The BR1 Scheme is Stable for the Compressible Navier–Stokes Equations

Abstract

In this work we prove that the original (Bassi and Rebay in J Comput Phys 131:267–279, 1997) scheme (BR1) for the discretization of second order viscous terms within the discontinuous Galerkin collocation spectral element method (DGSEM) with Gauss Lobatto nodes is stable. More precisely, we prove in the first part that the BR1 scheme preserves energy stability of the skew-symmetric advection term DGSEM discretization for the linearized compressible Navier–Stokes equations (NSE). In the second part, we prove that the BR1 scheme preserves the entropy stability of the recently developed entropy stable compressible Euler DGSEM discretization of Carpenter et al. (SIAM J Sci Comput 36:B835–B867, 2014) for the non-linear compressible NSE, provided that the auxiliary gradient equations use the entropy variables. Both parts are presented for fully three-dimensional, unstructured curvilinear hexahedral grids. Although the focus of this work is on the BR1 scheme, we show that the proof naturally includes the Local DG scheme of Cockburn and Shu.

Appendices

An Entropy Conserving Euler Flux

We explicitly present the Kinetic Energy Preserving and Entropy Conservative (KEPEC) numerical flux function for the compressible Euler equations. The KEPEC flux was first derived by Chandrashekar [8]

$$\begin{aligned} \mathbf {F}_1^{*,\mathrm{ec}} = {\small \begin{bmatrix} \rho ^{\ln }\{\!\{v_1\}\!\} \\ \rho ^{\ln }\{\!\{v_1\}\!\}^2 + \{\!\{p\}\!\} \\ \rho ^{\ln }\{\!\{v_1\}\!\}\{\!\{v_2\}\!\} \\ \rho ^{\ln }\{\!\{v_1\}\!\}\{\!\{v_3\}\!\} \\ \frac{p^{\ln }\{\!\{v_1\}\!\}}{(\gamma -1)} +\{\!\{p\}\!\}\{\!\{v_1\}\!\} + \frac{1}{2}\rho ^{\ln }\{\!\{v_1\}\!\}\overline{||\mathbf {v} ||^2}\end{bmatrix}}, \end{aligned}$$

(A.1)

where

$$\begin{aligned} \{\!\{p\}\!\}= & {} \frac{\{\!\{\rho \}\!\}}{2\{\!\{\beta \}\!\}},\qquad {p}^{\ln } = \frac{{\rho }^{\ln }}{2{\beta }^{\ln }},\nonumber \\ \overline{||\mathbf {v} ||^2}= & {} 2\left( \{\!\{v_1\}\!\}^2 + \{\!\{v_2\}\!\}^2 + \{\!\{v_3\}\!\}^2\right) \nonumber \\&-\left( \{\!\{v_1^2\}\!\} + \{\!\{v_2^2\}\!\} + \{\!\{v_3^2\}\!\}\right) , \end{aligned}$$

(A.2)

and $(\cdot )^{\ln }$ is the logarithmic mean

$$\begin{aligned} (\cdot )^{\mathrm{ln}} = \frac{\llbracket \cdot \rrbracket }{\llbracket \ln (\cdot )\rrbracket }\,. \end{aligned}$$

(A.3)

A numerically stable method to compute the logarithmic mean when $(\cdot )_{\mathrm{R}} \approx (\cdot )_{\mathrm{L}}$ is given in [22, App. B].

It is also possible to add additional dissipation to the entropy conservative scheme and create an entropy stable (ES) approximation. One example is with a matrix dissipation entropy stable flux for the compressible Euler equations of the form

$$\begin{aligned} \mathbf {F}_1^{*,\mathrm{ec}} = \mathbf {F}_1^{*,\mathrm{ec}} - \frac{1}{2}{{\hat{\varvec{\mathcal {R}}_1}}}|\varvec{\hat{\varLambda }}|\hat{\varvec{\mathcal {T}}}{{\hat{\varvec{\mathcal {R}}_1}}}^T\llbracket \mathbf {w}\rrbracket . \end{aligned}$$

(A.4)

The average components of the dissipation term are given by

$$\begin{aligned} \begin{aligned} {{\hat{\varvec{\mathcal {R}}_1}}}&= {\begin{bmatrix} 1&1&0&0&1 \\ \{\!\{v_1\}\!\} - \bar{a}&\{\!\{v_1\}\!\}&0&0&\{\!\{v_1\}\!\} + \bar{a} \\ \{\!\{v_2\}\!\}&\{\!\{v_2\}\!\}&1&0&\{\!\{v_2\}\!\} \\ \{\!\{v_3\}\!\}&\{\!\{v_3\}\!\}&0&1&\{\!\{v_3\}\!\} \\ \bar{h} - \{\!\{v_1\}\!\}\bar{a}&\frac{1}{2}\overline{\Vert \mathbf v \Vert ^2}&\{\!\{v_2\}\!\}&\{\!\{v_3\}\!\}&\bar{h} + \{\!\{v_1\}\!\}\bar{a} \\ \end{bmatrix}},\\ \hat{\varvec{\varLambda }}&= \text {diag}\left( \{\!\{v_1\}\!\} - \bar{a},\{\!\{v_1\}\!\},\{\!\{v_1\}\!\},\{\!\{v_1\}\!\},\{\!\{v_1\}\!\} + \bar{a}\right) ,\\ \hat{\varvec{\mathcal {T}}}&= \text {diag}\left( \frac{\rho ^{\ln }}{2\gamma },\frac{\rho ^{\ln }(\gamma -1)}{\gamma },\{\!\{p\}\!\},\{\!\{p\}\!\},\frac{\rho ^{\ln }}{2\gamma }\right) , \end{aligned} \end{aligned}$$

(A.5)

where

$$\begin{aligned} \bar{a} = \sqrt{\frac{\gamma \{\!\{p\}\!\}}{\rho ^{\ln }}},\qquad \bar{h} = \frac{\gamma }{2\beta ^{\ln }(\gamma -1)} + \frac{1}{2}\overline{||\mathbf {v} ||^2}\,. \end{aligned}$$

(A.6)

We note that the selection of the discrete dissipation operator (A.5) creates a scheme that is able to exactly resolve stationary contact discontinuities. The proof of this property follows the same structure as that presented by Chandrashekar [8].

For completeness, we provide the entropy conserving (and stable) fluxes for the other Cartesian directions

$$\begin{aligned} \mathbf {F}_2^{*,{\mathrm{ec}}}= & {} {\begin{bmatrix} \rho ^{\ln }\{\!\{v_2\}\!\} \\ \rho ^{\ln }\{\!\{v_1\}\!\}\{\!\{v_2\}\!\} \\ \rho ^{\ln }\{\!\{v_2\}\!\}^2 + \{\!\{p\}\!\} \\ \rho ^{\ln }\{\!\{v_2\}\!\}\{\!\{v_3\}\!\} \\ \frac{p^{\ln }\{\!\{v_2\}\!\}}{(\gamma -1)} +\{\!\{p\}\!\}\{\!\{v_2\}\!\} + \frac{1}{2}\rho ^{\ln }\{\!\{v_2\}\!\}\overline{||\mathbf {v} ||^2}\end{bmatrix}},\nonumber \\ \mathbf {F}_3^{*,\mathrm{ec}}= & {} {\begin{bmatrix} \rho ^{\ln }\{\!\{v_3\}\!\} \\ \rho ^{\ln }\{\!\{v_1\}\!\}\{\!\{v_3\}\!\} \\ \rho ^{\ln }\{\!\{v_2\}\!\}\{\!\{v_3\}\!\} \\ \rho ^{\ln }\{\!\{v_3\}\!\}^2 + \{\!\{p\}\!\} \\ \frac{p^{\ln }\{\!\{v_3\}\!\}}{(\gamma -1)} +\{\!\{p\}\!\}\{\!\{v_3\}\!\} + \frac{1}{2}\rho ^{\ln }\{\!\{v_3\}\!\}\overline{||\mathbf {v} ||^2}\end{bmatrix}}. \end{aligned}$$

(A.7)

The matrix dissipation term remains similar where the matrix of right eigenvectors are given by

$$\begin{aligned} {{\hat{\varvec{\mathcal {R}}_2}}}= & {} {\begin{bmatrix} 1&\quad 0&\quad 1&\quad 0&\quad 1 \\ \{\!\{v_1\}\!\}&\quad 1&\quad \{\!\{v_1\}\!\}&\quad 0&\quad \{\!\{v_1\}\!\} \\ \{\!\{v_2\}\!\} - \bar{a}&\quad 0&\quad \{\!\{v_2\}\!\}&\quad 0&\quad \{\!\{v_2\}\!\} + \bar{a} \\ \{\!\{v_3\}\!\}&\quad 0&\quad \{\!\{v_3\}\!\}&\quad 1&\quad \{\!\{v_3\}\!\} \\ \bar{h} - \{\!\{v_2\}\!\}\bar{a}&\quad \{\!\{v_1\}\!\}&\quad \frac{1}{2}\overline{\Vert \mathbf v \Vert ^2}&\quad \{\!\{v_3\}\!\}&\quad \bar{h} + \{\!\{v_2\}\!\}\bar{a} \\ \end{bmatrix}} ,\nonumber \\ {{\hat{\varvec{\mathcal {R}}_3}}}= & {} {\begin{bmatrix} 1&\quad 0&\quad 0&\quad 1&\quad 1 \\ \{\!\{v_1\}\!\}&\quad 1&\quad 0&\quad \{\!\{v_1\}\!\}&\quad \{\!\{v_1\}\!\} \\ \{\!\{v_2\}\!\}&\quad 0&\quad 1&\quad \{\!\{v_2\}\!\}&\quad \{\!\{v_2\}\!\} \\ \{\!\{v_3\}\!\} - \bar{a}&\quad 0&\quad 0&\quad \{\!\{v_3\}\!\}&\quad \{\!\{v_3\}\!\} + \bar{a} \\ \bar{h} - \{\!\{v_3\}\!\}\bar{a}&\quad \{\!\{v_1\}\!\}&\quad \{\!\{v_2\}\!\}&\quad \frac{1}{2}\overline{\Vert \mathbf v \Vert ^2}&\quad \bar{h} + \{\!\{v_3\}\!\}\bar{a} \\ \end{bmatrix}} \end{aligned}$$

(A.8)

respectively. The diagonal matrix of eigenvalues uses the appropriate value of the velocity depending on the spatial direction.

To create the contravariant entropy conservative fluxes we incorperate the average of the metric terms, e.g. at the $(\xi =1)$ element face

$$\begin{aligned} \tilde{\mathbf {F}}^{*,\mathrm{ec}} = \{\!\{Ja_1^1\}\!\}\mathbf {F}_1^{*,\mathrm{ec}} + \{\!\{Ja_2^1\}\!\}\mathbf {F}_2^{*,\mathrm{ec}} + \{\!\{Ja_3^1\}\!\}\mathbf {F}_3^{*,\mathrm{ec}}. \end{aligned}$$

(A.9)

The dissipation terms remain unchanged for the contravariant entropy stable approximations.

Proofs of Entropy Conservation for Non-Linear Advection Terms

1.1 Proof of Entropy Conservation for the One-Dimensional Volume Integral

We first show the property (4.21), rewritten here as

$$\begin{aligned} \left\langle \mathbb {D}(F)^{\mathrm {ec}},W\right\rangle _N = \left. F^{\,\epsilon }\right| _{-1}^1. \end{aligned}$$

(B.1)

For convenience we introduce the entropy potential

$$\begin{aligned} \varPsi ^f_i \equiv W_i\,F_i - F^{\,\epsilon }_i \end{aligned}$$

(B.2)

at each GL node $i=0,...,N$ to rewrite the entropy-conservation condition (4.18) on the two-point flux

$$\begin{aligned} F^{\mathrm {ec}}\,\llbracket W\rrbracket - \llbracket F\,W\rrbracket +\llbracket F^{\,\epsilon }\rrbracket = 0 \end{aligned}$$

(B.3)

as

$$\begin{aligned} F^{\mathrm {ec}}\,\llbracket W\rrbracket = \llbracket \varPsi ^f\rrbracket . \end{aligned}$$

(B.4)

We then explicitly write the volume integral in (B.1) as a sum

$$\begin{aligned} \left\langle \mathbb {D}(F)^{\mathrm {ec}},W\right\rangle _N = \sum \limits _{i=0}^N \omega _i W_i\,2\,\sum _{m=0}^N D_{im} F^{\mathrm {ec}}(U_i,U_m). \end{aligned}$$

(B.5)

Let us now introduce the summation-by-parts matrix $Q = MD$ with entries $Q_{im}\equiv \omega _{i}D_{im}$, which has the property

$$\begin{aligned} Q+Q^T=B\,, \end{aligned}$$

(B.6)

where $B=\mathrm {diag}([-1,0,...,0,1])$ is the boundary evaluation matrix. Because the derivative of a constant is exact, the Q matrix satisfies

$$\begin{aligned} \sum \limits _{m=0}^N Q_{im} V_i = 0 \end{aligned}$$

(B.7)

for all $V_{i}$.

In terms of Q,

$$\begin{aligned} \begin{aligned} \left\langle \mathbb {D}(F)^{\mathrm {ec}},W\right\rangle _N = \sum \limits _{i=0}^N W_i\,2\,\sum _{m=0}^N Q_{im} F^{\mathrm {ec}}(U_i,U_m)\,. \end{aligned} \end{aligned}$$

(B.8)

But from (B.6), $2\,Q_{im} = Q_{im} - Q_{mi} + B_{im}$ so

$$\begin{aligned} \begin{aligned} \left\langle \mathbb {D}(F)^{\mathrm {ec}},W\right\rangle _N&= \sum \limits _{i=0}^N \sum _{m=0}^N W_i\,(Q_{im} - Q_{mi} + B_{im}) F^{\mathrm {ec}}(U_i,U_m)\\ {}&= \sum \limits _{i,m} {{W_i}{Q_{im}}{F^{\mathrm {ec}}}\left( {{U_i},{U_m}} \right) } + \sum \limits _{i,m} {{W_i}{Q_{mi}}{F^{\mathrm {ec}}}\left( {{U_i},{U_m}} \right) } \\&\quad + \sum \limits _{i,m} {{B_{im}}{W_i}{F^{\mathrm {ec}}}\left( {{U_i},{U_m}} \right) }\,. \end{aligned} \end{aligned}$$

(B.9)

We now re-index the second sum, $i\leftrightarrow m$, use the fact that $F^{\mathrm {ec}}$ is symmetric in its arguments, and recombine the sums to get

$$\begin{aligned} \left\langle \mathbb {D}(F)^{\mathrm {ec}},W\right\rangle _N = \sum \limits _{i=0}^N \sum _{m=0}^N \left\{ (W_i - W_m)\,F^{\mathrm {ec}}(U_i,U_m)\,Q_{im} + B_{im}\,W_i\,F^{\mathrm {ec}}(U_i,U_m)\right\} \,.\nonumber \\ \end{aligned}$$

(B.10)

The definition of the entropy potential (B.4) says that we can write the argument

$$\begin{aligned} (W_i - W_m)\,F^{\mathrm {ec}}(U_i,U_m) = \varPsi ^f_i - \varPsi ^f_m\,. \end{aligned}$$

(B.11)

We further note that $B_{im}$ only has entries for $i=m=0$ and $i=m=N$ so with B.2 and the consistency condition on the entropy conserving flux $W_i\,F^{\mathrm {ec}}(U_i,U_i) = W_i\,F(U_i) = \varPsi ^f_i+F^{\,\epsilon }_i$,

$$\begin{aligned} B_{im}\,W_i\,F^{\mathrm {ec}}(U_i,U_m) = B_{im}\,\left( \varPsi ^f_i+F^{\,\epsilon }_i\right) \,. \end{aligned}$$

(B.12)

Inserting (B.11) and (B.12) into (B.10),

$$\begin{aligned} \begin{aligned} \left\langle \mathbb {D}(F)^{\mathrm {ec}},W\right\rangle _N&=\sum \limits _{i=0}^N \sum _{m=0}^N \left( \varPsi ^f_i - \varPsi ^f_m\right) \,Q_{im} + B_{im}\,\left( \varPsi ^f_i+F^{\,\epsilon }_i\right) \\ {}&= \sum \limits _i {\varPsi _i^f\sum \limits _m {{Q_{im}}} } - \sum \limits _{i,m}^{} {\varPsi _m^f{Q_{im}}} + \sum \limits _{i,m} {{B_{im}}\varPsi _i^f} + \sum \limits _{i,m} {{B_{im}}F_i^{\,\epsilon }} \,. \end{aligned} \end{aligned}$$

(B.13)

By (B.7), the first term in the second line is zero. Next, with $B_{im}=Q_{im}+Q_{mi}$, re-indexing $i\leftrightarrow m$, and using (B.7),

$$\begin{aligned} \sum \limits _{i,m} {{B_{im}}\varPsi _i^f} = \sum \limits _{i,m} {{Q_{im}}\varPsi _i^f} + \sum \limits _{i,m} {{Q_{mi}}\varPsi _i^f} = \sum \limits _{i,m} {{Q_{im}}\varPsi _m^f}\,. \end{aligned}$$

(B.14)

Therefore the second and third sums in the second line of (B.13) cancel. What is left is what we set out to show, namely

$$\begin{aligned} \begin{aligned} \left\langle \mathbb {D}(F)^{\mathrm {ec}},W\right\rangle _N&= \sum \limits _{i=0}^N \sum _{m=0}^N B_{im}\,F^{\,\epsilon }_i = F^{\,\epsilon }_N - F^{\,\epsilon }_0 = \left. F^{\,\epsilon }\right| _{-1}^1\,. \end{aligned} \end{aligned}$$

(B.15)

Remark 4

The result (B.1) is a general one that depends only on the summation-by-parts property. It is therefore not specific to the discontinuous Galerkin approximation, per se.

1.2 Proof of Entropy Conservation at Interelement Interfaces for Curvilinear Elements

We now show the property (4.80),

$$\begin{aligned} \sum \limits _{\begin{array}{c} {\mathrm {interior}} \\ {\mathrm {faces}} \end{array}} \int \limits _{N} \left( \left( \tilde{\mathbf {F}}_n^{\mathrm {ec},*}\right) ^T\llbracket \mathbf {W}\rrbracket - \llbracket \left( \tilde{\mathbf {F}}_n\right) ^T\,\mathbf {W}\rrbracket + \llbracket \tilde{ F}_n^{\,\epsilon }\rrbracket \right) \,\,{\text {dS}} = 0\,. \end{aligned}$$

(B.16)

For the approximate 2D surface integral, we evaluate the integrand at $(N+1)^2$ GL quadrature points, see (2.40). Therefore we only need to prove that the integrand vanishes discretely at each interior face quadrature point. We assume that the following derivations are restricted to one interior face quadrature point, and skip quadrature point indices.

The general three-dimensional conditions on the Cartesian components of the two-point numerical flux are

$$\begin{aligned} \begin{aligned}&\left( \mathbf {F}^{\mathrm {ec},*} \right) ^T\,\llbracket \mathbf {W}\rrbracket - \llbracket \mathbf {F}^T\,\mathbf {W}\rrbracket +\llbracket F^{\,\epsilon }\rrbracket = 0,\\&\left( \mathbf {G}^{\mathrm {ec},*} \right) ^T\,\llbracket \mathbf {W}\rrbracket - \llbracket \mathbf {G}^T\,\mathbf {W}\rrbracket +\llbracket G^{\,\epsilon }\rrbracket = 0,\\&\left( \mathbf {H}^{\mathrm {ec},*} \right) ^T\,\llbracket \mathbf {W}\rrbracket - \llbracket \mathbf {H}^T\,\mathbf {W}\rrbracket +\llbracket H^{\,\epsilon }\rrbracket = 0, \end{aligned} \end{aligned}$$

(B.17)

where we use the slave–master jump definition from (3.76),

$$\begin{aligned} \llbracket \mathbf {W}\rrbracket = \mathbf {W}_{\mathrm {slave}} - \mathbf {W}_ {\mathrm {master}} , \end{aligned}$$

(B.18)

and each interior face has the master element side orientation, so that quadrature points of the slave and the master map on each other.

We make the assumption that the mesh is watertight, i.e. that the normal vector and the surface element are continuous across element interfaces. For a conforming hexahedral mesh, the condition holds discretely at the surface quadrature points if we ensure the discrete metric identities (2.32) and if the unit outward facing normal vector and surface element on the element side are constructed from the element metrics by

$$\begin{aligned} \hat{s}= \left| \sum \limits _{l=1}^3\left( J\vec {a}^{\,l}\right) \hat{n}^l\right| \,,\quad \vec {n} = \frac{1}{\hat{s}} \sum \limits _{l=1}^3\left( J\vec {a}^{\,d}\right) \hat{n}^l. \end{aligned}$$

(B.19)

The continuity of the surface metric allows us to use only the metric of the master element side, so that we are able to move the metric into the jump.

We can write any normal contravariant surface flux using the Cartesian fluxes and the metric

$$\begin{aligned} \left( \tilde{\mathbf {F}}_n\right)= & {} \left( \hat{s}\vec {n}\right) \cdot {\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }} =\hat{s}\left( \mathbf {F} n_1 +\mathbf {G} n_2 + \mathbf {H} n_3\right) = \sum \limits _{l=1}^3\hat{n}^l \left( \left( Ja_1^{\,l}\right) \mathbf {F} +\left( Ja_2^{\,l}\right) \mathbf {G} + \left( Ja_3^{\,l}\right) \mathbf {H} \right) \nonumber \\= & {} \left( \mathcal {M}^T {\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }} \right) \cdot \hat{n}\,. \end{aligned}$$

(B.20)

We combine the three Cartesian equations (B.17) with the surface metric $\hat{s}\vec {n}$, defined in (B.19), leading to

$$\begin{aligned} \begin{aligned}&\hat{s}\Bigg (n_1 \left[ (\mathbf {F}^{\mathrm {ec},*} )^T\,\llbracket \mathbf {W}\rrbracket - \llbracket \mathbf {F}^T\,\mathbf {W}\rrbracket +\llbracket F^{\,\epsilon }\rrbracket \right] \\&\quad +n_2 \left[ (\mathbf {G}^{\mathrm {ec},*} )^T\,\llbracket \mathbf {W}\rrbracket - \llbracket \mathbf {G}^T\,\mathbf {W}\rrbracket +\llbracket G^{\,\epsilon }\rrbracket \right] \\&\quad +n_3 \left[ (\mathbf {H}^{\mathrm {ec},*} )^T\,\llbracket \mathbf {W}\rrbracket - \llbracket \mathbf {H}^T\,\mathbf {W}\rrbracket +\llbracket H^{\,\epsilon }\rrbracket \right] \Bigg )= 0\,. \end{aligned} \end{aligned}$$

(B.21)

If we define the contravariant numerical flux as

$$\begin{aligned} \tilde{\mathbf {F}}^{\mathrm {ec},*}_{n} \equiv \hat{s}\left( n_1\mathbf {F}^{\mathrm {ec},*} + n_2\mathbf {G}^{\mathrm {ec},*} + n_3\mathbf {H}^{\mathrm {ec},*} \right) \,, \end{aligned}$$

(B.22)

then

$$\begin{aligned} (\tilde{\mathbf {F}}^{\mathrm {ec},*}_{n})^T\,\llbracket \mathbf {W}\rrbracket +\left( \hat{s}\vec {n}\,\right) \cdot \left( - \llbracket {\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^T\,\mathbf {W}\rrbracket + \llbracket \vec {F}^{\,\epsilon }\rrbracket \right) = 0\,. \end{aligned}$$

(B.23)

Finally, using the continuity of the surface metric, we move it inside the jump terms, yielding

$$\begin{aligned} \begin{aligned} \left( \mathbf {F}^{\mathrm {ec},*}_{n}\right) ^T\,\llbracket \mathbf {W}\rrbracket - \llbracket \left( \left( \mathcal {M}^T{\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}\right) \cdot \hat{n}\right) ^T\,\mathbf {W}\rrbracket +\llbracket \left( \mathcal {M}^T\vec {F}^{\,\epsilon }\right) \cdot \hat{n}\rrbracket&\\ =\left( \tilde{\mathbf {F}}^{\mathrm {ec},*}_{n}\right) ^T\,\llbracket \mathbf {W}\rrbracket - \llbracket \left( \tilde{\mathbf {F}}_n\right) ^T\,\mathbf {W}\rrbracket +\llbracket \tilde{ F}_n^{\,\epsilon }\rrbracket&= 0 \,,\\ \end{aligned} \end{aligned}$$

(B.24)

proving that in (4.80), the integrand vanishes at each quadrature point of an interior face individually.

1.3 Proof of Entropy Conservation in 3D Curvilinear Coordinates

We show in this section that the property (4.75) holds, reproduced here for convenience as

$$\begin{aligned} \left\langle \vec {\mathbb {D}}({\mathop {\tilde{\mathbf {F}}}\limits ^{\leftrightarrow }})^{\mathrm {ec}},\mathbf {W}\right\rangle _N = \int \limits _{\partial E,N} \left( \vec {\tilde{F}}^{\,\epsilon }\cdot \hat{n}\right) \,{\text {dS}} \,, \end{aligned}$$

(B.25)

provided that the discrete metric identities (2.32) are satisfied.

Similar to what was done in “Appendix B.1”, we introduce three entropy potentials, one for each Cartesian space direction

$$\begin{aligned} \begin{aligned} \varPsi ^l_{ijk}&\equiv \left( \mathbf {F}^{l}_{ijk}\right) ^{T}\mathbf {W}_{ijk} - F^{l,{\,\epsilon }}_{ijk},\; l = 1,2,3, \end{aligned} \end{aligned}$$

(B.26)

for each GL node $i,j,k=0,...,N$. We use them to rewrite the entropy-conservation condition on the two-point fluxes $F^{l,\mathrm {ec},*}$, c.f. (B.17), into

$$\begin{aligned} \left( \mathbf {F}^{l,\mathrm {ec}}_{(i,m)jk}\right) ^T\,\llbracket \mathbf {W}\rrbracket _{(i,m)jk} = \llbracket \varPsi ^l\rrbracket _{(i,m)jk} ,\; l = 1,2,3, \end{aligned}$$

(B.27)

for $i,j,k,m=0,...,N$.

To match terms, we expand all the terms of the volume integral approximation

$$\begin{aligned} \begin{aligned} \left\langle \vec {\mathbb {D}}({\mathop {\tilde{\mathbf {F}}}\limits ^{\leftrightarrow }})^{\mathrm {ec}},\mathbf {W}\right\rangle _N&\equiv \,\sum \limits _{i,j,k=0}^N\omega _{ijk}\,\mathbf {W}^T_{ijk}\Bigg [ 2\sum _{m=0}^N D_{im}\left( {\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk}, U_{mjk})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}\right) \\&\quad +2\sum _{m=0}^N D_{jm}\left( {\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk}, U_{imk})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,2}\right\} \!\!\right\} _{i(j,m)k}\right) \\&\quad +2\sum _{m=0}^N D_{km}\left( {\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk}, U_{ijm})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,3}\right\} \!\!\right\} _{ij(k,m)}\right) \Bigg ]\,, \end{aligned} \end{aligned}$$

(B.28)

(c.f. (4.69)) and of the surface integral approximation

$$\begin{aligned} \begin{aligned} \int \limits _{\partial E,N} \left( \vec {\tilde{F}}^{\,\epsilon }\cdot \hat{n}\right) \,\,{\text {dS}} =&\quad \sum \limits _{j,k=0}^N\omega _{jk}\left[ \left( \left( J\vec {a}^{\,1}\right) _{Njk}\cdot \vec {F}^{\,\epsilon }_{Njk}\right) -\left( \left( J\vec {a}^{\,1}\right) _{0jk}\cdot \vec {F}^{\,\epsilon }_{0jk}\right) \right] \\&+\sum \limits _{i,k=0}^N\omega _{ik}\left[ \left( \left( J\vec {a}^{\,2}\right) _{iNk}\cdot \vec {F}^{\,\epsilon }_{iNk}\right) -\left( \left( J\vec {a}^{\,2}\right) _{i0k}\cdot \vec {F}^{\,\epsilon }_{i0k}\right) \right] \\&+\sum \limits _{i,j=0}^N\omega _{ij}\left[ \left( \left( J\vec {a}^{\,3}\right) _{ijN}\cdot \vec {F}^{\,\epsilon }_{ijN}\right) - \left( \left( J\vec {a}^{\,3}\right) _{ij0}\cdot \vec {F}^{\,\epsilon }_{ij0}\right) \right] \,. \end{aligned} \end{aligned}$$

(B.29)

We then focus on the first ($\xi$ direction) term of the volume integral approximation, which allows us to follow the one dimensional proof as closely as possible. The sum can be written in terms of $Q_{im}=w_{i}D_{im}$,

$$\begin{aligned} \begin{aligned}&\sum \limits _{i,j,k=0}^N\omega _{ijk}\,\mathbf {W}^T_{ijk} 2\sum _{m=0}^N D_{im}\, {\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk}, U_{mjk})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk} \\&\qquad \qquad =\sum \limits _{j,k=0}^N\omega _{jk}\sum \limits _{i=0}^N \mathbf {W}^T_{ijk}2 \sum _{m=0}^N \omega _iD_{im}\,{\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk}, U_{mjk})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk} \\&\qquad \qquad =\sum \limits _{j,k=0}^N\omega _{jk}\sum \limits _{i=0}^N \mathbf {W}^T_{ijk}\, \sum _{m=0}^N 2Q_{im}\,{\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk}, U_{mjk})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}\,. \end{aligned} \end{aligned}$$

(B.30)

Therefore, we can use the same steps as in one dimension: We use the summation-by-parts property $2\,Q_{im}=Q_{im} - Q_{mi} + B_{im}$, a re-indexing of i and m to subsume the $Q_{mi}$ term, and the facts that $F^{\mathrm {ec}}(U_{ijk}, U_{mjk})$ and the jump operator of the metric term $\left\{ \!\!\left\{ Ja^1_1\right\} \!\!\right\} _{(i,m)jk}$ are symmetric with respect to the index i and m to rewrite the $\xi$ direction contribution to the volume integral approximation as

$$\begin{aligned} \begin{aligned}&\sum \limits _{i=0}^N\mathbf {W}^T_{ijk}\,2\sum _{m=0}^N Q_{im}\, {\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk}, U_{mjk})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk} \\&\quad = \sum \limits _{i=0}^N \sum _{m=0}^N \mathbf {W}^T_{ijk}\,(Q_{im} - Q_{mi} + B_{im}) {\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk},U_{mjk})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}\\&\quad = \sum \limits _{i=0}^N \sum _{m=0}^N Q_{im}\left( \mathbf {W}_{ijk} - \mathbf {W}_{mjk}\right) ^T\, {\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk},U_{mjk})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}\, \\&\quad + B_{im}\,\mathbf {W}^T_{ijk}\,{\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk},U_{mjk})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}. \end{aligned} \end{aligned}$$

(B.31)

Next, we use the relations (B.27) of the Cartesian two-point entropy-conserving flux $\mathbf {F}^{l,\mathrm {ec}}$ to note that

$$\begin{aligned} \left( \mathbf {W}_{ijk} - \mathbf {W}_{mjk}\right) ^T\,\mathbf {F}^{l,\mathrm {ec}}(U_{ijk},U_{mjk}) = \varPsi ^l_{ijk} - \varPsi ^l_{mjk}\,, \; l = 1,2,3. \end{aligned}$$

(B.32)

We further note that $B_{im}$ only has entries for $i=m=0$ and $i=m=N$ (c.f. (B.12)) so

$$\begin{aligned} B_{im}\,\mathbf {W}^T_{ijk}\,\mathbf {F}^{l,\mathrm {ec}}(U_{ijk},U_{mjk}) = B_{im}\,\left( \varPsi ^l_{ijk}+ F^{l,{\,\epsilon }}_{ijk}\right) \,, \; l = 1,2,3. \end{aligned}$$

(B.33)

Finally, we can exploit the consistency of the two-point flux and (B.26) so that for $i=m=0$ and $i=m=N$,

$$\begin{aligned} \mathbf {W}^T_{ijk}\,\mathbf {F}^{l,\mathrm {ec}}(U_{ijk},U_{ijk}) = \mathbf {W}^T_{ijk}\,\mathbf {F}(U_{ijk}) = \varPsi ^l_{ijk}+ F^{l,{\,\epsilon }}_{ijk}\,, \; l = 1,2,3. \end{aligned}$$

(B.34)

Inserting relations (B.32)-(B.34) into the last line of (B.31) says that

$$\begin{aligned} \begin{aligned}&\sum \limits _{i=0}^N \mathbf {W}^T_{ijk}\,2\sum _{m=0}^N Q_{im}\,{\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk}, U_{mjk})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}\\&\quad = \sum \limits _{i=0}^N \sum _{m=0}^N Q_{im} \left( \vec {\varPsi }_{ijk} - \vec {\varPsi }_{mjk}\right) \cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk} +B_{im}\,\left( \vec {\varPsi }_{ijk}+\vec {F}^{\,\epsilon }_{ijk}\right) \cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}. \end{aligned} \end{aligned}$$

(B.35)

Now, since the derivative of a constant is zero,

$$\begin{aligned} \begin{aligned} \sum \limits _{i,m = 0}^N {{Q_{im}}{\vec {\varPsi }_{ijk}}\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}}&= \frac{1}{2}\sum \limits _{i = 0}^N {{\vec {\varPsi }_{ijk}}\cdot \left( J\vec {a}^{\,1}\right) _{ijk}\sum \limits _{m = 0}^N {{Q_{im}}} } \\&\quad + \frac{1}{2}\sum \limits _{i = 0}^N {\sum \limits _{m = 0}^N {{{\vec {\varPsi }_{ijk}}\cdot \left( J\vec {a}^{\,1}\right) _{mjk}Q_{im}}} } \\&= \frac{1}{2}\sum \limits _{i = 0}^N {\sum \limits _{m = 0}^N {{{\vec {\varPsi }_{ijk}}\cdot \left( J\vec {a}^{1}\right) _{mjk}Q_{im}}} }. \end{aligned}\end{aligned}$$

Next, since $B_{im}=0$ unless $i=m=0$ or $i=m=N$,

$$\begin{aligned} \sum \limits _{i=0}^N \sum _{m=0}^N B_{im}\vec {F}^{\,\epsilon }_{ijk}\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}=\sum \limits _{i=0}^N \sum _{m=0}^N B_{im}\vec {F}^{\,\epsilon }_{ijk}\cdot \left( J\vec {a}^{\,1}\right) _{ijk}. \end{aligned}$$

Finally, $B_{im}$ being diagonal and symmetric, we can swap the $i\leftrightarrow m$ in $B_{im}\vec {\varPsi }_{ijk}$ to rewrite the second line of (B.35)

$$\begin{aligned} \begin{aligned}&\sum \limits _{i=0}^N \sum \limits _{m=0}^N Q_{im}\left( \vec {\varPsi }_{ijk} - \vec {\varPsi }_{mjk}\right) \cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk} + B_{im}\,\left( \vec {\varPsi }_{ijk}+\vec {F}^{\,\epsilon }_{ijk}\right) \cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk} \\&\quad =\sum \limits _{i=0}^N \sum \limits _{m=0}^N \frac{1}{2}Q_{im}\vec {\varPsi }_{ijk} \cdot \left( J\vec {a}^{\,1}\right) _{mjk} + B_{im}\vec {F}^{\,\epsilon }_{ijk}\cdot \left( J\vec {a}^{\,1}\right) _{ijk}\\&\qquad + (B_{im}-Q_{im})\vec {\varPsi }_{mjk}\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}. \\ \end{aligned} \end{aligned}$$

(B.36)

The last step is to use the summation-by-parts property $B_{im}-Q_{im}=Q_{mi}$ and re-index to see that

$$\begin{aligned} \begin{aligned} \sum \limits _{i=0}^N \sum \limits _{m=0}^N(B_{im}-Q_{im})\vec {\varPsi }_{mjk}\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}&=\sum \limits _{i=0}^N \sum \limits _{m=0}^N \frac{1}{2}(Q_{mi})\vec {\varPsi }_{mjk}\cdot \left( J\vec {a}^{\,1}\right) _{ijk} \\&= \sum \limits _{i=0}^N \sum \limits _{m=0}^N \frac{1}{2}(Q_{im})\vec {\varPsi }_{ijk}\cdot \left( J\vec {a}^{\,1}\right) _{mjk}\,. \end{aligned} \end{aligned}$$

(B.37)

Therefore,

$$\begin{aligned} \begin{aligned}&\sum \limits _{i=0}^N \sum \limits _{m=0}^N Q_{im}\left( \vec {\varPsi }_{ijk} - \vec {\varPsi }_{mjk}\right) \cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk} + B_{im}\,\left( \vec {\varPsi }_{ijk}+\vec {F}^{\,\epsilon }_{ijk}\right) \cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk}\\&\quad =\sum \limits _{i=0}^N \sum \limits _{m=0}^N Q_{im}\vec {\varPsi }_{ijk} \cdot \left( J\vec {a}^{\,1}\right) _{mjk} + B_{im}\vec {F}^{\,\epsilon }_{ijk}\cdot \left( J\vec {a}^{\,1}\right) _{ijk} . \end{aligned} \end{aligned}$$

(B.38)

To summarize, the first ($\xi$ direction) part of the volume integral approximation (B.30) is

$$\begin{aligned} \begin{aligned} \sum \limits _{i,j,k=0}^N&\omega _{ijk}\,\mathbf {W}^T_{ijk} 2\sum _{m=0}^N D_{im}\, {\mathop {{\mathbf {F}}}\limits ^{\leftrightarrow }}^{\mathrm {ec}}(U_{ijk}, U_{mjk})\cdot \left\{ \!\!\left\{ J\vec {a}^{\,1}\right\} \!\!\right\} _{(i,m)jk} \\&= \sum \limits _{j,k=0}^N \omega _{jk}\left( \vec {F}^{\,\epsilon }_{Njk}\cdot \left( J\vec {a}^{\,1}\right) _{Njk}-\vec {F}^{\,\epsilon }_{0jk}\cdot \left( J\vec {a}^{\,1}\right) _{0jk}+ \sum \limits _{i,m=0}^N\omega _i D_{im}\vec {\varPsi }_{ijk} \cdot \left( J\vec {a}^{\,1}\right) _{mjk} \right) \,, \end{aligned} \end{aligned}$$

(B.39)

where we have returned $Q_{im}=\omega _i D_{im}$ and represented the boundary terms explicitly.

Similar results hold for the second and third parts of the volume integral approximation, leading to

$$\begin{aligned} \begin{aligned}&\left\langle \vec {\mathbb {D}}({\mathop {\tilde{\mathbf {F}}}\limits ^{\leftrightarrow }})^{\mathrm {ec}},\mathbf {W}\right\rangle _N\\&\quad = \sum \limits _{j,k=0}^N \omega _{jk}\left( \vec {F}^{\,\epsilon }_{Njk} \cdot \left( J\vec {a}^{\,1}\right) _{Njk} -\vec {F}^{\,\epsilon }_{0jk} \cdot \left( J\vec {a}^{\,1}\right) _{0jk} + \sum \limits _{i,m=0}^N \omega _i D_{im}\vec {\varPsi }_{ijk} \cdot \left( J\vec {a}^{\,1}\right) _{mjk} \right) \\&\quad + \sum \limits _{i,k=0}^N \omega _{ik}\left( \vec {F}^{\,\epsilon }_{iNk} \cdot \left( J\vec {a}^{\,2}\right) _{iNk} -\vec {F}^{\,\epsilon }_{i0k} \cdot \left( J\vec {a}^{\,2}\right) _{i0k} + \sum \limits _{j,m=0}^N \omega _j D_{jm}\vec {\varPsi }_{ijk} \cdot \left( J\vec {a}^{\,2}\right) _{imk} \right) \\&\quad + \sum \limits _{i,j=0}^N \omega _{ij}\left( \vec {F}^{\,\epsilon }_{ijN} \cdot \left( J\vec {a}^{\,3}\right) _{ijN} -\vec {F}^{\,\epsilon }_{ij0} \cdot \left( J\vec {a}^{\,3}\right) _{ij0} + \sum \limits _{k,m=0}^N \omega _k D_{km}\vec {\varPsi }_{ijk} \cdot \left( J\vec {a}^{\,3}\right) _{ijm} \right) \,. \end{aligned} \end{aligned}$$

(B.40)

Regrouping the boundary terms and the volume terms, we have shown that

$$\begin{aligned} \left\langle \vec {\mathbb {D}}({\mathop {\tilde{\mathbf {F}}}\limits ^{\leftrightarrow }})^{\mathrm {ec}},\mathbf {W}\right\rangle _N= & {} \int \limits _{\partial E,N} \left( \vec {\tilde{F}}^{\,\epsilon }\cdot \hat{n}\right) \,\,{\text {dS}} \nonumber \\&+ \sum \limits _{i,j,k=0}^N\omega _{ijk}\varPsi ^1_{ijk}\, \left\{ \sum _{m=0}^N D_{im}\,\left( Ja^1_1\right) _{mjk}+D_{jm}\,\left( Ja^2_1\right) _{imk}+D_{km}\,\left( Ja^3_1\right) _{ijm}\right\} \nonumber \\&+ \sum \limits _{i,j,k=0}^N\omega _{ijk}\varPsi ^2_{ijk}\, \left\{ \sum _{m=0}^N D_{im}\,\left( Ja^1_2\right) _{mjk}+D_{jm}\,\left( Ja^2_2\right) _{imk}+D_{km}\,\left( Ja^3_2\right) _{ijm}\right\} \nonumber \\&+ \sum \limits _{i,j,k=0}^N\omega _{ijk}\varPsi ^3_{ijk}\, \left\{ \sum _{m=0}^N D_{im}\,\left( Ja^1_3\right) _{mjk}+D_{jm}\,\left( Ja^2_3\right) _{imk}+D_{km}\,\left( Ja^3_3\right) _{ijm}\right\} ,\nonumber \\ \end{aligned}$$

(B.41)

which gives the desired entropy condition (B.25), provided that the metric identities (2.32) hold discretely, i.e. that

$$\begin{aligned} \sum _{m=0}^N D_{im}\,\left( J a^1_n\right) _{mjk}+D_{jm}\,\left( Ja^2_n\right) _{imk}+D_{km}\,\left( Ja^3_n\right) _{ijm}=\left( \sum _{l=1}^{3}\frac{\partial }{\partial \xi ^{l}}\mathbb {I}^{N}\left( Ja^l_n\right) \right) _{ijk} = 0 \,,\nonumber \\ \end{aligned}$$

(B.42)

for $n=1,2,3$ and all GL points $i,j,k,=0,...,N$ within an element. For the strategy on how to properly approximate the metric terms so that the metric identities are satisfied, see [25].