Entropy-Stable Multidimensional Summation-by-Parts Discretizations on hp-Adaptive Curvilinear Grids for Hyperbolic Conservation Laws

Abstract

We develop high-order entropy-conservative semi-discrete schemes for hyperbolic conservation laws applicable to non-conforming curvilinear grids arising from h-, p-, or hp-adaptivity. More precisely, building on previous work with conforming grids by Crean et al. (J Comput Phys 356:410–438, 2018) and Chan et al. (SIAM J Sci Comput 41:A2938–A2966, 2019), we present two schemes: the first couples neighbouring elements in a skew-symmetric method, the second in a pointwise fashion. The key ingredients are degree p diagonal-norm summation-by-parts operators equipped with interface quadrature rules of degree 2p or higher, a skew-symmetric geometric mapping procedure using suitable polynomial functions, and a numerical flux that conserves mathematical entropy. Furthermore, entropy-stable schemes are obtained when augmenting the original schemes with a stabilization term that dissipates mathematical entropy at element interfaces. We provide both theoretical and numerical analysis for the compressible Euler equations demonstrating the element-wise conservation, entropy conservation/dissipation, and accuracy properties of the schemes. While both methods produce comparable results, our studies suggest that the scheme coupling elements in a pointwise manner is more computationally efficient.

Appendices

Appendix A Condition 2 of Definition 2

Here we show that condition 2 of Definition 2, i.e.

$$\begin{aligned} \mathsf {R}_{\kappa \gamma }^\mathrm {T}\hat{\mathsf {B}}_{\kappa \gamma } \mathsf {R}_{\kappa \gamma }&= \mathsf {R}_{\kappa \gamma }^\mathrm {T}\mathsf {P}_{\kappa \gamma \rightarrow I}^\mathrm {T}\hat{\mathsf {B}}_I \mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }, \\ \mathsf {R}_{\nu \gamma }^\mathrm {T}\hat{\mathsf {B}}_{\nu \gamma } \mathsf {R}_{\nu \gamma }&= \mathsf {R}_{\nu \gamma }^\mathrm {T}\mathsf {P}_{\nu \gamma \rightarrow I}^\mathrm {T}\hat{\mathsf {B}}_I \mathsf {P}_{\nu \gamma \rightarrow I}\mathsf {R}_{\nu \gamma }, \end{aligned}$$

holds under mild assumptions. For simplicity we show the proof for an arbitrary element and thus drop the subscripts $\kappa$ and $\nu$.

Assumption 2

We assume that the reference element’s facet quadrature rule and the intermediate reference interface’s facet quadrature rule exactly integrate polynomials of at least degree $s \ge 2r$. Furthermore, we assume that the volume nodal set $S_{\hat{\Omega }}$ and facet nodal set $S_{\hat{\gamma }}$ produce degree r full-column-rank Vandermonde matrices $\hat{\mathsf {V}}_{\Omega } \in \mathbb {R}^{N \times n_r}$ and $\hat{\mathsf {V}}_{\gamma } \in \mathbb {R}^{N_{\gamma } \times n_r}$, respectively.

Lemma 1

If Assumption 2 holds, then

$$\begin{aligned} \mathsf {R}_{\gamma }^\mathrm {T}\hat{\mathsf {B}}_{\gamma }\mathsf {R}_{\gamma }= \mathsf {R}_{\gamma }^\mathrm {T}\mathsf {P}_{\gamma \rightarrow I}^\mathrm {T}\hat{\mathsf {B}}_{I}\mathsf {P}_{\gamma \rightarrow I}\mathsf {R}_{\gamma }. \end{aligned}$$

Proof

Let $\hat{\mathsf {V}}_{I} \in \mathbb {R}^{N_I \times n_r}$ be the degree r Vandermonde matrix evaluated at the nodal set $S_{\hat{I}}$ (which does not necessarily coincide with $S_{\hat{\gamma }}$). Since $\hat{\mathsf {B}}_{I}$ and $\hat{\mathsf {B}}_{\gamma }$ integrate polynomials of at least degree $s \ge 2r$ and the Vandermonde matrices $\hat{\mathsf {V}}_{\gamma }$ and $\hat{\mathsf {V}}_{I}$ are each of degree r, then

$$\begin{aligned} \hat{\mathsf {V}}_{\gamma }^\mathrm {T}\hat{\mathsf {B}}_{\gamma }\hat{\mathsf {V}}_{\gamma } = \hat{\mathsf {V}}_{I}^\mathrm {T}\hat{\mathsf {B}}_{I}\hat{\mathsf {V}}_{I}. \end{aligned}$$

Left-multiplying the right-hand side by $(\hat{\mathsf {V}}_{\gamma }^{\dagger } \hat{\mathsf {V}}_{\gamma })^\mathrm {T}= \mathsf {I}$ and right-multiplying it by $(\hat{\mathsf {V}}_{\gamma }^{\dagger } \hat{\mathsf {V}}_{\gamma })= \mathsf {I}$, we obtain

$$\begin{aligned} \hat{\mathsf {V}}_{\gamma }^\mathrm {T}\hat{\mathsf {B}}_{\gamma }\hat{\mathsf {V}}_{\gamma } = (\hat{\mathsf {V}}_{I} \hat{\mathsf {V}}_{\gamma }^{\dagger } \hat{\mathsf {V}}_{\gamma })^\mathrm {T}\hat{\mathsf {B}}_{I}(\hat{\mathsf {V}}_{I} \hat{\mathsf {V}}_{\gamma }^{\dagger } \hat{\mathsf {V}}_{\gamma }). \end{aligned}$$

Left-multiplying both sides by $(\hat{\mathsf {V}}_{\Omega }^{\dagger })^\mathrm {T}$ and right-multiplying them by $\hat{\mathsf {V}}_{\Omega }^{\dagger }$ gives

$$\begin{aligned} (\hat{\mathsf {V}}_{\gamma } \hat{\mathsf {V}}_{\Omega }^{\dagger })^\mathrm {T}\hat{\mathsf {B}}_{\gamma }(\hat{\mathsf {V}}_{\gamma } \hat{\mathsf {V}}_{\Omega }^{\dagger }) = \Big ( (\hat{\mathsf {V}}_{I} \hat{\mathsf {V}}_{\gamma }^{\dagger }) (\hat{\mathsf {V}}_{\gamma } \hat{\mathsf {V}}_{\Omega }^{\dagger }) \Big )^\mathrm {T}\hat{\mathsf {B}}_{I}\Big ((\hat{\mathsf {V}}_{I} \hat{\mathsf {V}}_{\gamma }^{\dagger })(\hat{\mathsf {V}}_{\gamma } \hat{\mathsf {V}}_{\Omega }^{\dagger }) \Big ). \end{aligned}$$

Substituting $\mathsf {R}_{\gamma }\equiv \hat{\mathsf {V}}_{\gamma } \hat{\mathsf {V}}_{\Omega }^{\dagger }$ and $\mathsf {P}_{\gamma \rightarrow I}\equiv \hat{\mathsf {V}}_{I} \hat{\mathsf {V}}_{\gamma }^{\dagger }$ in the above equation leads to

$$\begin{aligned} \mathsf {R}_{\gamma }^\mathrm {T}\hat{\mathsf {B}}_{\gamma }\mathsf {R}_{\gamma }&= (\mathsf {P}_{\gamma \rightarrow I}\mathsf {R}_{\gamma })^\mathrm {T}\hat{\mathsf {B}}_{I}(\mathsf {P}_{\gamma \rightarrow I}\mathsf {R}_{\gamma }), \\ \mathsf {R}_{\gamma }^\mathrm {T}\hat{\mathsf {B}}_{\gamma }\mathsf {R}_{\gamma }&= \mathsf {R}_{\gamma }^\mathrm {T}\mathsf {P}_{\gamma \rightarrow I}^\mathrm {T}\hat{\mathsf {B}}_{I}\mathsf {P}_{\gamma \rightarrow I}\mathsf {R}_{\gamma }, \end{aligned}$$

which is the desired result. $\square$

Appendix B Proof of Theorem 1

To show that the diagonal-$\mathsf {E}$ entropy-conservative scheme (18) and the dense-coupling entropy-conservative scheme (13) are equivalent on affine meshes for SBP operators equipped with facet quadrature rules of degree $s \ge 2r \ge 2p$, we individually deal with the transient, volumetric, and inter-element coupling terms.

The transient terms in (18) and (13) are the same. Furthermore, noting that on affine meshes $\mathsf {N}_{\kappa \gamma , {x_i} }= n_{\gamma ,x_i} \mathsf {I}_{N_{\kappa \gamma }}$ and invoking condition 2 of Definition 2, we can regain the volumetric terms of (13) from the volumetric terms of (18) as follows:

$$\begin{aligned}&\Bigg \{ \bigg (- 2 \tilde{\mathsf {Q}}_{x_i}^\mathrm {T}+ \sum _{\kappa \gamma } (\tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\tilde{\mathsf {N}}_{\kappa \gamma , {x_i} }\tilde{\mathsf {B}}_{\kappa \gamma }\tilde{\mathsf {R}}_{\kappa \gamma }) \bigg ) \circ \mathsf {F}_{x_i} (\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \Bigg \} \varvec{1}_{\kappa } \\&\quad = \Bigg \{ \bigg (- 2 \tilde{\mathsf {Q}}_{x_i}^\mathrm {T}+ \sum _{\kappa \gamma } \tilde{\mathsf {E}}_{x_i}^{(\kappa \kappa )}\bigg ) \circ \mathsf {F}_{x_i} (\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \Bigg \} \varvec{1}_{\kappa } \\&\quad = \Bigg \{ \bigg (- 2 \tilde{\mathsf {Q}}_{x_i}^\mathrm {T}+ \tilde{\mathsf {E}}_{x_i}\bigg ) \circ \mathsf {F}_{x_i} (\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \Bigg \} \varvec{1}_{\kappa } \\&\quad = \Bigg \{ \bigg (2\tilde{\mathsf {S}}_{x_i}\bigg ) \circ \mathsf {F}_{x_i} (\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \Bigg \} \varvec{1}_{\kappa }, \end{aligned}$$

where we used the property $\mathsf {E}_{x_i}= \sum _{\kappa \gamma } \mathsf {E}_{x_i}^{(\kappa \kappa )}$ in the penultimate step and $\mathsf {S}_{x_i}= -2 \mathsf {Q}_{x_i}^\mathrm {T}+ \mathsf {E}_{x_i}$ in the last step. The last term to consider is the facet coupling terms of the two schemes. We start with the coupling term of the dense-coupling entropy-stable scheme and utilize the property $[\mathsf {R}_{\gamma }]_{lj} = \delta _{lj} , \ \forall \,\, \ j\in \{1,\ldots ,N\}, \ l\in \{1,\ldots ,N_{\gamma }\}$ of diagonal-$\mathsf {E}$ SBP operators. For $j\in \{1,\ldots ,N_{\kappa } \}$, we have

$$\begin{aligned}&\Bigg [ \bigg (\tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \bigg ) \varvec{1}_{\nu } \Bigg ]_j\\&\quad = \Bigg [ \bigg ((\tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}^\mathrm {T}\tilde{\mathsf {N}}^{(\kappa )}_{I,{x_i}}\tilde{\mathsf {B}}_{I}\tilde{\mathsf {P}}_{\nu \gamma \rightarrow I}\tilde{\mathsf {R}}_{\nu \gamma }) \circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \bigg ) \varvec{1}_{\nu } \Bigg ]_j \\&\quad = \sum _{l=1}^{N_{\kappa \gamma }} \sum _{k=1}^{N_{\kappa I}} \sum _{m=1}^{N_{\nu \gamma }} \delta _{lj} [\mathsf {P}_{\kappa \gamma \rightarrow I}]_{kl} [\mathsf {N}^{(\kappa )}_{I,{x_i}}]_{kk} [\mathsf {B}_{I}]_{kk} [\mathsf {P}_{\nu \gamma \rightarrow I}]_{km} \sum _{n=1}^{N_{\nu }} \delta _{mn} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_{\nu }]_n) \\&\quad = \sum _{l=1}^{N_{\kappa \gamma }} \sum _{k=1}^{N_{\kappa I}} \sum _{m=1}^{N_{\nu \gamma }} \delta _{lj} [\mathsf {P}_{\kappa \gamma \rightarrow I}]_{kl} [\mathsf {N}^{(\kappa )}_{I,{x_i}}]_{kk} [\mathsf {B}_{I}]_{kk} [\mathsf {P}_{\nu \gamma \rightarrow I}]_{km} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa \gamma }]_l,[\varvec{u}_{\nu \gamma }]_m), \end{aligned}$$

where we used the properties of the Kronecker delta and substituted $\delta _{lj} \sum _{n=1}^{N_{\nu }} \delta _{mn} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_{\nu }]_n) = \delta _{lj} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa \gamma }]_l,[\varvec{u}_{\nu \gamma }]_m)$. On affine meshes $\mathsf {N}^{(\kappa )}_{I,{x_i}}= n_{\gamma ,x_i} \mathsf {I}_{N_I}$ and we can write the above expression in matrix form as

$$\begin{aligned} \bigg (\tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \bigg ) \varvec{1}_{\nu }&= n_{\gamma ,x_i} \tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\bigg ( (\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}^\mathrm {T}\tilde{\mathsf {B}}_{I}\tilde{\mathsf {P}}_{\nu \gamma \rightarrow I}) \circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa \gamma },\varvec{u}_{\nu \gamma }) \bigg ) \varvec{1}_{\nu \gamma } \\&= n_{\gamma ,x_i} \tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\bigg ( (\tilde{\mathsf {B}}_{\kappa \gamma }\tilde{\mathsf {P}}_{I \rightarrow \kappa \gamma }\tilde{\mathsf {P}}_{\nu \gamma \rightarrow I}) \circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa \gamma },\varvec{u}_{\nu \gamma }) \bigg ) \varvec{1}_{\nu \gamma } \\&= n_{\gamma ,x_i} \tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\bigg ( (\tilde{\mathsf {B}}_{\kappa \gamma }\tilde{\mathsf {P}}_{\nu \gamma \rightarrow \kappa \gamma }) \circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa \gamma },\varvec{u}_{\nu \gamma }) \bigg ) \varvec{1}_{\nu \gamma }, \end{aligned}$$

where we used (20), $\mathsf {P}_{\kappa \gamma \rightarrow I}^\mathrm {T}\mathsf {B}_{I}= \mathsf {B}_{\kappa \gamma }\mathsf {P}_{I \rightarrow \kappa \gamma }$, and (19), $\mathsf {P}_{\nu \gamma \rightarrow \kappa \gamma }\equiv \mathsf {P}_{I \rightarrow \kappa \gamma }\mathsf {P}_{\nu \gamma \rightarrow I}$, in the second and last steps, respectively. Finally, we can pull out the diagonal mass matrix and write the normal as a matrix to obtain

$$\begin{aligned} \bigg (\tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \bigg ) \varvec{1}_{\nu }&= \tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\tilde{\mathsf {N}}_{\kappa \gamma , {x_i} }\tilde{\mathsf {B}}_{\kappa \gamma }\bigg ( \tilde{\mathsf {P}}_{\nu \gamma \rightarrow \kappa \gamma }\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa \gamma },\varvec{u}_{\nu \gamma }) \bigg ) \varvec{1}_{\nu \gamma }, \end{aligned}$$

which is exactly the coupling term in (18).

Appendix C Dense-Coupling Entropy-Conservative Scheme: Conservation Proof (Theorems 2 and 3)

1.1 Element-Wise Conservation

Lemma 2

The spatial derivative term in (21) simplifies to

$$\begin{aligned} \varvec{1}_{\kappa }^\mathrm {T}\bigg (2\mathsf {S}_{x_i}\circ \mathsf {F}_{x_i}^{(\rho )}(\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \bigg ) \varvec{1}_{\kappa } = 0, \quad \forall \,\, \ i\in \{1,\ldots ,d\}. \end{aligned}$$

Proof

Since $\mathsf {S}_{x_i}$ is a skew-symmetric matrix and $\mathsf {F}_{x_i}^{(\rho )}(\varvec{u}_{\kappa },\varvec{u}_{\kappa })$ is a symmetric matrix, their Hadamard product, i.e. $\mathsf {S}_{x_i}~\circ ~\mathsf {F}_{x_i}^{(\rho )}(\varvec{u}_{\kappa },\varvec{u}_{\kappa })$, is a skew-symmetric matrix as well. Therefore, the term above is zero. $\square$

1.2 Global Conservation

The proof of Theorem 3 follows from the element-wise conservation theorem (Theorem 2), the symmetry of the flux functions, and the skew-symmetric property $\Big (\mathsf {E}_{x_i}^{(\nu \kappa )}\Big )^\mathrm {T}= - \mathsf {E}_{x_i}^{(\kappa \nu )}$. Using Theorem 2, we add the element-wise conservation equation for element $\kappa$ to the equivalent equation for its neighbouring element $\nu$ (and drop the terms not associated with the shared facet) to obtain

$$\begin{aligned} \varvec{1}_{\kappa }^\mathrm {T}\mathsf {H}_{\kappa } \frac{ \mathrm {d} \varvec{\rho }_{\kappa }}{\mathrm {d} t} + \varvec{1}_{\nu }^\mathrm {T}\mathsf {H}_{\nu } \frac{ \mathrm {d} \varvec{\rho }_{\nu }}{\mathrm {d} t}&= \sum _{i=1}^d \varvec{1}_{\kappa }^\mathrm {T}\Bigg \{ \mathsf {E}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}^{(\rho )}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{1}_{\nu }\\&\quad + \varvec{1}_{\nu }^\mathrm {T}\Bigg \{ \mathsf {E}_{x_i}^{(\nu \kappa )}\circ \mathsf {F}^{(\rho )}_{x_i}(\varvec{u}_{\nu },\varvec{u}_{\kappa }) \Bigg \} \varvec{1}_{\kappa } \\&= \sum _{i=1}^d \varvec{1}_{\kappa }^\mathrm {T}\Bigg \{ \mathsf {E}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}^{(\rho )}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{1}_{\nu } \\&\quad + \varvec{1}_{\kappa }^\mathrm {T}\Bigg \{ \Big (\mathsf {E}_{x_i}^{(\nu \kappa )}\Big )^\mathrm {T}\circ \Big (\mathsf {F}^{(\rho )}_{x_i}(\varvec{u}_{\nu },\varvec{u}_{\kappa }) \Big )^\mathrm {T}\Bigg \} \varvec{1}_{\nu } \\&= \sum _{i=1}^d \varvec{1}_{\kappa }^\mathrm {T}\Bigg \{ \mathsf {E}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}^{(\rho )}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{1}_{\nu } \\&\quad - \varvec{1}_{\kappa }^\mathrm {T}\Bigg \{ \mathsf {E}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}^{(\rho )}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{1}_{\nu } \\&= 0, \end{aligned}$$

where we took the transpose of the second (scalar) term in the second step and used the properties $\Big ( \mathsf {F}^{(\rho )}_{x_i}(\varvec{u}_{\nu },\varvec{u}_{\kappa }) \Big )^\mathrm {T}= \mathsf {F}^{(\rho )}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu })$ and $\Big (\mathsf {E}_{x_i}^{(\nu \kappa )}\Big )^\mathrm {T}= - \mathsf {E}_{x_i}^{(\kappa \nu )}$ in the penultimate step. Repeating these steps for all facets in $\Gamma _h = \Gamma _{h,i}$, we arrive at the desired result.

Appendix D Dense-Coupling Entropy-Conservative Scheme: Entropy Conservation Proof (Theorems 4 and 5)

1.1 Element-Wise Entropy Conservation

Lemma 3

The transient term in (22) simplifies to

$$\begin{aligned} \varvec{w}_{\kappa }^\mathrm {T}\tilde{\mathsf {H}}\frac{ \mathrm {d} \varvec{u}_{\kappa }}{\mathrm {d} t} = \varvec{1}_{\kappa }^\mathrm {T}\mathsf {H}\frac{ \mathrm {d} \varvec{s}_{\kappa }}{\mathrm {d} t}. \end{aligned}$$

Proof

Writing the term in index notation and noting that the $\mathsf {H}$ matrix is diagonal, we obtain

$$\begin{aligned} \varvec{w}_{\kappa }^\mathrm {T}\tilde{\mathsf {H}}\frac{ \mathrm {d} \varvec{u}_{\kappa }}{\mathrm {d} t}&= \sum _{j=1}^{N_{\kappa }} [\varvec{w}_{\kappa }]_j^\mathrm {T}[\mathsf {H}]_{jj} \left[ \frac{ \mathrm {d} \varvec{u}_{\kappa }}{\mathrm {d} t}\right] _j = \sum _{j=1}^{N_{\kappa }} [\mathsf {H}]_{jj} \left[ \frac{ \mathrm {d} \varvec{s}_{\kappa }}{\mathrm {d} t}\right] _j = \varvec{1}_{\kappa }^\mathrm {T}\mathsf {H}\frac{ \mathrm {d} \varvec{s}_{\kappa }}{\mathrm {d} t}, \end{aligned}$$

where, for the second equality, we invoked the relation $\varvec{\mathcal {W}}^\mathrm {T}\frac{ \mathrm {d} \varvec{\mathcal {U}}}{\mathrm {d} t} = \frac{ \partial \mathcal {S}}{\partial \varvec{\mathcal {U}}} \cdot \frac{ \mathrm {d} \varvec{\mathcal {U}}}{\mathrm {d} t} =\frac{ \mathrm {d} \mathcal {S}}{\mathrm {d} t}$ for each node $j\in \{1,\ldots ,N_{\kappa }\}$. $\square$

Lemma 4

The spatial derivative term in (22) simplifies to

$$\begin{aligned} \varvec{w}_{\kappa }^\mathrm {T}\bigg (2\tilde{\mathsf {S}}_{x_i}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \bigg ) \varvec{1}_{\kappa } = -\varvec{1}_{\kappa }^\mathrm {T}\mathsf {E}_{x_i}\varvec{\psi }_{\kappa ,x_i} , \ \forall \,\, \ i \in \{1,\ldots ,d\}. \end{aligned}$$

Proof

The proof relies on the SBP property, the symmetry and the entropy conservation property of the entropy-conservative flux function, and the exactness of the derivative operator for constant functions. Substituting $2\mathsf {S}_{x_i}= \mathsf {S}_{x_i}- \mathsf {S}_{x_i}^\mathrm {T}= \mathsf {Q}_{x_i}- \mathsf {Q}_{x_i}^\mathrm {T}, \ \forall \,\, i\in \{1,\ldots ,d\}$, we obtain,

$$\begin{aligned} \varvec{w}_{\kappa }^\mathrm {T}\bigg (2\tilde{\mathsf {S}}_{x_i}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \bigg ) \varvec{1}_{\kappa }&= \sum _{j=1}^{N_{\kappa }} \sum _{k=1}^{N_{\kappa }} 2 [\varvec{w}_{\kappa }]_j^\mathrm {T}[\mathsf {S}_{x_i}]_{jk} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_{\kappa }]_k) \\&= \sum _{j=1}^{N_{\kappa }} \sum _{k=1}^{N_{\kappa }} [\varvec{w}_{\kappa }]_j^\mathrm {T}\Big ([\mathsf {Q}_{x_i}]_{jk} - [\mathsf {Q}_{x_i}]_{kj} \Big ) \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_{\kappa }]_k) \\&= \sum _{j=1}^{N_{\kappa }} \sum _{k=1}^{N_{\kappa }} [\varvec{w}_{\kappa }]_j^\mathrm {T}[\mathsf {Q}_{x_i}]_{jk} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_{\kappa }]_k) \\&\quad -\sum _{j=1}^{N_{\kappa }} \sum _{k=1}^{N_{\kappa }} [\varvec{w}_{\kappa }]_k^\mathrm {T}[\mathsf {Q}_{x_i}]_{jk} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_k,[\varvec{u}_{\kappa }]_j), \end{aligned}$$

where we flipped the indices of the second (scalar) term in the last line. Invoking the symmetry of the entropy-conservative flux and Tadmor’s condition (from Definition 3), we find

$$\begin{aligned} \varvec{w}_{\kappa }^\mathrm {T}\bigg (2\tilde{\mathsf {S}}_{x_i}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \bigg ) \varvec{1}_{\kappa }&= \sum _{j=1}^{N_{\kappa }} \sum _{k=1}^{N_{\kappa }} \Big ([\varvec{w}_{\kappa }]_j - [\varvec{w}_{\kappa }]_k \Big )^\mathrm {T}\varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_{\kappa }]_k) [\mathsf {Q}_{x_i}]_{jk} \\&= \sum _{j=1}^{N_{\kappa }} \sum _{k=1}^{N_{\kappa }} \Big ([\psi _{\kappa ,x_i}]_j - [\psi _{\kappa ,x_i}]_k \Big ) [\mathsf {Q}_{x_i}]_{jk} \\&= \varvec{1}_{\kappa }^\mathrm {T}\mathsf {Q}_{x_i}^\mathrm {T}\varvec{\psi }_{\kappa ,x_i} - \varvec{1}_{\kappa }^\mathrm {T}\mathsf {Q}_{x_i}\varvec{\psi }_{\kappa ,x_i}. \\ \end{aligned}$$

Since SBP operators exactly differentiate constant functions, i.e. $\mathsf {Q}_{x_i}\varvec{1} = \varvec{0}$, we can drop the first term and add $-\varvec{1}^\mathrm {T}\mathsf {Q}_{x_i}^\mathrm {T}\varvec{\psi }_{\kappa ,x_i} = 0$ to obtain

$$\begin{aligned} \varvec{w}_{\kappa }^\mathrm {T}\bigg (2\tilde{\mathsf {S}}_{x_i}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \bigg ) \varvec{1}_{\kappa }&= - \varvec{1}_{\kappa }^\mathrm {T}(\mathsf {Q}_{x_i}^\mathrm {T}+ \mathsf {Q}_{x_i}) \varvec{\psi }_{\kappa ,x_i} = -\varvec{1}_{\kappa }^\mathrm {T}\mathsf {E}_{x_i}\varvec{\psi }_{\kappa ,x_i}, \end{aligned}$$

where we used the SBP property for the last equality. $\square$

1.2 Global Entropy Conservation

Lemma 5

Adding the interface coupling term of two neighbouring elements $\kappa$ and $\nu$ in each direction, we obtain

$$\begin{aligned}&\varvec{w}_{\kappa }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{1}_{\nu } + \varvec{w}_{\nu }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\nu \kappa )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\nu },\varvec{u}_{\kappa }) \Bigg \} \varvec{1}_{\kappa } \\&\quad = \varvec{1}_{\kappa }^\mathrm {T}\mathsf {E}_{x_i}^{(\kappa \kappa )}\varvec{\psi }_{\kappa ,x_i} + \varvec{1}_{\nu }^\mathrm {T}{\mathsf {E}}^{(\nu \nu )}_{x_i} \varvec{\psi }_{\nu ,x_i} , \ \forall \,\, i\in \{1,\dots ,d\}. \end{aligned}$$

Proof

The proof relies on the property $\Big (\mathsf {E}_{x_i}^{(\nu \kappa )}\Big )^\mathrm {T}= - \mathsf {E}_{x_i}^{(\kappa \nu )}$, the symmetry and the entropy conservation property of the entropy-conservative flux function, and the exactness of the extrapolation operators for constant functions. Taking the transpose of the second (scalar) term on the left-hand side, we obtain

$$\begin{aligned}&\varvec{w}_{\kappa }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{1}_{\nu } + \varvec{w}_{\nu }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\nu \kappa )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\nu },\varvec{u}_{\kappa }) \Bigg \} \varvec{1}_{\kappa } \\&\quad = \varvec{w}_{\kappa }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{1}_{\nu } + \varvec{1}_{\kappa } \Bigg \{ \Big (\tilde{\mathsf {E}}_{x_i}^{(\nu \kappa )}\Big )^\mathrm {T}\circ \Big ( \mathsf {F}_{x_i}(\varvec{u}_{\nu },\varvec{u}_{\kappa }) \Big )^\mathrm {T}\Bigg \} \varvec{w}_{\nu } \\&\quad = \varvec{w}_{\kappa }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{1}_{\nu } - \varvec{1}_{\kappa } \Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{w}_{\nu }, \end{aligned}$$

where we used the properties $\Big ( \mathsf {F}_{x_i}(\varvec{u}_{\nu },\varvec{u}_{\kappa }) \Big )^\mathrm {T}= \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu })$ and $\Big (\mathsf {E}_{x_i}^{(\nu \kappa )}\Big )^\mathrm {T}= - \mathsf {E}_{x_i}^{(\kappa \nu )}$ in the last step. Switching to index notation and invoking Tadmor’s condition, we obtain

$$\begin{aligned}&\varvec{w}_{\kappa }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{1}_{\nu } + \varvec{w}_{\nu }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\nu \kappa )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\nu },\varvec{u}_{\kappa }) \Bigg \} \varvec{1}_{\kappa } \\&\quad = \sum _{j=1}^{N_{\kappa }} \sum _{k=1}^{N_{\nu }} \Bigg ([\varvec{w}_{\kappa }]_j - [\varvec{w}_{\nu }]_k \Bigg )^\mathrm {T}\varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_{\nu }]_k) \Big [ \mathsf {E}_{x_i}^{(\kappa \nu )}\Big ]_{jk} \\&\quad = \sum _{j=1}^{N_{\kappa }} \sum _{k=1}^{N_{\nu }} \Bigg ([\psi _{\kappa ,x_i}]_j - [\psi _{\nu ,x_i}]_k \Bigg ) \Big [ \mathsf {E}_{x_i}^{(\kappa \nu )}\Big ]_{jk}. \end{aligned}$$

Switching back to matrix notation and using the property $\Big (\mathsf {E}_{x_i}^{(\kappa \nu )}\Big )^\mathrm {T}= - \mathsf {E}_{x_i}^{(\nu \kappa )}$ again, we have

$$\begin{aligned} \varvec{w}_{\kappa }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{1}_{\nu } + \varvec{w}_{\nu }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\nu \kappa )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\nu },\varvec{u}_{\kappa }) \Bigg \} \varvec{1}_{\kappa } =&\varvec{\psi }_{\kappa ,x_i}^\mathrm {T}\mathsf {E}_{x_i}^{(\kappa \nu )}\varvec{1}_{\nu } - \varvec{1}_{\kappa }^\mathrm {T}\mathsf {E}_{x_i}^{(\kappa \nu )}\varvec{\psi }_{\nu ,x_i} \\ =&\varvec{\psi }_{\kappa ,x_i}^\mathrm {T}\mathsf {E}_{x_i}^{(\kappa \nu )}\varvec{1}_{\nu } + \varvec{\psi }_{\nu ,x_i}^\mathrm {T}\mathsf {E}_{x_i}^{(\nu \kappa )}\varvec{1}_{\kappa } \\ =&\varvec{\psi }_{\kappa ,x_i}^\mathrm {T}\mathsf {E}_{x_i}^{(\kappa \kappa )}\varvec{1}_{\kappa } + \varvec{\psi }_{\nu ,x_i}^\mathrm {T}\mathsf {E}^{(\nu \nu )}_{x_i} \varvec{1}_{\nu } \\ =&\varvec{1}_{\kappa }^\mathrm {T}\mathsf {E}_{x_i}^{(\kappa \kappa )}\varvec{\psi }_{\kappa ,x_i} + \varvec{1}_{\nu }^\mathrm {T}\mathsf {E}^{(\nu \nu )}_{x_i} \varvec{\psi }_{\nu ,x_i}, \end{aligned}$$

where we invoked the properties $\mathsf {E}_{x_i}^{(\kappa \nu )}\varvec{1}_{\nu } = \mathsf {E}_{x_i}^{(\kappa \kappa )}\varvec{1}_{\kappa }$ and $\mathsf {E}_{x_i}^{(\nu \kappa )}\varvec{1}_{\kappa } = \mathsf {E}^{(\nu \nu )}_{x_i} \varvec{1}_{\nu }$ (from Sect. 3.6) in the third line, and the symmetry of $\mathsf {E}_{x_i}^{(\kappa \kappa )}$ and ${\mathsf {E}}_{x_i}^{(\nu \nu )}$ in the last line. $\square$

Adding the element-wise entropy conservation equation of Theorem 4 for 2 neighbouring elements and dropping the terms not associated with the shared facet, we obtain

$$\begin{aligned} \varvec{1}_{\kappa }^\mathrm {T}\mathsf {H}_{\kappa } \frac{ \mathrm {d} s_{\kappa }}{\mathrm {d} t} + \varvec{1}_{\nu }^\mathrm {T}\mathsf {H}_{\nu } \frac{ \mathrm {d} s_{\nu }}{\mathrm {d} t}&=- \sum _{i=1}^d \varvec{w}_{\kappa }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \Bigg \} \varvec{1}_{\nu } -\sum _{i=1}^d \varvec{w}_{\nu }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\nu \kappa )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\nu },\varvec{u}_{\kappa }) \Bigg \} \varvec{1}_{\kappa } \\&\quad +\sum _{i=1}^d \varvec{1}_{\kappa }^\mathrm {T}\mathsf {E}_{x_i}^{(\kappa \kappa )}\varvec{\psi }_{\kappa ,x_i} +\sum _{i=1}^d \varvec{1}_{\nu }^\mathrm {T}\mathsf {E}_{x_i}^{(\nu \nu )} \varvec{\psi }_{\nu ,x_i}. \end{aligned}$$

Invoking Lemma 5, we see that the right-hand side is zero. Repeating this step for all facets in $\Gamma _h = \Gamma _{h,i}$, we obtain the desired result of Theorem 5.

Appendix E Dense-Coupling Entropy-Conservative Scheme: Design-Order Accuracy (Theorem 6)

To prove Theorem 6, it is sufficient to prove that the penalty term vanishes for $\varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}(\varvec{\mathcal {U}}(\cdot ),\varvec{\mathcal {U}}(\cdot )) \in \mathbb {P}_{r_{\text {min}}}(\Omega _{\kappa } \cup \Omega _{\nu })$ w.r.t. $x_i , \ \forall \,\, \ i\in \{1,\ldots ,d\}$. Noting that $\mathsf {E}_{x_i}^{(\kappa \kappa )}= \mathsf {E}^{(\kappa I)}_{x_i}\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }$ and $\mathsf {E}_{x_i}^{(\kappa \nu )}= \mathsf {E}^{(\kappa I)}_{x_i}\mathsf {P}_{\nu \gamma \rightarrow I}\mathsf {R}_{\nu \gamma }$ (see Sect. 3.6), for $i\in \{1,\ldots ,d\}$, we can show that

$$\begin{aligned}&\Bigg [ \bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\kappa \nu )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \bigg \} \varvec{1}_{\nu } - \bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\kappa \kappa )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \bigg \} \varvec{1}_{\kappa } \Bigg ]_j \\&\quad = \Bigg [ \bigg \{ (\tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\tilde{\mathsf {P}}_{\nu \gamma \rightarrow I}\tilde{\mathsf {R}}_{\nu \gamma }) \circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\nu }) \bigg \} \varvec{1}_{\nu } \\&\qquad - \bigg \{ (\tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}\tilde{\mathsf {R}}_{\kappa \gamma }) \circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \bigg \} \varvec{1}_{\kappa } \Bigg ]_j \\&\quad = \sum _{l=1}^{N_I} [\mathsf {E}^{(\kappa I)}_{x_i}]_{jl} \sum _{k=1}^{N_{\nu }} [\mathsf {P}_{\nu \gamma \rightarrow I}\mathsf {R}_{\nu \gamma }]_{lk} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_{\nu }]_k) \\&\qquad - \sum _{l=1}^{N_I} [\mathsf {E}^{(\kappa I)}_{x_i}]_{jl} \sum _{m=1}^{N_{\kappa }} [\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }]_{lm} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_{\kappa }]_m) \\&\quad = \sum _{l=1}^{N_I} [\mathsf {E}^{(\kappa I)}_{x_i}]_{jl} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_I]_l) \\&\qquad - \sum _{l=1}^{N_I} [\mathsf {E}^{(\kappa I)}_{x_i}]_{jl} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_I]_l) \\&\quad = 0, \end{aligned}$$

where we used the accuracy properties of $\mathsf {R}_{\kappa \gamma }$, $\mathsf {R}_{\nu \gamma }$, $\mathsf {P}_{\kappa \gamma \rightarrow I}$, and $\mathsf {P}_{\nu \gamma \rightarrow I}$ to arrive at the penultimate step.

Appendix F Pointwise-Coupling Entropy-Conservative Scheme: Conservation Proof (Theorems 7 and 8)

1.1 Element-Wise Conservation

The proof of Theorem 7 is as follows. Taking the transpose of the second term on the right-hand side of (23) and noting that $\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }\varvec{1}_{\kappa } = \varvec{1}_{I}$ and $\bigg ( \mathsf {F}^{(\rho )}_{x_i}(\overline{\varvec{u}}_{\kappa I},\varvec{u}_{\kappa }) \bigg )^\mathrm {T}= \mathsf {F}^{(\rho )}_{x_i}(\varvec{u}_{\kappa },\overline{\varvec{u}}_{\kappa I})$, this term cancels out with the first term. Furthermore, invoking Lemma 2 for the volumetric terms, we arrive at the desired result.

1.2 Global Conservation

The proof of Theorem 8 follows from the element-wise conservation theorem (Theorem 7) and the symmetry of the entropy-conservative numerical flux. Using Theorem 7, we add the element-wise conservation equation for two neighbouring elements $\kappa$ and $\nu$, drop the terms not associated with the shared facet, and use the properties $\varvec{1}_{\kappa }^\mathrm {T}\mathsf {E}^{(\kappa I)}_{x_i}= \varvec{1}_I^\mathrm {T}\mathsf {B}_{I}\mathsf {N}^{(\kappa )}_{I,{x_i}}$ and $\varvec{1}_{\nu }^\mathrm {T}\mathsf {E}^{(\nu I)}_{x_i}= \varvec{1}_I^\mathrm {T}\mathsf {B}_{I}\mathsf {N}^{(\nu )}_{I,{x_i}}$ (from Sect. 3.6) to obtain

$$\begin{aligned} \varvec{1}_{\kappa }^\mathrm {T}\mathsf {H}_{\kappa } \frac{ \mathrm {d} \varvec{\rho }_{\kappa }}{\mathrm {d} t} + \varvec{1}_{\nu }^\mathrm {T}\mathsf {H}_{\nu } \frac{ \mathrm {d} \varvec{\rho }_{\nu }}{\mathrm {d} t}&=- \sum _{i=1}^d \varvec{1}^\mathrm {T}_{\kappa } \mathsf {E}^{(\kappa I)}_{x_i}\varvec{f}_{x_i}^{*,\text {EC},\rho }(\overline{\varvec{u}}_{\kappa I},\overline{\varvec{u}}_{\nu I}) \\&\quad - \sum _{i=1}^d\varvec{1}_{\nu }^\mathrm {T}\mathsf {E}^{(\nu I)}_{x_i}\varvec{f}_{x_i}^{*,\text {EC},\rho }(\overline{\varvec{u}}_{\nu I},\overline{\varvec{u}}_{\kappa I}) \\&=- \sum _{i=1}^d \varvec{1}^\mathrm {T}_{I} \mathsf {N}^{(\kappa )}_{I,{x_i}}\mathsf {B}_{I}\varvec{f}_{x_i}^{*,\text {EC},\rho }(\overline{\varvec{u}}_{\kappa I},\overline{\varvec{u}}_{\nu I}) \\&\quad - \sum _{i=1}^d\varvec{1}_{I}^\mathrm {T}\mathsf {N}^{(\nu )}_{I,{x_i}}\mathsf {B}_{I}\varvec{f}_{x_i}^{*,\text {EC},\rho }(\overline{\varvec{u}}_{\nu I},\overline{\varvec{u}}_{\kappa I}) \\&= 0, \end{aligned}$$

where, to arrive at the last step, we used the symmetry of the numerical flux and $\mathsf {N}^{(\nu )}_{I,{x_i}}= - \mathsf {N}^{(\kappa )}_{I,{x_i}}$. Repeating these steps for all facets in $\Gamma _h = \Gamma _{h,i}$, we arrive at the desired result.

Appendix G Pointwise-Coupling Entropy-Conservative Scheme: Entropy Conservation Proof (Theorems 9 and 10)

1.1 Element-Wise Entropy Conservation

Lemma 6

The first and second terms on the right-hand side of (24) simplify to

$$\begin{aligned}&\sum _{\kappa \gamma } -\varvec{w}_{\kappa }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\overline{\varvec{u}}_{\kappa I})\Bigg \} \varvec{1}_{I} + \varvec{w}_{\kappa }^\mathrm {T}\tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}^\mathrm {T}\Bigg \{ \big (\tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\big )^\mathrm {T}\circ \mathsf {F}_{x_i}(\overline{\varvec{u}}_{\kappa I},\varvec{u}_{\kappa })\Bigg \} \varvec{1}_{\kappa } \\&\quad = - \varvec{1}_{\kappa }^\mathrm {T}\mathsf {E}_{x_i}\varvec{\psi }_{\kappa ,x_i} + \sum _{\kappa \gamma } \varvec{1}_{I}^\mathrm {T}\mathsf {N}^{(\kappa )}_{I,{x_i}}\mathsf {B}_{I}\overline{\varvec{\psi }}_{\kappa I,x_i}, \quad \forall \,\, \ i\in \{1,\ldots ,d\}, \end{aligned}$$

where, as previously defined, $[\overline{\varvec{\psi }}_{\kappa I,x_i}]_l \equiv \psi _{x_i} \big ([\overline{\varvec{u}}_{\kappa I}]_l \big ) , \ \forall \,\, l\in \{1,\ldots ,N_{I} \}$ is the potential flux vector in the $x_i$-direction corresponding to the extrapolated entropy variables $\varvec{w}_{\kappa I}$.

Proof

The proof relies on the symmetry and the entropy conservation property of the entropy-conservative flux function and the exactness of the extrapolation operators for constant functions. Using $\varvec{w}_{\kappa I} \equiv \tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}\tilde{\mathsf {R}}_{\kappa \gamma }\varvec{w}_{\kappa }$, we can show that, for $i\in \{1,\ldots ,d\}$,

$$\begin{aligned}&\sum _{\kappa \gamma } -\varvec{w}_{\kappa }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\overline{\varvec{u}}_{\kappa I})\Bigg \} \varvec{1}_{I} + \varvec{w}_{\kappa }^\mathrm {T}\tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}^\mathrm {T}\Bigg \{ \big (\tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\big )^\mathrm {T}\circ \mathsf {F}_{x_i}(\overline{\varvec{u}}_{\kappa I},\varvec{u}_{\kappa })\Bigg \} \varvec{1}_{\kappa } \\&\quad = \sum _{\kappa \gamma } -\varvec{w}_{\kappa }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\overline{\varvec{u}}_{\kappa I})\Bigg \} \varvec{1}_{I} \\&\quad \quad + \varvec{w}_{\kappa I}^\mathrm {T}\Bigg \{ \big (\tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\big )^\mathrm {T}\circ \mathsf {F}_{x_i}(\overline{\varvec{u}}_{\kappa I},\varvec{u}_{\kappa })\Bigg \} \varvec{1}_{\kappa } \\&\quad =\sum _{\kappa \gamma } \sum _{j=1}^{N_{\kappa }} \sum _{l=1}^{N_I} \big ([\varvec{w}_{\kappa I}]_l - [\varvec{w}_{\kappa }]_j \big )^\mathrm {T}\varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\overline{\varvec{u}}_{\kappa I}]_l,[\varvec{u}_{\kappa }]_j) [\mathsf {E}^{(\kappa I)}_{x_i}]_{jl} \\&\quad = \sum _{\kappa \gamma } \sum _{j=1}^{N_{\kappa }} \sum _{l=1}^{N_I} ([\overline{\psi }_{\kappa I,x_i}]_l - [\psi _{\kappa , x_i}]_j) [\mathsf {E}^{(\kappa I)}_{x_i}]_{jl} \\&\quad = \sum _{\kappa \gamma } \varvec{1}_{\kappa }^\mathrm {T}\mathsf {E}^{(\kappa I)}_{x_i}\overline{\varvec{\psi }}_{\kappa I,x_i} - \varvec{1}_{I}^\mathrm {T}(\mathsf {E}^{(\kappa I)}_{x_i})^\mathrm {T}\varvec{\psi }_{\kappa , x_i}, \end{aligned}$$

where we invoked the symmetry of the numerical flux in the second step, and Tadmor’s condition in the penultimate step. Using $\varvec{1}_{\kappa }^\mathrm {T}\mathsf {E}^{(\kappa I)}_{x_i}= \varvec{1}_I^\mathrm {T}\mathsf {N}^{(\kappa )}_{I,{x_i}}\mathsf {B}_{I}$ and $\sum _{\kappa \gamma } \mathsf {E}^{(\kappa I)}_{x_i}\varvec{1}_{I} = \mathsf {E}_{x_i}^\mathrm {T}\varvec{1}_{\kappa }$, we obtain

$$\begin{aligned}&\sum _{\kappa \gamma } -\varvec{w}_{\kappa }^\mathrm {T}\Bigg \{ \tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\overline{\varvec{u}}_{\kappa I})\Bigg \} \varvec{1}_{I} + \varvec{w}_{\kappa }^\mathrm {T}\tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}^\mathrm {T}\Bigg \{ \big (\tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\big )^\mathrm {T}\circ \mathsf {F}_{x_i}(\overline{\varvec{u}}_{\kappa I},\varvec{u}_{\kappa })\Bigg \} \varvec{1}_{\kappa } \\&\quad = \sum _{\kappa \gamma } \varvec{1}_{I}^\mathrm {T}\mathsf {N}^{(\kappa )}_{I,{x_i}}\mathsf {B}_{I}\overline{\varvec{\psi }}_{\kappa I,x_i} - \varvec{1}_{\kappa }^\mathrm {T}\mathsf {E}_{x_i}\varvec{\psi }_{\kappa ,x_i}, \end{aligned}$$

which is the desired result. $\square$

1.2 Global Entropy Conservation

The proof of Theorem 10 follows from the element-wise entropy conservation theorem (Theorem 9), and the symmetry and the entropy conservation property of the entropy-conservative numerical flux. Using Theorem 9, we add the element-wise entropy conservation equation for two neighbouring elements $\kappa$ and $\nu$ and, dropping the terms not associated with the shared facet, we obtain

$$\begin{aligned} \varvec{1}_{\kappa }^\mathrm {T}\mathsf {H}_{\kappa } \frac{ \mathrm {d} \varvec{s}_{\kappa }}{\mathrm {d} t} + \varvec{1}_{\nu }^\mathrm {T}\mathsf {H}_{\nu } \frac{ \mathrm {d} \varvec{s}_{\nu }}{\mathrm {d} t}&= \sum _{i=1}^d \varvec{1}_I^\mathrm {T}\mathsf {N}^{(\kappa )}_{I,{x_i}}\mathsf {B}_{I}\overline{\varvec{\psi }}_{\kappa I,x_i} + \sum _{i=1}^d\varvec{1}_I^\mathrm {T}\mathsf {N}^{(\nu )}_{I,{x_i}}\mathsf {B}_{I}\overline{\varvec{\psi }}_{\nu I,x_i} \\&\quad \quad -\sum _{i=1}^d \varvec{w}_{\kappa I}^\mathrm {T}\tilde{\mathsf {N}}^{(\kappa )}_{I,{x_i}}\tilde{\mathsf {B}}_{I}\varvec{f}_{x_i}^{*,\mathrm {EC}}(\overline{\varvec{u}}_{\kappa I},\overline{\varvec{u}}_{\nu I}) \\&\quad \quad - \sum _{i=1}^d\varvec{w}_{\kappa I}^\mathrm {T}\tilde{\mathsf {N}}^{(\nu )}_{I,{x_i}}\tilde{\mathsf {B}}_{I}\varvec{f}_{x_i}^{*,\mathrm {EC}}(\overline{\varvec{u}}_{\nu I},\overline{\varvec{u}}_{\kappa I}) \\&= \sum _{i=1}^d \varvec{1}_I^\mathrm {T}\mathsf {N}^{(\kappa )}_{I,{x_i}}\mathsf {B}_{I}(\overline{\varvec{\psi }}_{\kappa I,x_i} - \overline{\varvec{\psi }}_{\nu I,x_i}) \\&\quad \quad -\sum _{i=1}^d (\varvec{w}_{\kappa I} - \varvec{w}_{\nu I})^\mathrm {T}\tilde{\mathsf {N}}^{(\kappa )}_{I,{x_i}}\tilde{\mathsf {B}}_{I}\varvec{f}_{x_i}^{*,\mathrm {EC}}(\overline{\varvec{u}}_{\kappa I},\overline{\varvec{u}}_{\nu I}), \end{aligned}$$

where we used $\mathsf {N}^{(\nu )}_{I,{x_i}}= - \mathsf {N}^{(\kappa )}_{I,{x_i}}$ and the symmetry of the numerical flux. Writing the last term in index notation and invoking Tadmor’s condition, we see that

$$\begin{aligned}&(\varvec{w}_{\kappa I} - \varvec{w}_{\nu I})^\mathrm {T}\tilde{\mathsf {N}}^{(\kappa )}_{I,{x_i}}\tilde{\mathsf {B}}_{I}\varvec{f}_{x_i}^{*,\mathrm {EC}}(\overline{\varvec{u}}_{\kappa I},\overline{\varvec{u}}_{\nu I})\\&\quad = \sum _{l=1}^{N_I} [\mathsf {N}^{(\kappa )}_{I,{x_i}}]_l [\mathsf {B}_{I}]_l ([\varvec{w}_{\kappa I}]_l - [\varvec{w}_{\nu I}]_l)^\mathrm {T}\varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\overline{\varvec{u}}_{\kappa I}]_l,[\overline{\varvec{u}}_{\nu I}]_l) \\&\quad = \sum _{l=1}^{N_I} [\mathsf {N}^{(\kappa )}_{I,{x_i}}]_l [\mathsf {B}_{I}]_l ([\overline{\varvec{\psi }}_{\kappa I,x_i}]_l - [\overline{\varvec{\psi }}_{\nu I,x_i}]_l) \\&\quad = \varvec{1}_I^\mathrm {T}\mathsf {N}^{(\kappa )}_{I,{x_i}}\mathsf {B}_{I}(\overline{\varvec{\psi }}_{\kappa I,x_i} - \overline{\varvec{\psi }}_{\nu I,x_i}). \end{aligned}$$

Plugging this back into the first equation gives

$$\begin{aligned} \varvec{1}_{\kappa }^\mathrm {T}\mathsf {H}_{\kappa } \frac{ \mathrm {d} \varvec{s}_{\kappa }}{\mathrm {d} t} + \varvec{1}_{\nu }^\mathrm {T}\mathsf {H}_{\nu } \frac{ \mathrm {d} \varvec{s}_{\nu }}{\mathrm {d} t} = 0. \end{aligned}$$

Repeating these steps for all facets in $\Gamma _h = \Gamma _{h,i}$, we arrive at the desired result:

$$\begin{aligned} \sum _{\Omega _{\kappa } \in \mathcal {T}_h} \varvec{1}_{\kappa }^\mathrm {T}\mathsf {H}_{\kappa } \frac{ \mathrm {d} \varvec{s}_{\kappa }}{\mathrm {d} t} = 0. \end{aligned}$$

Appendix H Pointwise-Coupling Entropy-Conservative Scheme: Design-Order Accuracy (Theorem 11)

To prove Theorem 11, it is sufficient to show that the penalty term of (14) vanishes for $\varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}(\varvec{\mathcal {U}}(\cdot ),\varvec{\mathcal {U}}(\cdot ))$, $\varvec{\mathcal {W}}(\varvec{\mathcal {U}}(\cdot ))$, $\varvec{\mathcal {U}}(\cdot ) \in \mathbb {P}_{r_{\text {min}}}(\Omega _{\kappa } \cup \Omega _{\nu })$ w.r.t. $x_i , \ \forall \,\, i\in \{1,\ldots ,d\}$. First, we show that the first and last terms in the right-hand side of (14) cancel each other. Using $\mathsf {E}_{x_i}^{(\kappa \kappa )}= \mathsf {E}^{(\kappa I)}_{x_i}\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }$, the first and last terms can be simplified as follows for $i\in \{1,\dots ,d\}$:

$$\begin{aligned}&\Bigg [ - \bigg \{ \tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\overline{\varvec{u}}_{\kappa I})\bigg \} \varvec{1}_{I} + \bigg \{ \tilde{\mathsf {E}}_{x_i}^{(\kappa \kappa )}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \bigg \} \varvec{1}_{\kappa } \Bigg ]_j \\&\quad = \Bigg [ - \bigg \{ \tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\overline{\varvec{u}}_{\kappa I})\bigg \} \varvec{1}_{I} +\bigg \{ (\tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}\tilde{\mathsf {R}}_{\kappa \gamma }) \circ \mathsf {F}_{x_i}(\varvec{u}_{\kappa },\varvec{u}_{\kappa }) \bigg \} \varvec{1}_{\kappa } \Bigg ]_j \\&\quad = -\sum _{l=1}^{N_I} [\mathsf {E}^{(\kappa I)}_{x_i}]_{jl} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\overline{\varvec{u}}_{\kappa I}]_l)\\&\quad \quad + \sum _{l=1}^{N_I} [\mathsf {E}^{(\kappa I)}_{x_i}]_{jl} \sum _{k=1}^{N_{\kappa }} (\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma })_{lk} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_{\kappa }]_k) \\&\quad = -\sum _{l=1}^{N_I} [\mathsf {E}^{(\kappa I)}_{x_i}]_{jl} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\overline{\varvec{u}}_{\kappa I}]_l)\\&\quad \quad + \sum _{l=1}^{N_I} [\mathsf {E}^{(\kappa I)}_{x_i}]_{jl} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\varvec{u}_{\kappa }]_j,[\varvec{u}_I]_l), \end{aligned}$$

where we used the accuracy properties of the extrapolation operators $\mathsf {R}_{\kappa \gamma }$ and $\mathsf {P}_{\kappa \gamma \rightarrow I}$ in the last step. Furthermore, since both $\varvec{\mathcal {W}}(\varvec{\mathcal {U}}(\cdot ))$ and $\varvec{\mathcal {U}}(\cdot )$ are in the degree $r_{\text {min}}$ polynomial space, the conservative variables evaluated at the extrapolated entropy variables are equal to the extrapolated conservative variables, i.e. $\overline{\varvec{u}}_{\kappa I} = \varvec{u}_I$. Therefore, the two terms in the last line above cancel each other.

Continuing, the second and third term in the right-hand side of (14) simplify, for $i\in \{1,\dots ,d\}$, to

$$\begin{aligned}&\Bigg [\tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}^\mathrm {T}\bigg \{ \big (\tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\big )^\mathrm {T}\circ \mathsf {F}_{x_i}(\overline{\varvec{u}}_{\kappa I},\varvec{u}_{\kappa })\bigg \} \varvec{1}_{\kappa } - \tilde{\mathsf {E}}^{(\kappa I)}_{x_i}\varvec{f}_{x_i}^{*,\mathrm {EC}}(\overline{\varvec{u}}_{\kappa I},\overline{\varvec{u}}_{\nu I})\Bigg ]_j \\&\quad = \Bigg [\tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}^\mathrm {T}\bigg \{ \big (\tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}^\mathrm {T}\tilde{\mathsf {N}}^{(\kappa )}_{I,{x_i}}\tilde{\mathsf {B}}_{I}\big )^\mathrm {T}\circ \mathsf {F}_{x_i}(\overline{\varvec{u}}_{\kappa I},\varvec{u}_{\kappa })\bigg \} \varvec{1}_{\kappa } \\&\quad \quad - \tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}^\mathrm {T}\tilde{\mathsf {N}}^{(\kappa )}_{I,{x_i}}\tilde{\mathsf {B}}_{I}\varvec{f}_{x_i}^{*,\mathrm {EC}}(\overline{\varvec{u}}_{\kappa I},\overline{\varvec{u}}_{\nu I})\Bigg ]_j \\&\quad = \sum _{l=1}^{N_I} [\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }]_{lj} [\mathsf {N}^{(\kappa )}_{I,{x_i}}\mathsf {B}_{I}]_{ll} \sum _{k=1}^{N_{\kappa }} [\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }]_{lk} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\overline{\varvec{u}}_{\kappa I}]_l,[\varvec{u}_{\kappa }]_k)\\&\quad \quad - \sum _{l=1}^{N_I} [\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }]_{lj} [\mathsf {N}^{(\kappa )}_{I,{x_i}}\mathsf {B}_{I}]_{ll} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\overline{\varvec{u}}_{\kappa I}]_l,[\overline{\varvec{u}}_{\nu I}]_l) \\&\quad = \sum _{l=1}^{N_I} [\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }]_{lj} [\mathsf {N}^{(\kappa )}_{I,{x_i}}\mathsf {B}_{I}]_{ll} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\overline{\varvec{u}}_{\kappa I}]_l,[\varvec{u}_I]_l) \\&\quad \quad - \sum _{l=1}^{N_I} [\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }]_{lj} [\mathsf {N}^{(\kappa )}_{I,{x_i}}\mathsf {B}_{I}]_{ll} \varvec{\mathcal {F}}_{x_i}^{*,\mathrm {EC}}([\overline{\varvec{u}}_{\kappa I}]_l,[\overline{\varvec{u}}_{\nu I}]_l), \end{aligned}$$

where we used the accuracy properties of $\mathsf {R}_{\kappa \gamma }$ and $\mathsf {P}_{\kappa \gamma \rightarrow I}$ in the last step. Again, noting that $\overline{\varvec{u}}_{\kappa I} = \overline{\varvec{u}}_{\nu I} = \varvec{u}_I$ (since both $\varvec{\mathcal {W}}(\varvec{\mathcal {U}}(\cdot ))$ and $\varvec{\mathcal {U}}(\cdot )$ are in the degree $r_{\text {min}}$ polynomial space), the last line sums to zero.

Appendix I Entropy Dissipative Term: Conservation Proof (Theorem 12)

The proof of Theorem 12 is as follows. Invoking the exactness of the extrapolation operators for constant functions (i.e. $\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }\varvec{1}_{\kappa } = \mathsf {P}_{\nu \gamma \rightarrow I}\mathsf {R}_{\nu \gamma }\varvec{1}_{\nu } = \varvec{1}_I$) and the symmetry $\Lambda _I(\varvec{u}_{\kappa I}, \varvec{u}_{\nu I}; \varvec{n}_{I}^{(\kappa )}) = \Lambda _I(\varvec{u}_{\nu I}, \varvec{u}_{\kappa I}; \varvec{n}_I^{(\nu )})$, we show that

$$\begin{aligned}&\varvec{1}_{\kappa }^\mathrm {T}\mathsf {R}_{\kappa \gamma }^\mathrm {T}\mathsf {P}_{\kappa \gamma \rightarrow I}^\mathrm {T}\Lambda _I(\varvec{u}_{\kappa I}, \varvec{u}_{\nu I}; \varvec{n}_{I}^{(\kappa )})^{(\rho )} \tilde{\mathsf {B}}_{I}(\varvec{w}_{\kappa I} - \varvec{w}_{\nu I}) \\&\quad \quad + \varvec{1}_{\nu }^\mathrm {T}\mathsf {R}_{\nu \gamma }^\mathrm {T}\mathsf {P}_{\nu \gamma \rightarrow I}^\mathrm {T}\Lambda _I(\varvec{u}_{\nu I}, \varvec{u}_{\kappa I}; \varvec{n}_I^{(\nu )})^{(\rho )} \tilde{\mathsf {B}}_{I}(\varvec{w}_{\nu I} - \varvec{w}_{\kappa I})\\&\quad = \varvec{1}_{I}^\mathrm {T}\Lambda _I(\varvec{u}_{\kappa I}, \varvec{u}_{\nu I}; \varvec{n}_{I}^{(\kappa )})^{(\rho )} \tilde{\mathsf {B}}_{I}\bigg [ (\varvec{w}_{\kappa I} - \varvec{w}_{\nu I}) + (\varvec{w}_{\nu I} - \varvec{w}_{\kappa I})\bigg ] \\&\quad = 0, \end{aligned}$$

which is the desired result.

Appendix J Entropy Dissipative Term: Entropy Dissipation Proof (Theorem 13)

The proof of Theorem 13 is as follows. Using the definitions $\varvec{w}_{\kappa I} \equiv \tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}\tilde{\mathsf {R}}_{\kappa \gamma }\varvec{w}_{\kappa }$ and $\varvec{w}_{\nu I} \equiv \tilde{\mathsf {P}}_{\nu \gamma \rightarrow I}\tilde{\mathsf {R}}_{\nu \gamma }\varvec{w}_{\nu }$ and $\Lambda _I(\varvec{u}_{\kappa I}, \varvec{u}_{\nu I}; \varvec{n}_{I}^{(\kappa )}) = \Lambda _I(\varvec{u}_{\nu I}, \varvec{u}_{\kappa I}; \varvec{n}_I^{(\nu )})$, we show that

$$\begin{aligned}&-\varvec{w}_{\kappa }^\mathrm {T}\tilde{\mathsf {R}}_{\kappa \gamma }^\mathrm {T}\tilde{\mathsf {P}}_{\kappa \gamma \rightarrow I}^\mathrm {T}\Lambda _I(\varvec{u}_{\kappa I}, \varvec{u}_{\nu I}; \varvec{n}_{I}^{(\kappa )}) \tilde{\mathsf {B}}_{I}(\varvec{w}_{\kappa I} - \varvec{w}_{\nu I}) \\&-\varvec{w}_{\nu }^\mathrm {T}\tilde{\mathsf {R}}_{\nu \gamma }^\mathrm {T}\tilde{\mathsf {P}}_{\nu \gamma \rightarrow I}^\mathrm {T}\Lambda _I(\varvec{u}_{\nu I}, \varvec{u}_{\kappa I}; \varvec{n}_I^{(\nu )}) \tilde{\mathsf {B}}_{I}(\varvec{w}_{\nu I} - \varvec{w}_{\kappa I})\\&\quad = -(\varvec{w}_{\kappa I} - \varvec{w}_{\nu I})^\mathrm {T}\Lambda _I(\varvec{u}_{\kappa I}, \varvec{u}_{\nu I}; \varvec{n}_{I}^{(\kappa )}) \tilde{\mathsf {B}}_{I}(\varvec{w}_{\kappa I} - \varvec{w}_{\nu I}) \\&\quad \le 0, \end{aligned}$$

since the matrix $\Lambda _I(\varvec{u}_{\kappa I}, \varvec{u}_{\nu I}; \varvec{n}_{I}^{(\kappa )}) \tilde{\mathsf {B}}_{I}$ is symmetric positive semi-definite.

Appendix K Entropy Dissipative Term: Design-Order Accuracy Proof (Theorem 14)

To prove Theorem 14, it is sufficient to show that the interface penalty term (17) vanishes for $\varvec{\mathcal {W}}(\varvec{\mathcal {U}}(\cdot )) \in \mathbb {P}_{r_{\text {min}}}(\Omega _{\kappa } \cup \Omega _{\nu })$ w.r.t. $x_i , \ \forall \,\, i \in \{1,\ldots ,d\}$. For $l\in \{1,\ldots ,N_I\}$, we can show that

$$\begin{aligned}{}[\varvec{w}_{\kappa I} - \varvec{w}_{\nu I}]_l&= \sum _{j=1}^{N_{\kappa }} [\mathsf {P}_{\kappa \gamma \rightarrow I}\mathsf {R}_{\kappa \gamma }]_{lj} [\varvec{w}_{\kappa }]_j - \sum _{k=1}^{N_{\nu }} [\mathsf {P}_{\nu \gamma \rightarrow I}\mathsf {R}_{\nu \gamma }]_{lk} [\varvec{w}_{\nu }]_k \\&= [\varvec{w}_I]_l - [\varvec{w}_I]_l \\&= 0, \end{aligned}$$

where we used the accuracy properties of $\mathsf {R}_{\kappa \gamma }$, $\mathsf {R}_{\nu \gamma }$, $\mathsf {P}_{\kappa \gamma \rightarrow I}$, and $\mathsf {P}_{\nu \gamma \rightarrow I}$.