From 2012 Petrova, "Finite Volume Methods- Powerful Means of Engineering Design"

$\overline{\varphi }=\frac{1}{\mathrm{\Delta }t}{\int }_{t_0}^{t_0+\mathrm{\Delta }t}\varphi dt=$ Reynolds averaging over time.

$\tilde{\varphi }=\frac{1}{\overline{\rho }}\frac{1}{\mathrm{\Delta }t}{\int }_{t_0}^{t_0+\mathrm{\Delta }t}(\rho \varphi )dt=$ Favre averaging.


variables:
$n^i$ = flux surface normal, in units of $[1]$
$\overline{\rho }$ = Reynolds-averaged density, in units of $\frac{kg}{m^3}$
${\tilde{v}}^i$ = Favre-averaged velocity, in units of $\frac{1}{s}m$
${\overline{m}}^i=\overline{\rho }\cdot {\tilde{v}}^i$ $=\overline{\rho v^i}=$ Reynolds-averaged momentum, in units of $\frac{kg}{m^{2}s}$
$k$ = turbulent kinetic energy, in units of $\frac{m^2}{s^2}$
$\omega$ = specific turbulent dissipation rate, in units of ???
$\tilde{T}$ = Favre-averaged temperature, in units of $K$
$C_v$ = constant-volume heat capacity, in units of $\frac{K{s}^2}{m^2}$
$C_p$ = constant-pressure heat capacity, in units of $\frac{K{s}^2}{m^2}$
$R$ = specific heat constant, in units of $\frac{m^2}{K{s}^2}$
$R=C_p-C_v$
$\gamma$ = heat capacity ratio, unitless
$\gamma =\frac{1}{C_v}{C}_p$
$\overline{P}=\overline{\rho }\cdot R\tilde{T}$ = Reynolds-averaged pressure, in units of $\frac{kg}{m{s}^2}$
$P^{\ast }$ = $\overline{P}+\frac{2}{3}\overline{\rho }\cdot k$ = $\frac{1}{3}\overline{\rho }\cdot \left(2k+3R\tilde{T}\right)$ = static pressure, in units of $\frac{kg}{m{s}^2}$
$c_s=\frac{1}{\sqrt{C_v}}\frac{1}{\sqrt{\overline{\rho }}}\sqrt{C_v\cdot P^{\ast }\cdot {|n|}^2+P^{\ast }\cdot R{|n|}^2-R\overline{\rho }\cdot {\tilde{v}}^c{\tilde{v}}^d{n}_c{n}_d}$ = speed of sound in units of $\frac{1}{s}m$
${g_i}_j$ = metric tensor, in units of $[1]$
$(\tilde{v}{)}^2={\tilde{v}}^k{\tilde{v}}^l{g_k}_l$ = Favre-averaged velocity, norm squared, in units of $\frac{m^2}{s^2}$

${\tilde{e}}_{kin}=\frac{1}{2}{\tilde{v}}^k{\tilde{v}}^l{g_k}_l$ = Favre-averaged specific kinetic energy, in units of $\frac{m^2}{s^2}$

${\tilde{e}}_{int}$ = $C_v\cdot \tilde{T}$ = $\frac{C_v\cdot \left(3P^{\ast }-2\overline{\rho }\cdot k\right)}{3R\overline{\rho }}$ = Favre-averaged specific internal energy, in units of $\frac{m^2}{s^2}$

${\tilde{e}}_{total}={\tilde{e}}_{int}+{\tilde{e}}_{kin}$ = Favre-averaged densitized total energy, in units of $\frac{kg}{m{s}^2}$
${\tilde{e}}_{total}=C_v\cdot P^{\ast }\cdot \frac{1}{R}\frac{1}{\overline{\rho }}+\frac{1}{2}(\tilde{v}{)}^2+-2\cdot \frac{1}{3}{C}_v\cdot k\frac{1}{R}$

Conservative and primitive variables:
$W^I=\left[\begin{array}{c}\overline{\rho }\\ {\tilde{v}}^i\\ P^{\ast }\\ k\\ \omega \end{array}\right]$
$U^I=\left[\begin{array}{c}\overline{\rho }\\ \overline{\rho }\cdot {\tilde{v}}^i\\ \overline{\rho }\cdot {\tilde{e}}_{total}\\ \overline{\rho }\cdot k\\ \overline{\rho }\cdot \omega \end{array}\right]$
Partial of conservative quantities wrt primitives:
$\frac{\partial U^I}{\partial W^J}=\left[\begin{array}{ccccc}\frac{\partial \overline{\rho }}{\partial \overline{\rho }}& \frac{\partial \overline{\rho }}{\partial {\tilde{v}}^j}& \frac{\partial \overline{\rho }}{\partial P^{\ast }}& \frac{\partial \overline{\rho }}{\partial k}& \frac{\partial \overline{\rho }}{\partial \omega }\\ \frac{\partial }{\partial \overline{\rho }}\left(\overline{\rho }\cdot {\tilde{v}}^i\right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(\overline{\rho }\cdot {\tilde{v}}^i\right)& \frac{\partial }{\partial P^{\ast }}\left(\overline{\rho }\cdot {\tilde{v}}^i\right)& \frac{\partial }{\partial k}\left(\overline{\rho }\cdot {\tilde{v}}^i\right)& \frac{\partial }{\partial \omega }\left(\overline{\rho }\cdot {\tilde{v}}^i\right)\\ \frac{\partial }{\partial \overline{\rho }}\left(\overline{\rho }\cdot {\tilde{e}}_{total}\right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(\overline{\rho }\cdot {\tilde{e}}_{total}\right)& \frac{\partial }{\partial P^{\ast }}\left(\overline{\rho }\cdot {\tilde{e}}_{total}\right)& \frac{\partial }{\partial k}\left(\overline{\rho }\cdot {\tilde{e}}_{total}\right)& \frac{\partial }{\partial \omega }\left(\overline{\rho }\cdot {\tilde{e}}_{total}\right)\\ \frac{\partial }{\partial \overline{\rho }}\left(\overline{\rho }\cdot k\right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(\overline{\rho }\cdot k\right)& \frac{\partial }{\partial P^{\ast }}\left(\overline{\rho }\cdot k\right)& \frac{\partial }{\partial k}\left(\overline{\rho }\cdot k\right)& \frac{\partial }{\partial \omega }\left(\overline{\rho }\cdot k\right)\\ \frac{\partial }{\partial \overline{\rho }}\left(\overline{\rho }\cdot \omega \right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(\overline{\rho }\cdot \omega \right)& \frac{\partial }{\partial P^{\ast }}\left(\overline{\rho }\cdot \omega \right)& \frac{\partial }{\partial k}\left(\overline{\rho }\cdot \omega \right)& \frac{\partial }{\partial \omega }\left(\overline{\rho }\cdot \omega \right)\end{array}\right]$
$\frac{\partial U^I}{\partial W^J}=\left[\begin{array}{ccccc}\frac{\partial \overline{\rho }}{\partial \overline{\rho }}& \frac{\partial \overline{\rho }}{\partial {\tilde{v}}^j}& \frac{\partial \overline{\rho }}{\partial P^{\ast }}& \frac{\partial \overline{\rho }}{\partial k}& \frac{\partial \overline{\rho }}{\partial \omega }\\ \frac{\partial }{\partial \overline{\rho }}\left(\overline{\rho }\cdot {\tilde{v}}^i\right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(\overline{\rho }\cdot {\tilde{v}}^i\right)& \frac{\partial }{\partial P^{\ast }}\left(\overline{\rho }\cdot {\tilde{v}}^i\right)& \frac{\partial }{\partial k}\left(\overline{\rho }\cdot {\tilde{v}}^i\right)& \frac{\partial }{\partial \omega }\left(\overline{\rho }\cdot {\tilde{v}}^i\right)\\ \frac{\partial }{\partial \overline{\rho }}\left(\overline{\rho }\cdot \left(\frac{1}{2}{\tilde{v}}^k{\tilde{v}}^l{g_k}_l+\frac{C_v\cdot \left(3P^{\ast }-2\overline{\rho }\cdot k\right)}{3R\overline{\rho }}\right)\right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(\overline{\rho }\cdot \left(\frac{1}{2}{\tilde{v}}^k{\tilde{v}}^l{g_k}_l+\frac{C_v\cdot \left(3P^{\ast }-2\overline{\rho }\cdot k\right)}{3R\overline{\rho }}\right)\right)& \frac{\partial }{\partial P^{\ast }}\left(\overline{\rho }\cdot \left(\frac{1}{2}{\tilde{v}}^k{\tilde{v}}^l{g_k}_l+\frac{C_v\cdot \left(3P^{\ast }-2\overline{\rho }\cdot k\right)}{3R\overline{\rho }}\right)\right)& \frac{\partial }{\partial k}\left(\overline{\rho }\cdot \left(\frac{1}{2}{\tilde{v}}^k{\tilde{v}}^l{g_k}_l+\frac{C_v\cdot \left(3P^{\ast }-2\overline{\rho }\cdot k\right)}{3R\overline{\rho }}\right)\right)& \frac{\partial }{\partial \omega }\left(\overline{\rho }\cdot \left(\frac{1}{2}{\tilde{v}}^k{\tilde{v}}^l{g_k}_l+\frac{C_v\cdot \left(3P^{\ast }-2\overline{\rho }\cdot k\right)}{3R\overline{\rho }}\right)\right)\\ \frac{\partial }{\partial \overline{\rho }}\left(\overline{\rho }\cdot k\right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(\overline{\rho }\cdot k\right)& \frac{\partial }{\partial P^{\ast }}\left(\overline{\rho }\cdot k\right)& \frac{\partial }{\partial k}\left(\overline{\rho }\cdot k\right)& \frac{\partial }{\partial \omega }\left(\overline{\rho }\cdot k\right)\\ \frac{\partial }{\partial \overline{\rho }}\left(\overline{\rho }\cdot \omega \right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(\overline{\rho }\cdot \omega \right)& \frac{\partial }{\partial P^{\ast }}\left(\overline{\rho }\cdot \omega \right)& \frac{\partial }{\partial k}\left(\overline{\rho }\cdot \omega \right)& \frac{\partial }{\partial \omega }\left(\overline{\rho }\cdot \omega \right)\end{array}\right]$
$\frac{\partial U^I}{\partial W^J}=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ {\tilde{v}}^i& \overline{\rho }\cdot {{\delta }^i}_j& 0& 0& 0\\ -2\cdot \frac{1}{3}{C}_v\cdot k\frac{1}{R}+\frac{1}{2}(\tilde{v}{)}^2& \overline{\rho }\cdot {\tilde{v}}_j& C_v\cdot \frac{1}{R}& -2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot \frac{1}{R}& 0\\ k& 0& 0& \overline{\rho }& 0\\ \omega & 0& 0& 0& \overline{\rho }\end{array}\right]$
Expanded:
$\frac{\partial U^I}{\partial W^J}=\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ {\tilde{v}}^x& \overline{\rho }& 0& 0& 0& 0& 0\\ {\tilde{v}}^y& 0& \overline{\rho }& 0& 0& 0& 0\\ {\tilde{v}}^z& 0& 0& \overline{\rho }& 0& 0& 0\\ \frac{3(\tilde{v}{)}^2\cdot R-4C_v\cdot k}{6R}& \overline{\rho }\cdot {\tilde{v}}_x& \overline{\rho }\cdot {\tilde{v}}_y& \overline{\rho }\cdot {\tilde{v}}_z& \frac{1}{R}{C}_v& -\frac{2C_v\cdot \overline{\rho }}{3R}& 0\\ k& 0& 0& 0& 0& \overline{\rho }& 0\\ \omega & 0& 0& 0& 0& 0& \overline{\rho }\end{array}\right]$
$(\tilde{v}{)}^2={\tilde{v}}^x{\tilde{v}}_x+{\tilde{v}}^y{\tilde{v}}_y+{\tilde{v}}^z{\tilde{v}}_z$
$\frac{\partial W^I}{\partial U^J}=\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ -\frac{1}{\overline{\rho }}{\tilde{v}}^x& \frac{1}{\overline{\rho }}& 0& 0& 0& 0& 0\\ -\frac{1}{\overline{\rho }}{\tilde{v}}^y& 0& \frac{1}{\overline{\rho }}& 0& 0& 0& 0\\ -\frac{1}{\overline{\rho }}{\tilde{v}}^z& 0& 0& \frac{1}{\overline{\rho }}& 0& 0& 0\\ \frac{R(\tilde{v}{)}^2}{2C_v}& -\frac{1}{C_v}R{\tilde{v}}_x& -\frac{1}{C_v}R{\tilde{v}}_y& -\frac{1}{C_v}R{\tilde{v}}_z& \frac{1}{C_v}R& \frac{2}{3}& 0\\ -\frac{1}{\overline{\rho }}k& 0& 0& 0& 0& \frac{1}{\overline{\rho }}& 0\\ -\frac{1}{\overline{\rho }}\omega & 0& 0& 0& 0& 0& \frac{1}{\overline{\rho }}\end{array}\right]$
$\frac{\partial W^I}{\partial U^J}=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ \frac{1}{\overline{\rho }}\left(-{\tilde{v}}^i\right)& \frac{1}{\overline{\rho }}{{\delta }^i}_j& 0& 0& 0\\ \frac{1}{2}\frac{1}{C_v}R(\tilde{v}{)}^2& -\frac{1}{C_v}R{\tilde{v}}_j& \frac{1}{C_v}R& \frac{2}{3}& 0\\ \frac{1}{\overline{\rho }}\left(-k\right)& 0& 0& \frac{1}{\overline{\rho }}& 0\\ \frac{1}{\overline{\rho }}\left(-\omega \right)& 0& 0& 0& \frac{1}{\overline{\rho }}\end{array}\right]$

Flux:
$F^I=\left[\begin{array}{c}\overline{\rho }\cdot {\tilde{v}}^j{n}_j\\ \overline{\rho }\cdot {\tilde{v}}^i{\tilde{v}}^j{n}_j+n^i{P}^{\ast }\\ {\tilde{v}}^j{n}_j\left(\overline{\rho }\cdot {\tilde{e}}_{total}+P^{\ast }\right)\\ \overline{\rho }\cdot {\tilde{v}}^j{n}_{j}k\\ \overline{\rho }\cdot {\tilde{v}}^j{n}_j\omega \end{array}\right]$
$F^I=\left[\begin{array}{c}\overline{\rho }\cdot {\tilde{v}}^j{n}_j\\ P^{\ast }\cdot n^i+\overline{\rho }\cdot {\tilde{v}}^i{\tilde{v}}^j{n}_j\\ C_v\cdot P^{\ast }\cdot {\tilde{v}}^j{n}_j\frac{1}{R}+P^{\ast }\cdot {\tilde{v}}^j{n}_j+\frac{1}{2}(\tilde{v}{)}^2\cdot \overline{\rho }\cdot {\tilde{v}}^j{n}_j+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{\tilde{v}}^j{n}_j\frac{1}{R}\\ \overline{\rho }\cdot k{\tilde{v}}^j{n}_j\\ \overline{\rho }\cdot \omega \cdot {\tilde{v}}^j{n}_j\end{array}\right]$

Flux derivative wrt primitive variables:
$\frac{\partial F^I}{\partial W^J}=\left[\begin{array}{ccccc}\frac{\partial }{\partial \overline{\rho }}\left(\overline{\rho }\cdot {\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(\overline{\rho }\cdot {\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial P^{\ast }}\left(\overline{\rho }\cdot {\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial k}\left(\overline{\rho }\cdot {\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial \omega }\left(\overline{\rho }\cdot {\tilde{v}}^k{n}_k\right)\\ \frac{\partial }{\partial \overline{\rho }}\left(P^{\ast }\cdot n^i+\overline{\rho }\cdot {\tilde{v}}^i{\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(P^{\ast }\cdot n^i+\overline{\rho }\cdot {\tilde{v}}^i{\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial P^{\ast }}\left(P^{\ast }\cdot n^i+\overline{\rho }\cdot {\tilde{v}}^i{\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial k}\left(P^{\ast }\cdot n^i+\overline{\rho }\cdot {\tilde{v}}^i{\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial \omega }\left(P^{\ast }\cdot n^i+\overline{\rho }\cdot {\tilde{v}}^i{\tilde{v}}^k{n}_k\right)\\ \frac{\partial }{\partial \overline{\rho }}\left(C_v\cdot P^{\ast }\cdot {\tilde{v}}^k{n}_k\frac{1}{R}+P^{\ast }\cdot {\tilde{v}}^k{n}_k+\frac{1}{2}(\tilde{v}{)}^2\cdot \overline{\rho }\cdot {\tilde{v}}^k{n}_k+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{\tilde{v}}^k{n}_k\frac{1}{R}\right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(C_v\cdot P^{\ast }\cdot {\tilde{v}}^k{n}_k\frac{1}{R}+P^{\ast }\cdot {\tilde{v}}^k{n}_k+\frac{1}{2}(\tilde{v}{)}^2\cdot \overline{\rho }\cdot {\tilde{v}}^k{n}_k+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{\tilde{v}}^k{n}_k\frac{1}{R}\right)& \frac{\partial }{\partial P^{\ast }}\left(C_v\cdot P^{\ast }\cdot {\tilde{v}}^k{n}_k\frac{1}{R}+P^{\ast }\cdot {\tilde{v}}^k{n}_k+\frac{1}{2}(\tilde{v}{)}^2\cdot \overline{\rho }\cdot {\tilde{v}}^k{n}_k+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{\tilde{v}}^k{n}_k\frac{1}{R}\right)& \frac{\partial }{\partial k}\left(C_v\cdot P^{\ast }\cdot {\tilde{v}}^k{n}_k\frac{1}{R}+P^{\ast }\cdot {\tilde{v}}^k{n}_k+\frac{1}{2}(\tilde{v}{)}^2\cdot \overline{\rho }\cdot {\tilde{v}}^k{n}_k+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{\tilde{v}}^k{n}_k\frac{1}{R}\right)& \frac{\partial }{\partial \omega }\left(C_v\cdot P^{\ast }\cdot {\tilde{v}}^k{n}_k\frac{1}{R}+P^{\ast }\cdot {\tilde{v}}^k{n}_k+\frac{1}{2}(\tilde{v}{)}^2\cdot \overline{\rho }\cdot {\tilde{v}}^k{n}_k+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{\tilde{v}}^k{n}_k\frac{1}{R}\right)\\ \frac{\partial }{\partial \overline{\rho }}\left(\overline{\rho }\cdot k{\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(\overline{\rho }\cdot k{\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial P^{\ast }}\left(\overline{\rho }\cdot k{\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial k}\left(\overline{\rho }\cdot k{\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial \omega }\left(\overline{\rho }\cdot k{\tilde{v}}^k{n}_k\right)\\ \frac{\partial }{\partial \overline{\rho }}\left(\overline{\rho }\cdot \omega \cdot {\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial {\tilde{v}}^j}\left(\overline{\rho }\cdot \omega \cdot {\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial P^{\ast }}\left(\overline{\rho }\cdot \omega \cdot {\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial k}\left(\overline{\rho }\cdot \omega \cdot {\tilde{v}}^k{n}_k\right)& \frac{\partial }{\partial \omega }\left(\overline{\rho }\cdot \omega \cdot {\tilde{v}}^k{n}_k\right)\end{array}\right]$
$\frac{\partial F^I}{\partial W^J}=\left[\begin{array}{ccccc}{\tilde{v}}^k{n}_k& \overline{\rho }\cdot n_j& 0& 0& 0\\ {\tilde{v}}^i{\tilde{v}}^k{n}_k& \overline{\rho }\cdot {\tilde{v}}^i{n}_j+\overline{\rho }\cdot {\tilde{v}}^k{n}_k{{\delta }^i}_j& n^i& 0& 0\\ \frac{1}{2}(\tilde{v}{)}^2\cdot {\tilde{v}}^k{n}_k+-2\cdot \frac{1}{3}{C}_v\cdot k{\tilde{v}}^k{n}_k\frac{1}{R}& C_v\cdot P^{\ast }\cdot n_j\frac{1}{R}+P^{\ast }\cdot n_j+\frac{1}{2}(\tilde{v}{)}^2\cdot \overline{\rho }\cdot n_j+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{n}_j\frac{1}{R}& C_v\cdot {\tilde{v}}^k{n}_k\frac{1}{R}+{\tilde{v}}^k{n}_k& -2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot {\tilde{v}}^k{n}_k\frac{1}{R}& 0\\ k{\tilde{v}}^k{n}_k& \overline{\rho }\cdot k{n}_j& 0& \overline{\rho }\cdot {\tilde{v}}^k{n}_k& 0\\ \omega \cdot {\tilde{v}}^k{n}_k& \overline{\rho }\cdot \omega \cdot n_j& 0& 0& \overline{\rho }\cdot {\tilde{v}}^k{n}_k\end{array}\right]$
$\frac{\partial F^I}{\partial W^J}=\left[\begin{array}{ccccc}{\tilde{v}}^a{n}_a& \overline{\rho }\cdot n_j& 0& 0& 0\\ {\tilde{v}}^a{\tilde{v}}^i{n}_a& \overline{\rho }\cdot {\tilde{v}}^i{n}_j+\overline{\rho }\cdot {\tilde{v}}^a{n}_a{{\delta }^i}_j& n^i& 0& 0\\ \frac{1}{2}(\tilde{v}{)}^2\cdot {\tilde{v}}^a{n}_a+-2\cdot \frac{1}{3}{C}_v\cdot k{\tilde{v}}^a{n}_a\frac{1}{R}& C_v\cdot P^{\ast }\cdot n_j\frac{1}{R}+P^{\ast }\cdot n_j+\frac{1}{2}(\tilde{v}{)}^2\cdot \overline{\rho }\cdot n_j+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{n}_j\frac{1}{R}& C_v\cdot {\tilde{v}}^a{n}_a\frac{1}{R}+{\tilde{v}}^a{n}_a& -2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot {\tilde{v}}^a{n}_a\frac{1}{R}& 0\\ k{\tilde{v}}^a{n}_a& \overline{\rho }\cdot k{n}_j& 0& \overline{\rho }\cdot {\tilde{v}}^a{n}_a& 0\\ \omega \cdot {\tilde{v}}^a{n}_a& \overline{\rho }\cdot \omega \cdot n_j& 0& 0& \overline{\rho }\cdot {\tilde{v}}^a{n}_a\end{array}\right]$
$\frac{\partial F^I}{\partial W^J}=\left[\begin{array}{ccccc}{\tilde{v}}^a{n}_a& \overline{\rho }\cdot n_j& 0& 0& 0\\ {\tilde{v}}^a{\tilde{v}}^i{n}_a& \overline{\rho }\cdot {\tilde{v}}^i{n}_j+\overline{\rho }\cdot {\tilde{v}}^a{n}_a{{\delta }^i}_j& n^i& 0& 0\\ \frac{1}{2}(\tilde{v}{)}^2\cdot {\tilde{v}}^a{n}_a+-2\cdot \frac{1}{3}{C}_v\cdot k{\tilde{v}}^a{n}_a\frac{1}{R}& C_v\cdot P^{\ast }\cdot n_j\frac{1}{R}+P^{\ast }\cdot n_j+\frac{1}{2}(\tilde{v}{)}^2\cdot \overline{\rho }\cdot n_j+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{n}_j\frac{1}{R}& C_v\cdot {\tilde{v}}^a{n}_a\frac{1}{R}+{\tilde{v}}^a{n}_a& -2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot {\tilde{v}}^a{n}_a\frac{1}{R}& 0\\ k{\tilde{v}}^a{n}_a& \overline{\rho }\cdot k{n}_j& 0& \overline{\rho }\cdot {\tilde{v}}^a{n}_a& 0\\ \omega \cdot {\tilde{v}}^a{n}_a& \overline{\rho }\cdot \omega \cdot n_j& 0& 0& \overline{\rho }\cdot {\tilde{v}}^a{n}_a\end{array}\right]$
$\frac{\partial F^I}{\partial W^J}=\left[\begin{array}{ccccc}{\tilde{v}}^a{n}_a& \overline{\rho }\cdot n_j& 0& 0& 0\\ {\tilde{v}}^a{\tilde{v}}^i{n}_a& \overline{\rho }\cdot {\tilde{v}}^i{n}_j+\overline{\rho }\cdot {\tilde{v}}^a{n}_a{{\delta }^i}_j& n^i& 0& 0\\ -2\cdot \frac{1}{3}{C}_v\cdot k{\tilde{v}}^a{n}_a\frac{1}{R}+\frac{1}{2}{\tilde{v}}^a{\tilde{v}}^b{\tilde{v}}_b{n}_a& C_v\cdot P^{\ast }\cdot n_j\frac{1}{R}+P^{\ast }\cdot n_j+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{n}_j\frac{1}{R}+\frac{1}{2}\overline{\rho }\cdot {\tilde{v}}_a{\tilde{v}}^a{n}_j& C_v\cdot {\tilde{v}}^a{n}_a\frac{1}{R}+{\tilde{v}}^a{n}_a& -2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot {\tilde{v}}^a{n}_a\frac{1}{R}& 0\\ k{\tilde{v}}^a{n}_a& \overline{\rho }\cdot k{n}_j& 0& \overline{\rho }\cdot {\tilde{v}}^a{n}_a& 0\\ \omega \cdot {\tilde{v}}^a{n}_a& \overline{\rho }\cdot \omega \cdot n_j& 0& 0& \overline{\rho }\cdot {\tilde{v}}^a{n}_a\end{array}\right]$

Flux derivative wrt conserved variables:
$\frac{\partial F^I}{\partial U^J}=\frac{\partial F^I}{\partial W^L}\frac{\partial W^L}{\partial U^J}$
$\frac{\partial F^I}{\partial U^J}=\left[\begin{array}{ccccc}{\tilde{v}}^a{n}_a& \overline{\rho }\cdot n_k& 0& 0& 0\\ {\tilde{v}}^a{\tilde{v}}^i{n}_a& \overline{\rho }\cdot {\tilde{v}}^i{n}_k+\overline{\rho }\cdot {\tilde{v}}^a{n}_a{{\delta }^i}_k& n^i& 0& 0\\ -2\cdot \frac{1}{3}{C}_v\cdot k{\tilde{v}}^a{n}_a\frac{1}{R}+\frac{1}{2}{\tilde{v}}^a{\tilde{v}}^b{\tilde{v}}_b{n}_a& C_v\cdot P^{\ast }\cdot n_k\frac{1}{R}+P^{\ast }\cdot n_k+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{n}_k\frac{1}{R}+\frac{1}{2}\overline{\rho }\cdot {\tilde{v}}_a{\tilde{v}}^a{n}_k& C_v\cdot {\tilde{v}}^a{n}_a\frac{1}{R}+{\tilde{v}}^a{n}_a& -2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot {\tilde{v}}^a{n}_a\frac{1}{R}& 0\\ k{\tilde{v}}^a{n}_a& \overline{\rho }\cdot k{n}_k& 0& \overline{\rho }\cdot {\tilde{v}}^a{n}_a& 0\\ \omega \cdot {\tilde{v}}^a{n}_a& \overline{\rho }\cdot \omega \cdot n_k& 0& 0& \overline{\rho }\cdot {\tilde{v}}^a{n}_a\end{array}\right]\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ \frac{1}{\overline{\rho }}\left(-{\tilde{v}}^k\right)& \frac{1}{\overline{\rho }}{{\delta }^k}_j& 0& 0& 0\\ \frac{1}{2}\frac{1}{C_v}R(\tilde{v}{)}^2& -\frac{1}{C_v}R{\tilde{v}}_j& \frac{1}{C_v}R& \frac{2}{3}& 0\\ \frac{1}{\overline{\rho }}\left(-k\right)& 0& 0& \frac{1}{\overline{\rho }}& 0\\ \frac{1}{\overline{\rho }}\left(-\omega \right)& 0& 0& 0& \frac{1}{\overline{\rho }}\end{array}\right]$
$\frac{\partial F^I}{\partial U^J}=\left[\begin{array}{ccccc}{\tilde{v}}^a{n}_a+-1{\tilde{v}}^k{n}_k& n_k{{\delta }^k}_j& 0& 0& 0\\ \frac{1}{2}(\tilde{v}{)}^2\cdot R{n}^i\frac{1}{C_v}+-1{\tilde{v}}^i{\tilde{v}}^k{n}_k+{\tilde{v}}^a{\tilde{v}}^i{n}_a+-1{\tilde{v}}^a{\tilde{v}}^k{n}_a{{\delta }^i}_k& -1R{\tilde{v}}_j{n}^i\frac{1}{C_v}+{\tilde{v}}^i{n}_k{{\delta }^k}_j+{\tilde{v}}^a{n}_a{{\delta }^i}_k{{\delta }^k}_j& R{n}^i\frac{1}{C_v}& 2\cdot \frac{1}{3}{n}^i& 0\\ -1C_v\cdot P^{\ast }\cdot {\tilde{v}}^k{n}_k\frac{1}{R}\frac{1}{\overline{\rho }}+\frac{1}{2}(\tilde{v}{)}^2\cdot R{\tilde{v}}^a{n}_a\frac{1}{C_v}+2\cdot \frac{1}{3}{C}_v\cdot k{\tilde{v}}^k{n}_k\frac{1}{R}+-1P^{\ast }\cdot {\tilde{v}}^k{n}_k\frac{1}{\overline{\rho }}+\frac{1}{2}(\tilde{v}{)}^2\cdot {\tilde{v}}^a{n}_a+-1\cdot \frac{1}{2}{\tilde{v}}^a{\tilde{v}}_a{\tilde{v}}^k{n}_k+\frac{1}{2}{\tilde{v}}^a{\tilde{v}}^b{\tilde{v}}_b{n}_a& C_v\cdot P^{\ast }\cdot n_k{{\delta }^k}_j\frac{1}{R}\frac{1}{\overline{\rho }}+-2\cdot \frac{1}{3}{C}_v\cdot k{n}_k{{\delta }^k}_j\frac{1}{R}+P^{\ast }\cdot n_k{{\delta }^k}_j\frac{1}{\overline{\rho }}+-1R{\tilde{v}}^a{\tilde{v}}_j{n}_a\frac{1}{C_v}+-1{\tilde{v}}^a{\tilde{v}}_j{n}_a+\frac{1}{2}{\tilde{v}}_a{\tilde{v}}^a{n}_k{{\delta }^k}_j& {\tilde{v}}^a{n}_a+R{\tilde{v}}^a{n}_a\frac{1}{C_v}& 2\cdot \frac{1}{3}{\tilde{v}}^a{n}_a& 0\\ -1k{\tilde{v}}^k{n}_k& k{n}_k{{\delta }^k}_j& 0& {\tilde{v}}^a{n}_a& 0\\ -1\omega \cdot {\tilde{v}}^k{n}_k& \omega \cdot n_k{{\delta }^k}_j& 0& 0& {\tilde{v}}^a{n}_a\end{array}\right]$
$\frac{\partial F^I}{\partial U^J}=\left[\begin{array}{ccccc}0& n_j& 0& 0& 0\\ \frac{1}{2}(\tilde{v}{)}^2\cdot R{n}^i\frac{1}{C_v}+-1{\tilde{v}}^a{\tilde{v}}^i{n}_a& {\tilde{v}}^i{n}_j+-1R{\tilde{v}}_j{n}^i\frac{1}{C_v}+{\tilde{v}}^a{n}_a{{\delta }^i}_j& R{n}^i\frac{1}{C_v}& 2\cdot \frac{1}{3}{n}^i& 0\\ -1C_v\cdot P^{\ast }\cdot {\tilde{v}}^a{n}_a\frac{1}{R}\frac{1}{\overline{\rho }}+\frac{1}{2}(\tilde{v}{)}^2\cdot R{\tilde{v}}^a{n}_a\frac{1}{C_v}+2\cdot \frac{1}{3}{C}_v\cdot k{\tilde{v}}^a{n}_a\frac{1}{R}+-1P^{\ast }\cdot {\tilde{v}}^a{n}_a\frac{1}{\overline{\rho }}+\frac{1}{2}(\tilde{v}{)}^2\cdot {\tilde{v}}^a{n}_a+-1\cdot \frac{1}{2}{\tilde{v}}_a{\tilde{v}}^a{\tilde{v}}^b{n}_b+\frac{1}{2}{\tilde{v}}^a{\tilde{v}}^b{\tilde{v}}_b{n}_a& C_v\cdot P^{\ast }\cdot n_j\frac{1}{R}\frac{1}{\overline{\rho }}+-2\cdot \frac{1}{3}{C}_v\cdot k{n}_j\frac{1}{R}+P^{\ast }\cdot n_j\frac{1}{\overline{\rho }}+-1R{\tilde{v}}^a{\tilde{v}}_j{n}_a\frac{1}{C_v}+\frac{1}{2}{\tilde{v}}_a{\tilde{v}}^a{n}_j+-1{\tilde{v}}^a{\tilde{v}}_j{n}_a& {\tilde{v}}^a{n}_a+R{\tilde{v}}^a{n}_a\frac{1}{C_v}& 2\cdot \frac{1}{3}{\tilde{v}}^a{n}_a& 0\\ -1k{\tilde{v}}^a{n}_a& k{n}_j& 0& {\tilde{v}}^a{n}_a& 0\\ -1\omega \cdot {\tilde{v}}^a{n}_a& \omega \cdot n_j& 0& 0& {\tilde{v}}^a{n}_a\end{array}\right]$

Acoustic matrix:
${A^I}_J+{\tilde{v}}_n\cdot {{\delta }^I}_J=\frac{\partial W^I}{\partial U^K}\frac{\partial F^K}{\partial W^J}$
${A^I}_J+{\tilde{v}}_n\cdot {{\delta }^I}_J=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ \frac{1}{\overline{\rho }}\left(-{\tilde{v}}^i\right)& \frac{1}{\overline{\rho }}{{\delta }^i}_k& 0& 0& 0\\ \frac{1}{2}\frac{1}{C_v}R(\tilde{v}{)}^2& -\frac{1}{C_v}R{\tilde{v}}_k& \frac{1}{C_v}R& \frac{2}{3}& 0\\ \frac{1}{\overline{\rho }}\left(-k\right)& 0& 0& \frac{1}{\overline{\rho }}& 0\\ \frac{1}{\overline{\rho }}\left(-\omega \right)& 0& 0& 0& \frac{1}{\overline{\rho }}\end{array}\right]\left[\begin{array}{ccccc}{\tilde{v}}^a{n}_a& \overline{\rho }\cdot n_j& 0& 0& 0\\ {\tilde{v}}^a{\tilde{v}}^k{n}_a& \overline{\rho }\cdot {\tilde{v}}^k{n}_j+\overline{\rho }\cdot {\tilde{v}}^a{n}_a{{\delta }^k}_j& n^k& 0& 0\\ -2\cdot \frac{1}{3}{C}_v\cdot k{\tilde{v}}^a{n}_a\frac{1}{R}+\frac{1}{2}{\tilde{v}}^a{\tilde{v}}^b{\tilde{v}}_b{n}_a& C_v\cdot P^{\ast }\cdot n_j\frac{1}{R}+P^{\ast }\cdot n_j+-2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot k{n}_j\frac{1}{R}+\frac{1}{2}\overline{\rho }\cdot {\tilde{v}}_a{\tilde{v}}^a{n}_j& C_v\cdot {\tilde{v}}^a{n}_a\frac{1}{R}+{\tilde{v}}^a{n}_a& -2\cdot \frac{1}{3}{C}_v\cdot \overline{\rho }\cdot {\tilde{v}}^a{n}_a\frac{1}{R}& 0\\ k{\tilde{v}}^a{n}_a& \overline{\rho }\cdot k{n}_j& 0& \overline{\rho }\cdot {\tilde{v}}^a{n}_a& 0\\ \omega \cdot {\tilde{v}}^a{n}_a& \overline{\rho }\cdot \omega \cdot n_j& 0& 0& \overline{\rho }\cdot {\tilde{v}}^a{n}_a\end{array}\right]$
${A^I}_J+{\tilde{v}}_n\cdot {{\delta }^I}_J=\left[\begin{array}{ccccc}{\tilde{v}}^a{n}_a& \overline{\rho }\cdot n_j& 0& 0& 0\\ -1{\tilde{v}}^a{\tilde{v}}^i{n}_a\frac{1}{\overline{\rho }}+{\tilde{v}}^a{\tilde{v}}^k{n}_a{{\delta }^i}_k\frac{1}{\overline{\rho }}& -1{\tilde{v}}^i{n}_j+{\tilde{v}}^k{n}_j{{\delta }^i}_k+{\tilde{v}}^a{n}_a{{\delta }^i}_k{{\delta }^k}_j& n^k{{\delta }^i}_k\frac{1}{\overline{\rho }}& 0& 0\\ \frac{1}{2}(\tilde{v}{)}^2\cdot R{\tilde{v}}^a{n}_a\frac{1}{C_v}+\frac{1}{2}R{\tilde{v}}^a{\tilde{v}}^b{\tilde{v}}_b{n}_a\frac{1}{C_v}+-1R{\tilde{v}}^a{\tilde{v}}^k{\tilde{v}}_k{n}_a\frac{1}{C_v}& P^{\ast }\cdot n_j+P^{\ast }\cdot R{n}_j\frac{1}{C_v}+\frac{1}{2}(\tilde{v}{)}^2\cdot R\overline{\rho }\cdot n_j\frac{1}{C_v}+\frac{1}{2}R\overline{\rho }\cdot {\tilde{v}}^a{\tilde{v}}_a{n}_j\frac{1}{C_v}+-1R\overline{\rho }\cdot {\tilde{v}}_k{\tilde{v}}^k{n}_j\frac{1}{C_v}+-1R\overline{\rho }\cdot {\tilde{v}}^a{\tilde{v}}_k{n}_a{{\delta }^k}_j\frac{1}{C_v}& {\tilde{v}}^a{n}_a+R{\tilde{v}}^a{n}_a\frac{1}{C_v}+-1R{\tilde{v}}_k{n}^k\frac{1}{C_v}& 0& 0\\ 0& 0& 0& {\tilde{v}}^a{n}_a& 0\\ 0& 0& 0& 0& {\tilde{v}}^a{n}_a\end{array}\right]$
${A^I}_J+{\tilde{v}}_n\cdot {{\delta }^I}_J=\left[\begin{array}{ccccc}{\tilde{v}}_n& \overline{\rho }\cdot n_j& 0& 0& 0\\ 0& {\tilde{v}}_n\cdot {{\delta }^i}_j& n^i\frac{1}{\overline{\rho }}& 0& 0\\ 0& P^{\ast }\cdot n_j+P^{\ast }\cdot R{n}_j\frac{1}{C_v}+-1R\overline{\rho }\cdot {\tilde{v}}_n\cdot {\tilde{v}}_j\frac{1}{C_v}& {\tilde{v}}_n& 0& 0\\ 0& 0& 0& {\tilde{v}}_n& 0\\ 0& 0& 0& 0& {\tilde{v}}_n\end{array}\right]$
${A^I}_J=\left[\begin{array}{ccccc}0& \overline{\rho }\cdot n_j& 0& 0& 0\\ 0& 0& \frac{1}{\overline{\rho }}{n}^i& 0& 0\\ 0& \frac{1}{C_v}\left(C_v\cdot P^{\ast }\cdot n_j+P^{\ast }\cdot R{n}_j-R\overline{\rho }\cdot {\tilde{v}}_n\cdot {\tilde{v}}_j\right)& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

$c_s=\frac{\sqrt{P^{\ast }\cdot \gamma \cdot {|n|}^2-\overline{\rho }\cdot {\gamma }_1\cdot {{\tilde{v}}_n}^2}}{\sqrt{\overline{\rho }}}$
$P^{\ast }=\frac{\overline{\rho }\cdot \left({c_s}^2+{\gamma }_1\cdot {{\tilde{v}}_n}^2\right)}{\gamma \cdot {|n|}^2}$
Acoustic matrix, expanded:
${A^I}_J=\left[\begin{array}{ccccccc}0& \overline{\rho }\cdot n_x& \overline{\rho }\cdot n_y& \overline{\rho }\cdot n_z& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\overline{\rho }}{n}^x& 0& 0\\ 0& 0& 0& 0& \frac{1}{\overline{\rho }}{n}^y& 0& 0\\ 0& 0& 0& 0& \frac{1}{\overline{\rho }}{n}^z& 0& 0\\ 0& \frac{1}{C_v}\left(C_v\cdot P^{\ast }\cdot n_x+P^{\ast }\cdot R{n}_x-R\overline{\rho }\cdot {\tilde{v}}_n\cdot {\tilde{v}}_x\right)& \frac{1}{C_v}\left(C_v\cdot P^{\ast }\cdot n_y+P^{\ast }\cdot R{n}_y-R\overline{\rho }\cdot {\tilde{v}}_n\cdot {\tilde{v}}_y\right)& \frac{1}{C_v}\left(C_v\cdot P^{\ast }\cdot n_z+P^{\ast }\cdot R{n}_z-R\overline{\rho }\cdot {\tilde{v}}_n\cdot {\tilde{v}}_z\right)& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

...in just the x-axis (using $n_x=1$, $n_y=0$, $n_z=0$ )
${A^I}_J=\left[\begin{array}{ccccccc}0& \overline{\rho }& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\overline{\rho }}{n}^x& 0& 0\\ 0& 0& 0& 0& \frac{1}{\overline{\rho }}{n}^y& 0& 0\\ 0& 0& 0& 0& \frac{1}{\overline{\rho }}{n}^z& 0& 0\\ 0& \frac{1}{C_v}\left(C_v\cdot P^{\ast }+P^{\ast }\cdot R-R\overline{\rho }\cdot {\tilde{v}}_n\cdot {\tilde{v}}_x\right)& -\frac{1}{C_v}R\overline{\rho }\cdot {\tilde{v}}_n\cdot {\tilde{v}}_y& -\frac{1}{C_v}R\overline{\rho }\cdot {\tilde{v}}_n\cdot {\tilde{v}}_z& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

...with a Cartesian metric (using $n^x=1$, $n^y=0$, $n^z=0$ )
${A^I}_J=\left[\begin{array}{ccccccc}0& \overline{\rho }& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\overline{\rho }}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& \frac{1}{C_v}\left(C_v\cdot P^{\ast }+P^{\ast }\cdot R-R\overline{\rho }\cdot {\tilde{v}}_x{\tilde{v}}_x\right)& -\frac{1}{C_v}R\overline{\rho }\cdot {\tilde{v}}_y{\tilde{v}}_x& -\frac{1}{C_v}R\overline{\rho }\cdot {\tilde{v}}_z{\tilde{v}}_x& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

speed of sound in Cartesian x-axis:
$c_s=\frac{\sqrt{P^{\ast }+P^{\ast }\cdot {\gamma }_1-\overline{\rho }\cdot {\gamma }_1\cdot {{\tilde{v}}_x}^2}}{\sqrt{\overline{\rho }}}$
using $R=C_p-C_v$ , $C_p=C_v\cdot \gamma$ , ${\gamma }_1=\gamma -1$
${A^I}_J=\left[\begin{array}{ccccccc}0& \overline{\rho }& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\overline{\rho }}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& P^{\ast }\cdot \gamma -\overline{\rho }\cdot {\gamma }_1\cdot {{\tilde{v}}_x}^2& -\overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_x{\tilde{v}}_y& -\overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_x{\tilde{v}}_z& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

Acoustic matrix eigen-decomposition:
${A^I}_J={{(R_A)}^I}_M{{({\mathrm{\Lambda }}_A)}^M}_N{{(L_A)}^N}_J$
A charpoly: $-\frac{1}{\overline{\rho }}{\lambda }^5\left(-P^{\ast }\cdot \gamma +\overline{\rho }\cdot {\gamma }_1\cdot {{\tilde{v}}_x}^2+\overline{\rho }\cdot {\lambda }^2\right)=0$
$R=\left[\begin{array}{ccccccc}0& 0& 0& 1& 0& \frac{1}{{c_s}^2}& \frac{1}{{c_s}^2}\\ 0& 0& 0& 0& \frac{1}{\overline{\rho }}& \frac{1}{c_s\cdot \overline{\rho }}& \frac{-1}{c_s\cdot \overline{\rho }}\\ 0& 0& \frac{-{\tilde{v}}_z}{{\tilde{v}}_x}& 0& \frac{{c_s}^2}{{\gamma }_1\cdot \overline{\rho }\cdot {\tilde{v}}_u{\tilde{v}}_v}& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 1\\ 0& 1& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\end{array}\right]$
$\mathrm{\Lambda }=\left[\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -c_s& 0\\ 0& 0& 0& 0& 0& 0& c_s\end{array}\right]$
$L=\left[\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 1& 0& 0& 0& -\frac{1}{{c_s}^2}& 0& 0\\ 0& 0& \frac{\overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_u{\tilde{v}}_v}{{c_s}^2}& \frac{\overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_u{\tilde{v}}_v{\tilde{v}}_z}{{\tilde{v}}_x{c_s}^2}& 0& 0& 0\\ 0& \frac{1}{2}\overline{\rho }\cdot c_s& -\frac{\overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_u{\tilde{v}}_v}{2c_s}& -\frac{\overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_u{\tilde{v}}_v{\tilde{v}}_z}{2c_s\cdot {\tilde{v}}_x}& \frac{1}{2}& 0& 0\\ 0& -\frac{1}{2}\overline{\rho }\cdot c_s& \frac{\overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_u{\tilde{v}}_v}{2c_s}& \frac{\overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_u{\tilde{v}}_v{\tilde{v}}_z}{2c_s\cdot {\tilde{v}}_x}& \frac{1}{2}& 0& 0\end{array}\right]$
reconstructed:
$A=\left[\begin{array}{ccccccc}0& -\overline{\rho }& \frac{\overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_u{\tilde{v}}_v}{{c_s}^2}& \frac{\overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_u{\tilde{v}}_v{\tilde{v}}_z}{{\tilde{v}}_x{c_s}^2}& 0& 0& 0\\ 0& 0& 0& 0& -\frac{1}{\overline{\rho }}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& -\overline{\rho }\cdot {c_s}^2& \overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_u{\tilde{v}}_v& \frac{\overline{\rho }\cdot {\gamma }_1\cdot {\tilde{v}}_u{\tilde{v}}_v{\tilde{v}}_z}{{\tilde{v}}_x}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$