{F\subset {\mathrm{\Gamma }}_{-}}{\sum }_q{\omega }_{F,q}\mathrm{\mid }{b}_{K,F}\mathrm{\mid }{g}_{F,q}^2\mathrm{.}$ Consequently, zero inflow data give a nonincreasing SBP norm.

Let $d\ge 1$. If $d=1$, let $\mathrm{\Omega }=(a,b)$, $a<b$, have a finite oriented interval mesh ${\mathcal{T}}_h$ from Oriented interval meshes and broken polynomial spaces; if $d\ge 2$, let ${\mathcal{T}}_h$ be a finite matching affine simplicial mesh of $\mathrm{\Omega }$. Let $\beta \in {\mathbb{R}}^d$ be constant, and let $u\in C^1([0,T]\times \bar{\mathrm{\Omega }})$ solve $u_t+\beta \cdot \mathrm{\nabla }u=0$ with inflow data equal to its boundary trace. Assume Countable Choice for the imported affine-simplex Green and Lebesgue-measure suppliers; the full-support and finite-measure properties on each cell are established below. Let $u_h$ be the semidiscrete upwind DG solution of degree $p\ge 0$, and let ${\mathrm{\Pi }}_h$ be the cellwise L2 projection with respect to Lebesgue measure on each cell. Use $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2={\sum }_K{\int }_K\mathrm{\mid }w{\mathrm{\mid }}^{2}dx$ and put $\eta ={\mathrm{\Pi }}_{h}u-u$. For an interior face with chosen normal $n_F$, let ${\eta }^{\mathrm{u}\mathrm{p}}$ be the trace of $\eta$ from the upwind side. In dimension one, face and boundary integrals are sums with respect to zero-dimensional surface measure. Define $R_h(t{)}^2={\sum }_{F\text{ interior}}{\int }_F\mathrm{\mid }\beta \cdot n_F\mathrm{\mid }\mathrm{\mid }{\eta }^{\mathrm{u}\mathrm{p}}{\mathrm{\mid }}^{2}dS+{\int }_{{\mathrm{\Gamma }}_{+}}(\beta \cdot n)\mathrm{\mid }\eta {\mathrm{\mid }}^{2}dS\mathrm{.}$ Then $\mathrm{\parallel }{u}_h(T)-u(T){\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le {(\mathrm{\parallel }{u}_h(0)-{\mathrm{\Pi }}_{h}u(0){\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2+2{\int }_0^T{R}_h(t{)}^{2}dt)}^{1\mathrm{/}2}+\mathrm{\parallel }{\mathrm{\Pi }}_{h}u(T)-u(T){\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\mathrm{.}$ This is an exact projection-trace residual bound. It asserts no mesh-order rate unless the displayed projection errors are separately estimated.

Assume the affine-element method and all directional entropy/SBP data of Conservative flux-differencing SBP method for nonlinear systems. Suppose all nodal states remain in $\mathcal{A}$. Define the discrete entropy and the boundary contribution by $E_h(t)={\sum }_{K\in {\mathcal{T}}_h}{\mathbf{1}}^T{H}_K\eta (U^K),{\mathcal{B}}_h(t)={\sum }_{F\subset \mathrm{\partial }\mathrm{\Omega }}{\sum }_q{\omega }_{F,q}(v(U_{F,q}{)}^T{\widehat{F}}_{n_F}^{\ast }-{\psi }_{n_F}(U_{F,q}))\mathrm{.}$ If ${\mathcal{B}}_h(t)\ge 0$, then $\frac{d}{dt}{E}_h(t)+{\sum }_{F\text{ interior}}{\sum }_q{\omega }_{F,q}\frac{{\alpha }_{F,q}}{2}\parallel [v{]}_{F,q}{\parallel }^2+{\mathcal{B}}_h(t)=0,$ so $E_h$ is nonincreasing. Here ${\widehat{F}}_{n_F,q}^{\ast }$ is the explicitly defined normal flux of Conservative flux-differencing SBP method for nonlinear systems, and the boundary sum is over physical boundary faces. For the periodic conclusion, assume each identified boundary-face pair has matching nodes and positive weights and opposite normals. Treat each pair as one interior face using the flux above on the paired traces, with the same ${\alpha }_{F,q}$ under orientation reversal; include it in the interior-face dissipation sum and omit it from ${\mathcal{B}}_h$. The conservative state-flux contributions of each pair cancel, while its entropy contribution is the same nonnegative interface dissipation in the displayed identity. The same semidiscretization conserves the quadrature-weighted total state on such a periodic mesh. No fully discrete entropy statement follows.