variables:
${ {(n_1)}} ^i$ = flux surface normal, in units of $[1]$
$\rho$ = density, in units of $\frac{kg}{{m}^{3}}$
${ v} ^i$ = velocity, in units of ${\frac{1}{s}} {m}$
${{ m} ^i} = {{{\rho}} \cdot {{{ v} ^i}}}$ = momentum, in units of $\frac{kg}{{{{m}^{2}}} {{s}}}$
$\gamma$ = heat capacity ratio, in units of $[1]$
${{(\gamma_{-1})}} = {{\gamma}{-{1}}}$
$P$ = pressure, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{c_s}} = {\sqrt{{\frac{1}{\rho}} {{{\gamma}} \cdot {{P}}}}}$ = speed of sound in units of ${\frac{1}{s}} {m}$
${{ g} _i} _j$ = metric tensor, in units of $[1]$
${{(v)^2}} = {{{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}$ = velocity norm squared, in units of $\frac{{m}^{2}}{{s}^{2}}$
${{e_{kin}}} = {{{\frac{1}{2}}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}$ = specific kinetic energy, in units of $\frac{{m}^{2}}{{s}^{2}}$
${{e_{int}}} = {\frac{P}{{{{(\gamma_{-1})}}} \cdot {{\rho}}}}$ = specific internal energy, in units of $\frac{{m}^{2}}{{s}^{2}}$
${{E_{total}}} = {{{\rho}} \cdot {{\left({{{e_{int}}} + {{e_{kin}}}}\right)}}}$ = densitized total energy, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{E_{total}}} = {{{{P}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{\frac{1}{2}}} {{{(v)^2}}} \cdot {{\rho}}}}$
${{H_{total}}} = {{{E_{total}}} + {P}}$ = total enthalpy, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{H_{total}}} = {{{{P}} {{\frac{1}{{(\gamma_{-1})}}}}} + {P} + {{{\frac{1}{2}}} {{{(v)^2}}} \cdot {{\rho}}}}$
${{h_{total}}} = {{\frac{1}{\rho}} {{H_{total}}}}$
${{h_{total}}} = {{{{P}} {{\frac{1}{{(\gamma_{-1})}}}} {{\frac{1}{\rho}}}} + {{{P}} {{\frac{1}{\rho}}}} + {{{\frac{1}{2}}} {{{(v)^2}}}}}$
Finite volume form:
${{{{ U} ^I} _{,t}} + {{{ {F(n^j)}} ^I} _{;j}}} = {0}$
This is ideal because we have the option of working with the flux vector, or working with its jacobian wrt conserved quantities, or working with its eigensystem.
As flux Jacobian:
${{{{ U} ^I} _{,t}} + {{{\frac{\partial { {F(n^j)}} ^I}{\partial { U} ^J}}} {{{{ U} ^J} _{;j}}}}} = {0}$
As eigensystem of jacobian:
${{{ U} _{,t}} + {{{{R_F(n^j)}}} \cdot {{{\Lambda_F(n^j)}}} \cdot {{{L_F(n^j)}}} \cdot {{{ U} _{;j}}}}} = {0}$
As characteristic variables:
${{{{{L_F(n^j)}}} \cdot {{{ U} _{,t}}}} + {{{{L_F(n^j)}}} \cdot {{{R_F(n^j)}}} \cdot {{{\Lambda_F(n^j)}}} \cdot {{{L_F(n^j)}}} \cdot {{{R_F(n^j)}}} \cdot {{{L_F(n^j)}}} \cdot {{{ U} _{;j}}}}} = {0}$
${{{{{L_F(n^j)}}} \cdot {{{ U} _{,t}}}} + {{{{\Lambda_F(n^j)}}} \cdot {{{L_F(n^j)}}} \cdot {{{ U} _{;j}}}}} = {0}$
Now you have 3 separate PDEs to solve, however they are dependent on flux normal, and not easily solvable.
As a PDE of primitive variables:
${{{ U} _{,t}} + {{{\frac{\partial {F(n^j)}}{\partial U}}} {{{ U} _{;j}}}}} = {0}$
${{{{\frac{\partial W}{\partial U}}} {{{ U} _{,t}}}} + {{{\frac{\partial W}{\partial U}}} {{\frac{\partial {F(n^j)}}{\partial U}}} {{{ U} _{;j}}}}} = {0}$
${{{{\frac{\partial W}{\partial U}}} {{{ U} _{,t}}}} + {{{\frac{\partial W}{\partial U}}} {{\frac{\partial {F(n^j)}}{\partial U}}} {{\frac{\partial U}{\partial W}}} {{\frac{\partial W}{\partial U}}} {{{ U} _{;j}}}}} = {0}$
${{{ W} _{,t}} + {{{\frac{\partial W}{\partial U}}} {{\frac{\partial {F(n^j)}}{\partial U}}} {{\frac{\partial U}{\partial W}}} {{{ W} _{;j}}}}} = {0}$
${{{ W} _{,t}} + {{{\frac{\partial W}{\partial U}}} {{\frac{\partial {F(n^j)}}{\partial W}}} {{{ W} _{;j}}}}} = {0}$
...where ${{\frac{\partial W}{\partial U}}} {{\frac{\partial {F(n^j)}}{\partial W}}}$ is equal to the acoustic matrix plus a diagonal of the velocity along the flux normal, as we will see below.
${{{ W} _{,t}} + {{{\left({{A} + {{{I}} {{{ v} ^j}}}}\right)}} {{{ W} _{;j}}}}} = {0}$
Now, even though the flux vector doesn't look so solvable as it does with conserved quantities, this system's eigendecomposition is much much more simple.
If you could find some flux vector such that the derivative of this flux vector wrt the primitive vector produced our "acoustic-plus-velocity-diagonal" then we could use flux-based numerical solvers...
${{{ W} _{,t}} + {{{\frac{\partial {F_W(n^j)}}{\partial W}}} {{{ W} _{;j}}}}} = {0}$
${{{ W} _{,t}} + {{ {F_W(n^j)}} _{;j}}} = {0}$
...but idk if this is possible.
So...
- The conserved form is able to be expressed as a flux vector: $\partial_t U + \partial_x F_U(n) = 0$.
From that, also as a linear relation of PDEs times x derivative $\partial_t U + \frac{\partial F_U(n)}{\partial U} \cdot \partial_x U = 0$,
and from that as an eigensystem $\partial_t U + R_U(n) \cdot \Lambda(n) \cdot L_U(n) \cdot \partial_x U = 0$.
- The primitive form is the simplest form of the linear relation of the PDEs $\partial_t W + B(n) \cdot \partial_x W = 0$
such that the state variables $W$ are independent of flux direction,
though the flux vector $F_W(n)$ may not be solvable from integrating the linear system times the state variable x derivative $B(n) \cdot \partial_x W$,
i.e. $\partial_t W + \partial_x F_W(n) = 0$ may not be a possible form to deduce.
- The characteristic form are completely separated PDEs, with diagonal flux and identity eigenvectors $\partial_t C(n) + \Lambda(n) \cdot \partial_x C(n) = 0$,
however the state variables $C(n)$ may be dependent on flux direction and may not be solvable.
Conserved and primitive variables:
${{ W} ^I} = {\left[\begin{array}{c} \rho\\ { v} ^i\\ P\end{array}\right]}$
${{ U} ^I} = {\left[\begin{array}{c} \rho\\ { m} ^i\\ {E_{total}}\end{array}\right]}$
${{ U} ^I} = {\left[\begin{array}{c} \rho\\ {{\rho}} \cdot {{{ v} ^x}}\\ {{\rho}} \cdot {{{ v} ^y}}\\ {{\rho}} \cdot {{{ v} ^z}}\\ {{{P}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{\frac{1}{2}}} {{{(v)^2}}} \cdot {{\rho}}}\end{array}\right]}$
${{ W} ^I} = {\left[\begin{array}{c} \rho\\ {\frac{1}{\rho}} {{ m} ^x}\\ {\frac{1}{\rho}} {{ m} ^y}\\ {\frac{1}{\rho}} {{ m} ^z}\\ {{{-1}} \cdot {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{(m)^2}}} \cdot {{\frac{1}{\rho}}}} + {{{{(\gamma_{-1})}}} \cdot {{{E_{total}}}}}\end{array}\right]}$
Partial of conserved quantities wrt primitives:
${\frac{\partial { U} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccc} \frac{\partial \rho}{\partial \rho}& \frac{\partial \rho}{\partial { v} ^j}& \frac{\partial \rho}{\partial P}\\ \frac{\partial { m} ^i}{\partial \rho}& \frac{\partial { m} ^i}{\partial { v} ^j}& \frac{\partial { m} ^i}{\partial P}\\ \frac{\partial {E_{total}}}{\partial \rho}& \frac{\partial {E_{total}}}{\partial { v} ^j}& \frac{\partial {E_{total}}}{\partial P}\end{array}\right]}$
${\frac{\partial { U} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccc} \frac{\partial \rho}{\partial \rho}& \frac{\partial \rho}{\partial { v} ^j}& \frac{\partial \rho}{\partial P}\\ {\frac{\partial}{\partial \rho}}\left({{{\rho}} \cdot {{{ v} ^i}}}\right)& {\frac{\partial}{\partial { v} ^j}}\left({{{\rho}} \cdot {{{ v} ^i}}}\right)& {\frac{\partial}{\partial P}}\left({{{\rho}} \cdot {{{ v} ^i}}}\right)\\ {\frac{\partial}{\partial \rho}}\left({{{\rho}} \cdot {{\left({{{{\frac{1}{2}}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}} + {\frac{P}{{{{(\gamma_{-1})}}} \cdot {{\rho}}}}}\right)}}}\right)& {\frac{\partial}{\partial { v} ^j}}\left({{{\rho}} \cdot {{\left({{{{\frac{1}{2}}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}} + {\frac{P}{{{{(\gamma_{-1})}}} \cdot {{\rho}}}}}\right)}}}\right)& {\frac{\partial}{\partial P}}\left({{{\rho}} \cdot {{\left({{{{\frac{1}{2}}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}} + {\frac{P}{{{{(\gamma_{-1})}}} \cdot {{\rho}}}}}\right)}}}\right)\end{array}\right]}$
${\frac{\partial { U} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccc} 1& 0& 0\\ { v} ^i& {{\rho}} \cdot {{{{ δ} ^i} _j}}& 0\\ {\frac{1}{2}} {{(v)^2}}& {{\rho}} \cdot {{{ v} _j}}& \frac{1}{{(\gamma_{-1})}}\end{array}\right]}$
${\frac{\partial { W} ^I}{\partial { U} ^J}} = {\left[\begin{array}{ccc} 1& 0& 0\\ {\frac{1}{\rho}}{\left({-{{ v} ^i}}\right)}& {\frac{1}{\rho}} {{{ δ} ^i} _j}& 0\\ {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}& {-{{(\gamma_{-1})}}} {{{ v} _j}}& {(\gamma_{-1})}\end{array}\right]}$
Flux:
${{ F} ^I} = {\left[\begin{array}{c} {{\rho}} \cdot {{{ v} ^j}} {{{ {(n_1)}} _j}}\\ {{{\rho}} \cdot {{{ v} ^i}} {{{ v} ^j}} {{{ {(n_1)}} _j}}} + {{{{ {(n_1)}} ^i}} {{P}}}\\ {{{ v} ^j}} {{{ {(n_1)}} _j}} {{{H_{total}}}}\end{array}\right]}$
${{ F} ^I} = {\left[\begin{array}{c} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^j}}\\ {{{P}} {{{ {(n_1)}} ^i}}} + {{{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^i}} {{{ v} ^j}}}\\ {{{P}} {{{ {(n_1)}} _j}} {{{ v} ^j}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{P}} {{{ {(n_1)}} _j}} {{{ v} ^j}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^j}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}\end{array}\right]}$
Flux derivative wrt primitive variables:
${\frac{\partial { F} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccc} {\frac{\partial}{\partial \rho}}\left({{{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}}}\right)& {\frac{\partial}{\partial { v} ^j}}\left({{{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}}}\right)& {\frac{\partial}{\partial P}}\left({{{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}}}\right)\\ {\frac{\partial}{\partial \rho}}\left({{{{P}} {{{ {(n_1)}} ^i}}} + {{{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^i}} {{{ v} ^k}}}}\right)& {\frac{\partial}{\partial { v} ^j}}\left({{{{P}} {{{ {(n_1)}} ^i}}} + {{{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^i}} {{{ v} ^k}}}}\right)& {\frac{\partial}{\partial P}}\left({{{{P}} {{{ {(n_1)}} ^i}}} + {{{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^i}} {{{ v} ^k}}}}\right)\\ {\frac{\partial}{\partial \rho}}\left({{{{P}} {{{ {(n_1)}} _k}} {{{ v} ^k}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{P}} {{{ {(n_1)}} _k}} {{{ v} ^k}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}} {{{ v} ^m}} {{{ v} ^l}} {{{{ g} _m} _l}}}}\right)& {\frac{\partial}{\partial { v} ^j}}\left({{{{P}} {{{ {(n_1)}} _k}} {{{ v} ^k}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{P}} {{{ {(n_1)}} _k}} {{{ v} ^k}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}} {{{ v} ^m}} {{{ v} ^l}} {{{{ g} _m} _l}}}}\right)& {\frac{\partial}{\partial P}}\left({{{{P}} {{{ {(n_1)}} _k}} {{{ v} ^k}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{P}} {{{ {(n_1)}} _k}} {{{ v} ^k}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}} {{{ v} ^m}} {{{ v} ^l}} {{{{ g} _m} _l}}}}\right)\end{array}\right]}$
${\frac{\partial { F} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccc} {{{ {(n_1)}} _k}} {{{ v} ^k}}& {{\rho}} \cdot {{{ {(n_1)}} _j}}& 0\\ {{{ {(n_1)}} _k}} {{{ v} ^i}} {{{ v} ^k}}& {{{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^i}}} + {{{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}} {{{{ δ} ^i} _j}}}& { {(n_1)}} ^i\\ {{\frac{1}{2}}} {{{ {(n_1)}} _k}} {{{ v} ^k}} {{{ v} ^l}} {{{ v} _l}}& {{{P}} {{{ {(n_1)}} _j}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{P}} {{{ {(n_1)}} _j}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^l}} {{{ v} _l}}} + {{{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} _j}} {{{ v} ^k}}}& {{{{ {(n_1)}} _k}} {{{ v} ^k}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{{ {(n_1)}} _k}} {{{ v} ^k}}}\end{array}\right]}$
${\frac{\partial { F} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccc} {{{ {(n_1)}} _a}} {{{ v} ^a}}& {{\rho}} \cdot {{{ {(n_1)}} _j}}& 0\\ {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^i}}& {{{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^i}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^i} _j}}}& { {(n_1)}} ^i\\ {{\frac{1}{2}}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^b}} {{{ v} _b}}& {{{P}} {{{ {(n_1)}} _j}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{P}} {{{ {(n_1)}} _j}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^a}} {{{ v} _a}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _j}}}& {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}}}\end{array}\right]}$
${\frac{\partial { F} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccc} {{{ {(n_1)}} _a}} {{{ v} ^a}}& {{\rho}} \cdot {{{ {(n_1)}} _j}}& 0\\ {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^i}}& {{{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^i}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^i} _j}}}& { {(n_1)}} ^i\\ {{\frac{1}{2}}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{(v)^2}}}& {{{P}} {{{ {(n_1)}} _j}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{P}} {{{ {(n_1)}} _j}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{(v)^2}}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _j}}}& {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}}}\end{array}\right]}$
${\frac{\partial { F} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccc} {{{ {(n_1)}} _a}} {{{ v} ^a}}& {{\rho}} \cdot {{{ {(n_1)}} _j}}& 0\\ {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^i}}& {{{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^i}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^i} _j}}}& { {(n_1)}} ^i\\ {{\frac{1}{2}}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^b}} {{{ v} _b}}& {{{{H_{total}}}} \cdot {{{ {(n_1)}} _j}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _j}}}& {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}}}\end{array}\right]}$
Flux derivative wrt conserved variables:
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {{{\frac{\partial { F} ^I}{\partial { W} ^L}}} {{\frac{\partial { W} ^L}{\partial { U} ^J}}}}$
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {{{\left[\begin{array}{ccc} {{{ {(n_1)}} _a}} {{{ v} ^a}}& {{\rho}} \cdot {{{ {(n_1)}} _k}}& 0\\ {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^i}}& {{{\rho}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^i}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^i} _k}}}& { {(n_1)}} ^i\\ {{\frac{1}{2}}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^b}} {{{ v} _b}}& {{{{H_{total}}}} \cdot {{{ {(n_1)}} _k}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _k}}}& {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}}}\end{array}\right]}} {{\left[\begin{array}{ccc} 1& 0& 0\\ {\frac{1}{\rho}}{\left({-{{ v} ^k}}\right)}& {\frac{1}{\rho}} {{{ δ} ^k} _j}& 0\\ {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ v} ^m}} {{{ v} ^l}} {{{{ g} _m} _l}}& {-{{(\gamma_{-1})}}} {{{ v} _j}}& {(\gamma_{-1})}\end{array}\right]}}}$
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {\left[\begin{array}{ccc} {{{{ {(n_1)}} _a}} {{{ v} ^a}}} + {{{-1}} {{{ {(n_1)}} _k}} {{{ v} ^k}}}& {{{ {(n_1)}} _k}} {{{{ δ} ^k} _j}}& 0\\ {{{-1}} {{{ {(n_1)}} _k}} {{{ v} ^i}} {{{ v} ^k}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^i}}} + {{{-1}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^k}} {{{{ δ} ^i} _k}}} + {{{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}} {{{ v} ^l}} {{{ v} ^m}} {{{{ g} _m} _l}}}& {{{-1}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}} {{{ v} _j}}} + {{{{ {(n_1)}} _k}} {{{ v} ^i}} {{{{ δ} ^k} _j}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^i} _k}} {{{{ δ} ^k} _j}}}& {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}}\\ {{{-1}} {{{H_{total}}}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}} {{\frac{1}{\rho}}}} + {{{\frac{1}{2}}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^b}} {{{ v} _b}}} + {{{-1}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^k}} {{{ v} _k}}} + {{{\frac{1}{2}}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^l}} {{{ v} ^m}} {{{{ g} _m} _l}}} + {{{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^l}} {{{ v} ^m}} {{{{ g} _m} _l}}}& {{{{H_{total}}}} \cdot {{{ {(n_1)}} _k}} {{{{ δ} ^k} _j}} {{\frac{1}{\rho}}}} + {{{-1}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _j}}} + {{{-1}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _j}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _k}} {{{{ δ} ^k} _j}}}& {{{{ {(n_1)}} _a}} {{{ v} ^a}}} + {{{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}}}\end{array}\right]}$
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {\left[\begin{array}{ccc} 0& { {(n_1)}} _j& 0\\ {{{-1}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^i}}} + {{{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}} {{{ v} ^a}} {{{ v} _a}}}& {{{{ {(n_1)}} _j}} {{{ v} ^i}}} + {{{-1}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}} {{{ v} _j}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^i} _j}}}& {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}}\\ {{{-1}} {{{H_{total}}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{\frac{1}{\rho}}}} + {{{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _b}} {{{ v} ^b}}}& {{{{H_{total}}}} \cdot {{{ {(n_1)}} _j}} {{\frac{1}{\rho}}}} + {{{-1}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _j}}}& {{{{ {(n_1)}} _a}} {{{ v} ^a}}} + {{{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}}}\end{array}\right]}$
Flux derivative matrix times state. If $\frac{\partial F}{\partial U} \cdot U = F$ then the equation is said to have the Homogeneity property (Toro proposition 3.4).
${{{\frac{\partial { F} ^I}{\partial { U} ^J}}} {{U}}} = {{{\left[\begin{array}{ccc} 0& { {(n_1)}} _j& 0\\ {{{-1}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^i}}} + {{{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}} {{{ v} ^a}} {{{ v} _a}}}& {{{{ {(n_1)}} _j}} {{{ v} ^i}}} + {{{-1}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}} {{{ v} _j}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^i} _j}}}& {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}}\\ {{{-1}} {{{H_{total}}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{\frac{1}{\rho}}}} + {{{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _b}} {{{ v} ^b}}}& {{{{H_{total}}}} \cdot {{{ {(n_1)}} _j}} {{\frac{1}{\rho}}}} + {{{-1}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _j}}}& {{{{ {(n_1)}} _a}} {{{ v} ^a}}} + {{{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}}}\end{array}\right]}} {{\left[\begin{array}{c} \rho\\ { m} ^j\\ {E_{total}}\end{array}\right]}}}$
${{{\frac{\partial { F} ^I}{\partial { U} ^J}}} {{U}}} = {{{\left[\begin{array}{ccc} 0& { {(n_1)}} _j& 0\\ {{{-1}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^i}}} + {{{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}} {{{ v} ^a}} {{{ v} _a}}}& {{{{ {(n_1)}} _j}} {{{ v} ^i}}} + {{{-1}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}} {{{ v} _j}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^i} _j}}}& {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^i}}\\ {{{-1}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{\frac{1}{\rho}}} {{\left({{{{P}} {{\frac{1}{{(\gamma_{-1})}}}}} + {P} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}}\right)}}} + {{{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _b}} {{{ v} ^b}}}& {{{{ {(n_1)}} _j}} {{\frac{1}{\rho}}} {{\left({{{{P}} {{\frac{1}{{(\gamma_{-1})}}}}} + {P} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}}\right)}}} + {{{-1}} {{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _j}}}& {{{{ {(n_1)}} _a}} {{{ v} ^a}}} + {{{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}}}\end{array}\right]}} {{\left[\begin{array}{c} \rho\\ {{\rho}} \cdot {{{ v} ^j}}\\ {{\rho}} \cdot {{\left({{{{\frac{1}{2}}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}} + {\frac{P}{{{{(\gamma_{-1})}}} \cdot {{\rho}}}}}\right)}}\end{array}\right]}}}$
${{{\frac{\partial { F} ^I}{\partial { U} ^J}}} {{U}}} = {\left[\begin{array}{c} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^j}}\\ {\frac{1}{2}}{\left({{{{2}} {{P}} {{{ {(n_1)}} ^i}}} + {{{2}} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^i}} {{{ v} ^j}}}{-{{{2}} {{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^i}}}} + {{{{(\gamma_{-1})}}} \cdot {{\rho}} \cdot {{{ {(n_1)}} ^i}} {{{ v} _a}} {{{ v} ^a}}}{-{{{2}} {{{(\gamma_{-1})}}} \cdot {{\rho}} \cdot {{{ {(n_1)}} ^i}} {{{ v} ^j}} {{{ v} _j}}}} + {{{2}} {{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^j}} {{{{ δ} ^i} _j}}} + {{{{(\gamma_{-1})}}} \cdot {{\rho}} \cdot {{{ {(n_1)}} ^i}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}}\right)}\\ \frac{{{{2}} {{P}} {{{ {(n_1)}} _j}} {{{ v} ^j}}} + {{{2}} {{{(\gamma_{-1})}}} \cdot {{P}} {{{ {(n_1)}} _j}} {{{ v} ^j}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _b}} {{{ v} ^b}} {{{{(\gamma_{-1})}}^{2}}}}{-{{{2}} {{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _j}} {{{ v} ^j}} {{{{(\gamma_{-1})}}^{2}}}}} + {{{{(\gamma_{-1})}}} \cdot {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^j}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}} {{{{(\gamma_{-1})}}^{2}}}}}{{{2}} {{{(\gamma_{-1})}}}}\end{array}\right]}$
${{{\frac{\partial { F} ^I}{\partial { U} ^J}}} {{U}}} = {\left[\begin{array}{c} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^j}}\\ {{{P}} {{{ {(n_1)}} ^i}}} + {{{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^i}} {{{ v} ^j}}}\\ {{{P}} {{{ {(n_1)}} _j}} {{{ v} ^j}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{P}} {{{ {(n_1)}} _j}} {{{ v} ^j}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^j}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}\end{array}\right]}$
${{ F} ^I} = {\left[\begin{array}{c} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^j}}\\ {{{P}} {{{ {(n_1)}} ^i}}} + {{{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^i}} {{{ v} ^j}}}\\ {{{P}} {{{ {(n_1)}} _j}} {{{ v} ^j}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{P}} {{{ {(n_1)}} _j}} {{{ v} ^j}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^j}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}\end{array}\right]}$
Looks like for the Euler fluid equations, $\frac{\partial F}{\partial U} \cdot U = F$.
Acoustic matrix:
${{{{ A} ^I} _J} + {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^I} _J}}}} = {{{\frac{\partial { W} ^I}{\partial { U} ^K}}} {{\frac{\partial { F} ^K}{\partial { W} ^J}}}}$
${{{{ A} ^I} _J} + {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^I} _J}}}} = {{{\left[\begin{array}{ccc} 1& 0& 0\\ {\frac{1}{\rho}}{\left({-{{ v} ^i}}\right)}& {\frac{1}{\rho}} {{{ δ} ^i} _k}& 0\\ {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ v} ^m}} {{{ v} ^l}} {{{{ g} _m} _l}}& {-{{(\gamma_{-1})}}} {{{ v} _k}}& {(\gamma_{-1})}\end{array}\right]}} {{\left[\begin{array}{ccc} {{{ {(n_1)}} _a}} {{{ v} ^a}}& {{\rho}} \cdot {{{ {(n_1)}} _j}}& 0\\ {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^k}}& {{{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^k}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^k} _j}}}& { {(n_1)}} ^k\\ {{\frac{1}{2}}} {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} ^b}} {{{ v} _b}}& {{{{H_{total}}}} \cdot {{{ {(n_1)}} _j}}} + {{{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _j}}}& {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}}}\end{array}\right]}}}$
${{{{ A} ^I} _J} + {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^I} _J}}}} = {\left[\begin{array}{ccc} {{{ {(n_1)}} _a}} {{{ v} ^a}}& {{\rho}} \cdot {{{ {(n_1)}} _j}}& 0\\ {\frac{1}{\rho}} {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{\left({{-{{ v} ^i}} + {{{{ v} ^k}} {{{{ δ} ^i} _k}}}}\right)}}}& {-{{{{ {(n_1)}} _j}} {{{ v} ^i}}}} + {{{{ {(n_1)}} _j}} {{{ v} ^k}} {{{{ δ} ^i} _k}}} + {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^i} _k}} {{{{ δ} ^k} _j}}}& {\frac{1}{\rho}} {{{{ {(n_1)}} ^k}} {{{{ δ} ^i} _k}}}\\ {\frac{1}{2}} {{{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{\left({{{{{ v} _b}} {{{ v} ^b}}}{-{{{2}} {{{ v} _k}} {{{ v} ^k}}}} + {{{{ v} ^l}} {{{ v} ^m}} {{{{ g} _m} _l}}}}\right)}}}& {\frac{1}{2}} {{{{(\gamma_{-1})}}} \cdot {{\left({{{{2}} {{{H_{total}}}} \cdot {{{ {(n_1)}} _j}}} + {{{2}} {{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _j}}}{-{{{2}} {{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^k}} {{{ v} _k}}}} + {{{\rho}} \cdot {{{ {(n_1)}} _j}} {{{ v} ^l}} {{{ v} ^m}} {{{{ g} _m} _l}}}{-{{{2}} {{\rho}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{ v} _k}} {{{{ δ} ^k} _j}}}}}\right)}}}& {{{{ {(n_1)}} _a}} {{{ v} ^a}}} + {{{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} _a}} {{{ v} ^a}}}{-{{{{(\gamma_{-1})}}} \cdot {{{ {(n_1)}} ^k}} {{{ v} _k}}}}\end{array}\right]}$
${{{{ A} ^I} _J} + {{{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^I} _J}}}} = {\left[\begin{array}{ccc} {{{ {(n_1)}} _a}} {{{ v} ^a}}& {{\rho}} \cdot {{{ {(n_1)}} _j}}& 0\\ 0& {{{ {(n_1)}} _a}} {{{ v} ^a}} {{{{ δ} ^i} _j}}& {\frac{1}{\rho}} {{ {(n_1)}} ^i}\\ 0& {{P}} {{\gamma}} \cdot {{{ {(n_1)}} _j}}& {{{ {(n_1)}} _a}} {{{ v} ^a}}\end{array}\right]}$
${{{ A} ^I} _J} = {\left[\begin{array}{ccc} 0& {{\rho}} \cdot {{{ {(n_1)}} _j}}& 0\\ 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^i}\\ 0& {{P}} {{\gamma}} \cdot {{{ {(n_1)}} _j}}& 0\end{array}\right]}$
Acoustic matrix, expanded:
${{{ A} ^I} _J} = {\left[\begin{array}{ccccc} 0& {{\rho}} \cdot {{{ {(n_1)}} _x}}& {{\rho}} \cdot {{{ {(n_1)}} _y}}& {{\rho}} \cdot {{{ {(n_1)}} _z}}& 0\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^x}\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^y}\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^z}\\ 0& {{P}} {{\gamma}} \cdot {{{ {(n_1)}} _x}}& {{P}} {{\gamma}} \cdot {{{ {(n_1)}} _y}}& {{P}} {{\gamma}} \cdot {{{ {(n_1)}} _z}}& 0\end{array}\right]}$
${{{ A} ^I} _J} = {\left[\begin{array}{ccccc} 0& {{\rho}} \cdot {{{ {(n_1)}} _x}}& {{\rho}} \cdot {{{ {(n_1)}} _y}}& {{\rho}} \cdot {{{ {(n_1)}} _z}}& 0\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^x}\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^y}\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^z}\\ 0& {{\rho}} \cdot {{{ {(n_1)}} _x}} {{{{c_s}}^{2}}}& {{\rho}} \cdot {{{ {(n_1)}} _y}} {{{{c_s}}^{2}}}& {{\rho}} \cdot {{{ {(n_1)}} _z}} {{{{c_s}}^{2}}}& 0\end{array}\right]}$
...in just the x-axis (using ${{ {(n_1)}} _x} = {1}$, ${{ {(n_1)}} _y} = {0}$, ${{ {(n_1)}} _z} = {0}$ )
${{{ A} ^I} _J} = {\left[\begin{array}{ccccc} 0& \rho& 0& 0& 0\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^x}\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^y}\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^z}\\ 0& {{\rho}} \cdot {{{{c_s}}^{2}}}& 0& 0& 0\end{array}\right]}$
...with a Cartesian metric (using ${{ {(n_1)}} ^x} = {1}$, ${{ {(n_1)}} ^y} = {0}$, ${{ {(n_1)}} ^z} = {0}$ )
${{{ A} ^I} _J} = {\left[\begin{array}{ccccc} 0& \rho& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\rho}\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& {{\rho}} \cdot {{{{c_s}}^{2}}}& 0& 0& 0\end{array}\right]}$
${{{ A} ^I} _J} = {\left[\begin{array}{ccccc} 0& \rho& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\rho}\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& {{P}} {{\gamma}}& 0& 0& 0\end{array}\right]}$
${{{{ W} ^I} _{,t}} + {{{\left({{{{ A} ^I} _J} + {{{{{ δ} ^I} _J}} {{{ v} ^k}}}}\right)}} {{{{ W} ^J} _{,k}}}}} = {0}$
${{{\frac{\partial}{\partial t}}\left({\left[\begin{array}{c} \rho\\ { v} ^x\\ { v} ^y\\ { v} ^z\\ P\end{array}\right]}\right)} + {{{\left({{\left[\begin{array}{ccccc} 0& \rho& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\rho}\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& {{P}} {{\gamma}}& 0& 0& 0\end{array}\right]} + {{{{ v} ^x}} {{\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}}}}\right)}} {{{\frac{\partial}{\partial x}}\left({\left[\begin{array}{c} \rho\\ { v} ^x\\ { v} ^y\\ { v} ^z\\ P\end{array}\right]}\right)}}}} = {\left[\begin{array}{c} 0\\ 0\\ 0\\ 0\\ 0\end{array}\right]}$
${{\frac{\partial \rho}{\partial t}} + {{{\rho}} \cdot {{\frac{\partial { v} ^x}{\partial x}}}} + {{{{ v} ^x}} {{\frac{\partial \rho}{\partial x}}}}} = {0}$
${{{{\frac{\partial P}{\partial x}}} {{\frac{1}{\rho}}}} + {\frac{\partial { v} ^x}{\partial t}} + {{{{ v} ^x}} {{\frac{\partial { v} ^x}{\partial x}}}}} = {0}$
${{\frac{\partial { v} ^y}{\partial t}} + {{{{ v} ^x}} {{\frac{\partial { v} ^y}{\partial x}}}}} = {0}$
${{\frac{\partial { v} ^z}{\partial t}} + {{{{ v} ^x}} {{\frac{\partial { v} ^z}{\partial x}}}}} = {0}$
${{\frac{\partial P}{\partial t}} + {{{{ v} ^x}} {{\frac{\partial P}{\partial x}}}} + {{{P}} {{\gamma}} \cdot {{\frac{\partial { v} ^x}{\partial x}}}}} = {0}$
${{{|n_1|}}^{2}} = {{{{{ {(n_1)}} _x}} {{{ {(n_1)}} ^x}}} + {{{{ {(n_1)}} _y}} {{{ {(n_1)}} ^y}}} + {{{{ {(n_1)}} _z}} {{{ {(n_1)}} ^z}}}}$
${{{|n_2|}}^{2}} = {{{{{ {(n_2)}} _x}} {{{ {(n_2)}} ^x}}} + {{{{ {(n_2)}} _y}} {{{ {(n_2)}} ^y}}} + {{{{ {(n_2)}} _z}} {{{ {(n_2)}} ^z}}}}$
${{{|n_3|}}^{2}} = {{{{{ {(n_3)}} _x}} {{{ {(n_3)}} ^x}}} + {{{{ {(n_3)}} _y}} {{{ {(n_3)}} ^y}}} + {{{{ {(n_3)}} _z}} {{{ {(n_3)}} ^z}}}}$
$(n_m)_i (n_n)^i = (n_m)_i (n_n)_j g^{ij} = \delta_{mn} |n_m|^2$
For $|n_i|$ is the metric-weighted norm.
${{{{{{ {N^\flat_{3x3}}} ^T}} {{{N^\sharp_{3x3}}}}} = {{{\left[\begin{array}{ccc} { {(n_1)}} _x& { {(n_1)}} _y& { {(n_1)}} _z\\ { {(n_2)}} _x& { {(n_2)}} _y& { {(n_2)}} _z\\ { {(n_3)}} _x& { {(n_3)}} _y& { {(n_3)}} _z\end{array}\right]}} {{\left[\begin{array}{ccc} { {(n_1)}} ^x& { {(n_2)}} ^x& { {(n_3)}} ^x\\ { {(n_1)}} ^y& { {(n_2)}} ^y& { {(n_3)}} ^y\\ { {(n_1)}} ^z& { {(n_2)}} ^z& { {(n_3)}} ^z\end{array}\right]}}}} = {\left[\begin{array}{ccc} {{{{ {(n_1)}} ^x}} {{{ {(n_1)}} _x}}} + {{{{ {(n_1)}} ^y}} {{{ {(n_1)}} _y}}} + {{{{ {(n_1)}} ^z}} {{{ {(n_1)}} _z}}}& {{{{ {(n_1)}} _x}} {{{ {(n_2)}} ^x}}} + {{{{ {(n_1)}} _y}} {{{ {(n_2)}} ^y}}} + {{{{ {(n_1)}} _z}} {{{ {(n_2)}} ^z}}}& {{{{ {(n_1)}} _x}} {{{ {(n_3)}} ^x}}} + {{{{ {(n_1)}} _y}} {{{ {(n_3)}} ^y}}} + {{{{ {(n_1)}} _z}} {{{ {(n_3)}} ^z}}}\\ {{{{ {(n_1)}} ^x}} {{{ {(n_2)}} _x}}} + {{{{ {(n_1)}} ^y}} {{{ {(n_2)}} _y}}} + {{{{ {(n_1)}} ^z}} {{{ {(n_2)}} _z}}}& {{{{ {(n_2)}} ^x}} {{{ {(n_2)}} _x}}} + {{{{ {(n_2)}} ^y}} {{{ {(n_2)}} _y}}} + {{{{ {(n_2)}} ^z}} {{{ {(n_2)}} _z}}}& {{{{ {(n_2)}} _x}} {{{ {(n_3)}} ^x}}} + {{{{ {(n_2)}} _y}} {{{ {(n_3)}} ^y}}} + {{{{ {(n_2)}} _z}} {{{ {(n_3)}} ^z}}}\\ {{{{ {(n_1)}} ^x}} {{{ {(n_3)}} _x}}} + {{{{ {(n_1)}} ^y}} {{{ {(n_3)}} _y}}} + {{{{ {(n_1)}} ^z}} {{{ {(n_3)}} _z}}}& {{{{ {(n_2)}} ^x}} {{{ {(n_3)}} _x}}} + {{{{ {(n_2)}} ^y}} {{{ {(n_3)}} _y}}} + {{{{ {(n_2)}} ^z}} {{{ {(n_3)}} _z}}}& {{{{ {(n_3)}} ^x}} {{{ {(n_3)}} _x}}} + {{{{ {(n_3)}} ^y}} {{{ {(n_3)}} _y}}} + {{{{ {(n_3)}} ^z}} {{{ {(n_3)}} _z}}}\end{array}\right]}} = {\left[\begin{array}{ccc} {{|n_1|}}^{2}& 0& 0\\ 0& {{|n_2|}}^{2}& 0\\ 0& 0& {{|n_3|}}^{2}\end{array}\right]}$
In terms of identity:
${{{{{ {N^\flat_{3x3}}} ^T}} {{{N^\sharp_{3x3}}}} \cdot {{{{N^\parallel_{3x3}}}^{-1}}}} = {{{\left[\begin{array}{ccc} { {(n_1)}} _x& { {(n_1)}} _y& { {(n_1)}} _z\\ { {(n_2)}} _x& { {(n_2)}} _y& { {(n_2)}} _z\\ { {(n_3)}} _x& { {(n_3)}} _y& { {(n_3)}} _z\end{array}\right]}} {{\left[\begin{array}{ccc} \frac{{ {(n_1)}} ^x}{{{|n_1|}}^{2}}& \frac{{ {(n_2)}} ^x}{{{|n_2|}}^{2}}& \frac{{ {(n_3)}} ^x}{{{|n_3|}}^{2}}\\ \frac{{ {(n_1)}} ^y}{{{|n_1|}}^{2}}& \frac{{ {(n_2)}} ^y}{{{|n_2|}}^{2}}& \frac{{ {(n_3)}} ^y}{{{|n_3|}}^{2}}\\ \frac{{ {(n_1)}} ^z}{{{|n_1|}}^{2}}& \frac{{ {(n_2)}} ^z}{{{|n_2|}}^{2}}& \frac{{ {(n_3)}} ^z}{{{|n_3|}}^{2}}\end{array}\right]}}}} = {\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}$
Now if $A A^{-1} = I$ then $A^{-1} A = I$:
${{{{{{N^\sharp_{3x3}}}} \cdot {{{{N^\parallel_{3x3}}}^{-1}}} {{{ {N^\flat_{3x3}}} ^T}}} = {{{\left[\begin{array}{ccc} \frac{{ {(n_1)}} ^x}{{{|n_1|}}^{2}}& \frac{{ {(n_2)}} ^x}{{{|n_2|}}^{2}}& \frac{{ {(n_3)}} ^x}{{{|n_3|}}^{2}}\\ \frac{{ {(n_1)}} ^y}{{{|n_1|}}^{2}}& \frac{{ {(n_2)}} ^y}{{{|n_2|}}^{2}}& \frac{{ {(n_3)}} ^y}{{{|n_3|}}^{2}}\\ \frac{{ {(n_1)}} ^z}{{{|n_1|}}^{2}}& \frac{{ {(n_2)}} ^z}{{{|n_2|}}^{2}}& \frac{{ {(n_3)}} ^z}{{{|n_3|}}^{2}}\end{array}\right]}} {{\left[\begin{array}{ccc} { {(n_1)}} _x& { {(n_1)}} _y& { {(n_1)}} _z\\ { {(n_2)}} _x& { {(n_2)}} _y& { {(n_2)}} _z\\ { {(n_3)}} _x& { {(n_3)}} _y& { {(n_3)}} _z\end{array}\right]}}}} = {\left[\begin{array}{ccc} {{{{ {(n_1)}} _x}} {{{ {(n_1)}} ^x}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{{ {(n_2)}} _x}} {{{ {(n_2)}} ^x}} {{\frac{1}{{{|n_2|}}^{2}}}}} + {{{{ {(n_3)}} ^x}} {{{ {(n_3)}} _x}} {{\frac{1}{{{|n_3|}}^{2}}}}}& {{{{ {(n_1)}} ^x}} {{{ {(n_1)}} _y}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{{ {(n_2)}} ^x}} {{{ {(n_2)}} _y}} {{\frac{1}{{{|n_2|}}^{2}}}}} + {{{{ {(n_3)}} ^x}} {{{ {(n_3)}} _y}} {{\frac{1}{{{|n_3|}}^{2}}}}}& {{{{ {(n_1)}} ^x}} {{{ {(n_1)}} _z}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{{ {(n_2)}} ^x}} {{{ {(n_2)}} _z}} {{\frac{1}{{{|n_2|}}^{2}}}}} + {{{{ {(n_3)}} ^x}} {{{ {(n_3)}} _z}} {{\frac{1}{{{|n_3|}}^{2}}}}}\\ {{{{ {(n_1)}} _x}} {{{ {(n_1)}} ^y}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{{ {(n_2)}} _x}} {{{ {(n_2)}} ^y}} {{\frac{1}{{{|n_2|}}^{2}}}}} + {{{{ {(n_3)}} _x}} {{{ {(n_3)}} ^y}} {{\frac{1}{{{|n_3|}}^{2}}}}}& {{{{ {(n_1)}} _y}} {{{ {(n_1)}} ^y}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{{ {(n_2)}} _y}} {{{ {(n_2)}} ^y}} {{\frac{1}{{{|n_2|}}^{2}}}}} + {{{{ {(n_3)}} ^y}} {{{ {(n_3)}} _y}} {{\frac{1}{{{|n_3|}}^{2}}}}}& {{{{ {(n_1)}} ^y}} {{{ {(n_1)}} _z}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{{ {(n_2)}} ^y}} {{{ {(n_2)}} _z}} {{\frac{1}{{{|n_2|}}^{2}}}}} + {{{{ {(n_3)}} ^y}} {{{ {(n_3)}} _z}} {{\frac{1}{{{|n_3|}}^{2}}}}}\\ {{{{ {(n_1)}} _x}} {{{ {(n_1)}} ^z}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{{ {(n_2)}} _x}} {{{ {(n_2)}} ^z}} {{\frac{1}{{{|n_2|}}^{2}}}}} + {{{{ {(n_3)}} _x}} {{{ {(n_3)}} ^z}} {{\frac{1}{{{|n_3|}}^{2}}}}}& {{{{ {(n_1)}} _y}} {{{ {(n_1)}} ^z}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{{ {(n_2)}} _y}} {{{ {(n_2)}} ^z}} {{\frac{1}{{{|n_2|}}^{2}}}}} + {{{{ {(n_3)}} _y}} {{{ {(n_3)}} ^z}} {{\frac{1}{{{|n_3|}}^{2}}}}}& {{{{ {(n_1)}} _z}} {{{ {(n_1)}} ^z}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{{ {(n_2)}} _z}} {{{ {(n_2)}} ^z}} {{\frac{1}{{{|n_2|}}^{2}}}}} + {{{{ {(n_3)}} ^z}} {{{ {(n_3)}} _z}} {{\frac{1}{{{|n_3|}}^{2}}}}}\end{array}\right]}} = {\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}$
Realigned, using the normal basis, which is inverses of one another, so that I can just apply it to the left and right eigenvector transforms.
${{{{ A} ^I} _J} = {{{\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& \frac{{ {(n_1)}} ^x}{{{|n_1|}}^{2}}& \frac{{ {(n_2)}} ^x}{{{|n_2|}}^{2}}& \frac{{ {(n_3)}} ^x}{{{|n_3|}}^{2}}& 0\\ 0& \frac{{ {(n_1)}} ^y}{{{|n_1|}}^{2}}& \frac{{ {(n_2)}} ^y}{{{|n_2|}}^{2}}& \frac{{ {(n_3)}} ^y}{{{|n_3|}}^{2}}& 0\\ 0& \frac{{ {(n_1)}} ^z}{{{|n_1|}}^{2}}& \frac{{ {(n_2)}} ^z}{{{|n_2|}}^{2}}& \frac{{ {(n_3)}} ^z}{{{|n_3|}}^{2}}& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} 0& \rho& 0& 0& 0\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{{|n_1|}}^{2}}\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& {{\rho}} \cdot {{{{c_s}}^{2}}}& 0& 0& 0\end{array}\right]}} {{\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& { {(n_1)}} _x& { {(n_1)}} _y& { {(n_1)}} _z& 0\\ 0& { {(n_2)}} _x& { {(n_2)}} _y& { {(n_2)}} _z& 0\\ 0& { {(n_3)}} _x& { {(n_3)}} _y& { {(n_3)}} _z& 0\\ 0& 0& 0& 0& 1\end{array}\right]}}}} = {\left[\begin{array}{ccccc} 0& {{\rho}} \cdot {{{ {(n_1)}} _x}}& {{\rho}} \cdot {{{ {(n_1)}} _y}}& {{\rho}} \cdot {{{ {(n_1)}} _z}}& 0\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^x}\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^y}\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^z}\\ 0& {{\rho}} \cdot {{{ {(n_1)}} _x}} {{{{c_s}}^{2}}}& {{\rho}} \cdot {{{ {(n_1)}} _y}} {{{{c_s}}^{2}}}& {{\rho}} \cdot {{{ {(n_1)}} _z}} {{{{c_s}}^{2}}}& 0\end{array}\right]}$
So now we know the left and right transforms to apply to our right and left eigennvector matrices to align them into an arbitrary normal frame, even with an arbitrary metric.
Now to eigen-decompose the normal magnitude diagonal matrix multiplied with the acoustic matrix.
Acoustic matrix eigen-decomposition:
${{{ A} ^I} _J} = {{{{{ {(R_A)}} ^I} _M}} {{{{ {(\Lambda_A)}} ^M} _N}} {{{{ {(L_A)}} ^N} _J}}}$
${{{ A} ^I} _J} = {{{\left[\begin{array}{ccccc} 1& 0& 0& \frac{1}{{{c_s}}^{2}}& \frac{1}{{{c_s}}^{2}}\\ 0& -{\frac{{ {(n_1)}} _y}{{ {(n_1)}} _x}}& -{\frac{{ {(n_1)}} _z}{{ {(n_1)}} _x}}& \frac{{ {(n_1)}} ^x}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}& -{\frac{{ {(n_1)}} ^x}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}}\\ 0& 1& 0& \frac{{ {(n_1)}} ^y}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}& -{\frac{{ {(n_1)}} ^y}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}}\\ 0& 0& 1& \frac{{ {(n_1)}} ^z}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}& -{\frac{{ {(n_1)}} ^z}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}}\\ 0& 0& 0& 1& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& {{{c_s}}} \cdot {{{|n_1|}}}& 0\\ 0& 0& 0& 0& -{{{{c_s}}} \cdot {{{|n_1|}}}}\end{array}\right]}} {{\left[\begin{array}{ccccc} 1& 0& 0& 0& -{\frac{1}{{{c_s}}^{2}}}\\ 0& -{\frac{{{{ {(n_1)}} _x}} {{{ {(n_1)}} ^y}}}{{{|n_1|}}^{2}}}& \frac{{{{{ {(n_1)}} ^z}} {{{ {(n_1)}} _z}}} + {{{{ {(n_1)}} ^x}} {{{ {(n_1)}} _x}}}}{{{|n_1|}}^{2}}& -{\frac{{{{ {(n_1)}} ^y}} {{{ {(n_1)}} _z}}}{{{|n_1|}}^{2}}}& 0\\ 0& -{\frac{{{{ {(n_1)}} _x}} {{{ {(n_1)}} ^z}}}{{{|n_1|}}^{2}}}& -{\frac{{{{ {(n_1)}} _y}} {{{ {(n_1)}} ^z}}}{{{|n_1|}}^{2}}}& \frac{{{{{ {(n_1)}} _y}} {{{ {(n_1)}} ^y}}} + {{{{ {(n_1)}} _x}} {{{ {(n_1)}} ^x}}}}{{{|n_1|}}^{2}}& 0\\ 0& \frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _x}}}{{{2}} {{{|n_1|}}}}& \frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _y}}}{{{2}} {{{|n_1|}}}}& \frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _z}}}{{{2}} {{{|n_1|}}}}& \frac{1}{2}\\ 0& -{\frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _x}}}{{{2}} {{{|n_1|}}}}}& -{\frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _y}}}{{{2}} {{{|n_1|}}}}}& -{\frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _z}}}{{{2}} {{{|n_1|}}}}}& \frac{1}{2}\end{array}\right]}}}$
${{{ A} ^I} _J} = {{{\left[\begin{array}{ccccc} 1& 0& 0& \frac{1}{{{c_s}}^{2}}& \frac{1}{{{c_s}}^{2}}\\ 0& -{\frac{{ {(n_1)}} _y}{{ {(n_1)}} _x}}& -{\frac{{ {(n_1)}} _z}{{ {(n_1)}} _x}}& \frac{{ {(n_1)}} ^x}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}& -{\frac{{ {(n_1)}} ^x}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}}\\ 0& 1& 0& \frac{{ {(n_1)}} ^y}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}& -{\frac{{ {(n_1)}} ^y}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}}\\ 0& 0& 1& \frac{{ {(n_1)}} ^z}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}& -{\frac{{ {(n_1)}} ^z}{{{\rho}} \cdot {{{c_s}}} \cdot {{{|n_1|}}}}}\\ 0& 0& 0& 1& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& {{{c_s}}} \cdot {{{|n_1|}}}& 0\\ 0& 0& 0& 0& -{{{{c_s}}} \cdot {{{|n_1|}}}}\end{array}\right]}} {{\left[\begin{array}{ccccc} 1& 0& 0& 0& -{\frac{1}{{{c_s}}^{2}}}\\ 0& -{\frac{{{{ {(n_1)}} _x}} {{{ {(n_1)}} ^y}}}{{{|n_1|}}^{2}}}& \frac{{{{{ {(n_1)}} ^z}} {{{ {(n_1)}} _z}}} + {{{{ {(n_1)}} ^x}} {{{ {(n_1)}} _x}}}}{{{|n_1|}}^{2}}& -{\frac{{{{ {(n_1)}} ^y}} {{{ {(n_1)}} _z}}}{{{|n_1|}}^{2}}}& 0\\ 0& -{\frac{{{{ {(n_1)}} _x}} {{{ {(n_1)}} ^z}}}{{{|n_1|}}^{2}}}& -{\frac{{{{ {(n_1)}} _y}} {{{ {(n_1)}} ^z}}}{{{|n_1|}}^{2}}}& \frac{{{{{ {(n_1)}} _y}} {{{ {(n_1)}} ^y}}} + {{{{ {(n_1)}} _x}} {{{ {(n_1)}} ^x}}}}{{{|n_1|}}^{2}}& 0\\ 0& \frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _x}}}{{{2}} {{{|n_1|}}}}& \frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _y}}}{{{2}} {{{|n_1|}}}}& \frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _z}}}{{{2}} {{{|n_1|}}}}& \frac{1}{2}\\ 0& -{\frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _x}}}{{{2}} {{{|n_1|}}}}}& -{\frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _y}}}{{{2}} {{{|n_1|}}}}}& -{\frac{{{\rho}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _z}}}{{{2}} {{{|n_1|}}}}}& \frac{1}{2}\end{array}\right]}}}$
permute by: $\left[\begin{array}{ccccc} 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0\end{array}\right]$ , scale by: $\left[\begin{array}{ccccc} {{{{c_s}}^{2}}} {{{|n_1|}}}& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} _x}& 0& 0\\ 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} _x}& 0\\ 0& 0& 0& 0& {{{{c_s}}^{2}}} {{{|n_1|}}}\end{array}\right]$
${{{ A} ^I} _J} = {{{\left[\begin{array}{ccccc} {|n_1|}& 1& 0& 0& {|n_1|}\\ -{{\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^x}}}}& 0& -{{\frac{1}{\rho}} {{ {(n_1)}} _y}}& -{{\frac{1}{\rho}} {{ {(n_1)}} _z}}& {\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^x}}}\\ -{{\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^y}}}}& 0& {\frac{1}{\rho}} {{ {(n_1)}} _x}& 0& {\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^y}}}\\ -{{\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^z}}}}& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} _x}& {\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^z}}}\\ {{{|n_1|}}} \cdot {{{{c_s}}^{2}}}& 0& 0& 0& {{{|n_1|}}} \cdot {{{{c_s}}^{2}}}\end{array}\right]}} {{\left[\begin{array}{ccccc} -{{{{c_s}}} \cdot {{{|n_1|}}}}& 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}}} \cdot {{{|n_1|}}}\end{array}\right]}} {{\left[\begin{array}{ccccc} 0& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} _x}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}}& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} _y}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}}& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} _z}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}}& \frac{1}{{{2}} {{{|n_1|}}} \cdot {{{{c_s}}^{2}}}}\\ 1& 0& 0& 0& -{\frac{1}{{{c_s}}^{2}}}\\ 0& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} ^y}}}{{{|n_1|}}^{2}}}& \frac{{{\rho}} \cdot {{\left({{{{|n_1|}}^{2}}{-{{{{ {(n_1)}} ^y}} {{{ {(n_1)}} _y}}}}}\right)}}}{{{{ {(n_1)}} _x}} {{{{|n_1|}}^{2}}}}& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} ^y}} {{{ {(n_1)}} _z}}}{{{{ {(n_1)}} _x}} {{{{|n_1|}}^{2}}}}}& 0\\ 0& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} ^z}}}{{{|n_1|}}^{2}}}& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} _y}} {{{ {(n_1)}} ^z}}}{{{{ {(n_1)}} _x}} {{{{|n_1|}}^{2}}}}}& \frac{{{\rho}} \cdot {{\left({{{{|n_1|}}^{2}}{-{{{{ {(n_1)}} ^z}} {{{ {(n_1)}} _z}}}}}\right)}}}{{{{ {(n_1)}} _x}} {{{{|n_1|}}^{2}}}}& 0\\ 0& \frac{{{\rho}} \cdot {{{ {(n_1)}} _x}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}& \frac{{{\rho}} \cdot {{{ {(n_1)}} _y}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}& \frac{{{\rho}} \cdot {{{ {(n_1)}} _z}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}& \frac{1}{{{2}} {{{|n_1|}}} \cdot {{{{c_s}}^{2}}}}\end{array}\right]}}}$
Acoustic matrix, reconstructed from eigen-decomposition:
${{{ A} ^I} _J} = {\left[\begin{array}{ccccc} 0& {{\rho}} \cdot {{{ {(n_1)}} _x}}& {{\rho}} \cdot {{{ {(n_1)}} _y}}& {{\rho}} \cdot {{{ {(n_1)}} _z}}& 0\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^x}\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^y}\\ 0& 0& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} ^z}\\ 0& {{\rho}} \cdot {{{ {(n_1)}} _x}} {{{{c_s}}^{2}}}& {{\rho}} \cdot {{{ {(n_1)}} _y}} {{{{c_s}}^{2}}}& {{\rho}} \cdot {{{ {(n_1)}} _z}} {{{{c_s}}^{2}}}& 0\end{array}\right]}$
Orthogonality of left and right eigenvectors:
$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$
$A$'s eigensystem is $R_A \Lambda_A L_A$.
$A + v I$'s eigensystem is $R_A \Lambda_A L_A + v I
= R_A \Lambda_A L_A + v R_A L_A
= R_A (\Lambda_A + v I) L_A$
.
$\frac{\partial W}{\partial U} \cdot \frac{\partial F}{\partial W} = A + v I$
$\frac{\partial F}{\partial U} = \frac{\partial U}{\partial W} \cdot \frac{\partial W}{\partial U} \cdot \frac{\partial F}{\partial W} \cdot \frac{\partial W}{\partial U}
= \frac{\partial U}{\partial W} \cdot (A + v I) \cdot \frac{\partial W}{\partial U}
= \frac{\partial U}{\partial W} \cdot R_A (\Lambda_A + v I) L_A \cdot \frac{\partial W}{\partial U}$.
Let $R_F = \frac{\partial U}{\partial W} \cdot R_A,
\Lambda_F = \Lambda_A + v I,
L_F = L_A \cdot \frac{\partial W}{\partial U}$.
$\frac{\partial F}{\partial U} = R_F \Lambda_F L_F$.
Flux Jacobian with respect to conserved variables:
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {{{\frac{\partial { U} ^I}{\partial { W} ^K}}} {{\frac{\partial { W} ^K}{\partial { U} ^L}}} {{\frac{\partial { F} ^L}{\partial { W} ^M}}} {{\frac{\partial { W} ^M}{\partial { U} ^J}}}}$
${{R_F}} = {{{\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ { v} ^x& \rho& 0& 0& 0\\ { v} ^y& 0& \rho& 0& 0\\ { v} ^z& 0& 0& \rho& 0\\ {\frac{1}{2}} {{(v)^2}}& {{\rho}} \cdot {{{ v} _x}}& {{\rho}} \cdot {{{ v} _y}}& {{\rho}} \cdot {{{ v} _z}}& \frac{1}{{(\gamma_{-1})}}\end{array}\right]}} {{\left[\begin{array}{ccccc} {|n_1|}& 1& 0& 0& {|n_1|}\\ -{{\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^x}}}}& 0& -{{\frac{1}{\rho}} {{ {(n_1)}} _y}}& -{{\frac{1}{\rho}} {{ {(n_1)}} _z}}& {\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^x}}}\\ -{{\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^y}}}}& 0& {\frac{1}{\rho}} {{ {(n_1)}} _x}& 0& {\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^y}}}\\ -{{\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^z}}}}& 0& 0& {\frac{1}{\rho}} {{ {(n_1)}} _x}& {\frac{1}{\rho}} {{{{c_s}}} \cdot {{{ {(n_1)}} ^z}}}\\ {{{|n_1|}}} \cdot {{{{c_s}}^{2}}}& 0& 0& 0& {{{|n_1|}}} \cdot {{{{c_s}}^{2}}}\end{array}\right]}}}$
${{R_F}} = {\left[\begin{array}{ccccc} {|n_1|}& 1& 0& 0& {|n_1|}\\ {{{-1}} {{{c_s}}} \cdot {{{ {(n_1)}} ^x}}} + {{{{|n_1|}}} \cdot {{{ v} ^x}}}& { v} ^x& {{-1}} {{{ {(n_1)}} _y}}& {{-1}} {{{ {(n_1)}} _z}}& {{{{c_s}}} \cdot {{{ {(n_1)}} ^x}}} + {{{{|n_1|}}} \cdot {{{ v} ^x}}}\\ {{{-1}} {{{c_s}}} \cdot {{{ {(n_1)}} ^y}}} + {{{{|n_1|}}} \cdot {{{ v} ^y}}}& { v} ^y& { {(n_1)}} _x& 0& {{{{c_s}}} \cdot {{{ {(n_1)}} ^y}}} + {{{{|n_1|}}} \cdot {{{ v} ^y}}}\\ {{{-1}} {{{c_s}}} \cdot {{{ {(n_1)}} ^z}}} + {{{{|n_1|}}} \cdot {{{ v} ^z}}}& { v} ^z& 0& { {(n_1)}} _x& {{{{c_s}}} \cdot {{{ {(n_1)}} ^z}}} + {{{{|n_1|}}} \cdot {{{ v} ^z}}}\\ {{{{|n_1|}}} \cdot {{{{c_s}}^{2}}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{\frac{1}{2}}} {{{(v)^2}}} \cdot {{{|n_1|}}}} + {{{-1}} {{{c_s}}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}}}& {{\frac{1}{2}}} {{{(v)^2}}}& {{{-1}} {{{ {(n_1)}} _y}} {{{ v} _x}}} + {{{{ {(n_1)}} _x}} {{{ v} _y}}}& {{{-1}} {{{ {(n_1)}} _z}} {{{ v} _x}}} + {{{{ {(n_1)}} _x}} {{{ v} _z}}}& {{{{|n_1|}}} \cdot {{{{c_s}}^{2}}} {{\frac{1}{{(\gamma_{-1})}}}}} + {{{\frac{1}{2}}} {{{(v)^2}}} \cdot {{{|n_1|}}}} + {{{{c_s}}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}}}\end{array}\right]}$
${{R_F}} = {\left[\begin{array}{ccccc} {|n_1|}& 1& 0& 0& {|n_1|}\\ {-{{{{c_s}}} \cdot {{{ {(n_1)}} ^x}}}} + {{{{|n_1|}}} \cdot {{{ v} ^x}}}& { v} ^x& -{{ {(n_1)}} _y}& -{{ {(n_1)}} _z}& {{{{c_s}}} \cdot {{{ {(n_1)}} ^x}}} + {{{{|n_1|}}} \cdot {{{ v} ^x}}}\\ {-{{{{c_s}}} \cdot {{{ {(n_1)}} ^y}}}} + {{{{|n_1|}}} \cdot {{{ v} ^y}}}& { v} ^y& { {(n_1)}} _x& 0& {{{{c_s}}} \cdot {{{ {(n_1)}} ^y}}} + {{{{|n_1|}}} \cdot {{{ v} ^y}}}\\ {-{{{{c_s}}} \cdot {{{ {(n_1)}} ^z}}}} + {{{{|n_1|}}} \cdot {{{ v} ^z}}}& { v} ^z& 0& { {(n_1)}} _x& {{{{c_s}}} \cdot {{{ {(n_1)}} ^z}}} + {{{{|n_1|}}} \cdot {{{ v} ^z}}}\\ {{{{h_{total}}}} \cdot {{{|n_1|}}}}{-{{{{c_s}}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}}}}& {\frac{1}{2}} {{(v)^2}}& {-{{{{ {(n_1)}} _y}} {{{ v} _x}}}} + {{{{ {(n_1)}} _x}} {{{ v} _y}}}& {-{{{{ {(n_1)}} _z}} {{{ v} _x}}}} + {{{{ {(n_1)}} _x}} {{{ v} _z}}}& {{{{h_{total}}}} \cdot {{{|n_1|}}}} + {{{{c_s}}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}}}\end{array}\right]}$
${{L_F}} = {{{\left[\begin{array}{ccccc} 0& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} _x}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}}& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} _y}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}}& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} _z}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}}& \frac{1}{{{2}} {{{|n_1|}}} \cdot {{{{c_s}}^{2}}}}\\ 1& 0& 0& 0& -{\frac{1}{{{c_s}}^{2}}}\\ 0& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} ^y}}}{{{|n_1|}}^{2}}}& \frac{{{\rho}} \cdot {{\left({{{{|n_1|}}^{2}}{-{{{{ {(n_1)}} ^y}} {{{ {(n_1)}} _y}}}}}\right)}}}{{{{ {(n_1)}} _x}} {{{{|n_1|}}^{2}}}}& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} ^y}} {{{ {(n_1)}} _z}}}{{{{ {(n_1)}} _x}} {{{{|n_1|}}^{2}}}}}& 0\\ 0& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} ^z}}}{{{|n_1|}}^{2}}}& -{\frac{{{\rho}} \cdot {{{ {(n_1)}} _y}} {{{ {(n_1)}} ^z}}}{{{{ {(n_1)}} _x}} {{{{|n_1|}}^{2}}}}}& \frac{{{\rho}} \cdot {{\left({{{{|n_1|}}^{2}}{-{{{{ {(n_1)}} ^z}} {{{ {(n_1)}} _z}}}}}\right)}}}{{{{ {(n_1)}} _x}} {{{{|n_1|}}^{2}}}}& 0\\ 0& \frac{{{\rho}} \cdot {{{ {(n_1)}} _x}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}& \frac{{{\rho}} \cdot {{{ {(n_1)}} _y}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}& \frac{{{\rho}} \cdot {{{ {(n_1)}} _z}}}{{{2}} {{{c_s}}} \cdot {{{{|n_1|}}^{2}}}}& \frac{1}{{{2}} {{{|n_1|}}} \cdot {{{{c_s}}^{2}}}}\end{array}\right]}} {{\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ -{{\frac{1}{\rho}} {{ v} ^x}}& \frac{1}{\rho}& 0& 0& 0\\ -{{\frac{1}{\rho}} {{ v} ^y}}& 0& \frac{1}{\rho}& 0& 0\\ -{{\frac{1}{\rho}} {{ v} ^z}}& 0& 0& \frac{1}{\rho}& 0\\ {\frac{1}{2}} {{{{(\gamma_{-1})}}} \cdot {{{(v)^2}}}}& -{{{{(\gamma_{-1})}}} \cdot {{{ v} _x}}}& -{{{{(\gamma_{-1})}}} \cdot {{{ v} _y}}}& -{{{{(\gamma_{-1})}}} \cdot {{{ v} _z}}}& {(\gamma_{-1})}\end{array}\right]}}}$
${{L_F}} = {\left[\begin{array}{ccccc} {{{\frac{1}{4}}} {{{(\gamma_{-1})}}} \cdot {{{(v)^2}}} \cdot {{\frac{1}{{|n_1|}}}} {{\frac{1}{{{c_s}}^{2}}}}} + {{{\frac{1}{2}}} {{{ {(n_1)}} _k}} {{{ v} ^k}} {{\frac{1}{{c_s}}}} {{\frac{1}{{{|n_1|}}^{2}}}}}& {{{-1}} \cdot {{\frac{1}{2}}} {{{ {(n_1)}} _x}} {{\frac{1}{{c_s}}}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ v} _x}} {{\frac{1}{{|n_1|}}}} {{\frac{1}{{{c_s}}^{2}}}}}& {{{-1}} \cdot {{\frac{1}{2}}} {{{ {(n_1)}} _y}} {{\frac{1}{{c_s}}}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ v} _y}} {{\frac{1}{{|n_1|}}}} {{\frac{1}{{{c_s}}^{2}}}}}& {{{-1}} \cdot {{\frac{1}{2}}} {{{ {(n_1)}} _z}} {{\frac{1}{{c_s}}}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ v} _z}} {{\frac{1}{{|n_1|}}}} {{\frac{1}{{{c_s}}^{2}}}}}& {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{\frac{1}{{|n_1|}}}} {{\frac{1}{{{c_s}}^{2}}}}\\ {1} + {{{-1}} \cdot {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{(v)^2}}} \cdot {{\frac{1}{{{c_s}}^{2}}}}}& {{{(\gamma_{-1})}}} \cdot {{{ v} _x}} {{\frac{1}{{{c_s}}^{2}}}}& {{{(\gamma_{-1})}}} \cdot {{{ v} _y}} {{\frac{1}{{{c_s}}^{2}}}}& {{{(\gamma_{-1})}}} \cdot {{{ v} _z}} {{\frac{1}{{{c_s}}^{2}}}}& {{-1}} {{{(\gamma_{-1})}}} \cdot {{\frac{1}{{{c_s}}^{2}}}}\\ {{{-1}} {{{ v} ^y}} {{\frac{1}{{ {(n_1)}} _x}}}} + {{{{ {(n_1)}} _k}} {{{ {(n_1)}} ^y}} {{{ v} ^k}} {{\frac{1}{{ {(n_1)}} _x}}} {{\frac{1}{{{|n_1|}}^{2}}}}}& {{-1}} {{{ {(n_1)}} ^y}} {{\frac{1}{{{|n_1|}}^{2}}}}& {\frac{1}{{ {(n_1)}} _x}} + {{{-1}} {{{ {(n_1)}} _y}} {{{ {(n_1)}} ^y}} {{\frac{1}{{ {(n_1)}} _x}}} {{\frac{1}{{{|n_1|}}^{2}}}}}& {{-1}} {{{ {(n_1)}} ^y}} {{{ {(n_1)}} _z}} {{\frac{1}{{ {(n_1)}} _x}}} {{\frac{1}{{{|n_1|}}^{2}}}}& 0\\ {{{-1}} {{{ v} ^z}} {{\frac{1}{{ {(n_1)}} _x}}}} + {{{{ {(n_1)}} _k}} {{{ {(n_1)}} ^z}} {{{ v} ^k}} {{\frac{1}{{ {(n_1)}} _x}}} {{\frac{1}{{{|n_1|}}^{2}}}}}& {{-1}} {{{ {(n_1)}} ^z}} {{\frac{1}{{{|n_1|}}^{2}}}}& {{-1}} {{{ {(n_1)}} _y}} {{{ {(n_1)}} ^z}} {{\frac{1}{{ {(n_1)}} _x}}} {{\frac{1}{{{|n_1|}}^{2}}}}& {\frac{1}{{ {(n_1)}} _x}} + {{{-1}} {{{ {(n_1)}} _z}} {{{ {(n_1)}} ^z}} {{\frac{1}{{ {(n_1)}} _x}}} {{\frac{1}{{{|n_1|}}^{2}}}}}& 0\\ {{{\frac{1}{4}}} {{{(\gamma_{-1})}}} \cdot {{{(v)^2}}} \cdot {{\frac{1}{{|n_1|}}}} {{\frac{1}{{{c_s}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ {(n_1)}} _k}} {{{ v} ^k}} {{\frac{1}{{c_s}}}} {{\frac{1}{{{|n_1|}}^{2}}}}}& {{{\frac{1}{2}}} {{{ {(n_1)}} _x}} {{\frac{1}{{c_s}}}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ v} _x}} {{\frac{1}{{|n_1|}}}} {{\frac{1}{{{c_s}}^{2}}}}}& {{{\frac{1}{2}}} {{{ {(n_1)}} _y}} {{\frac{1}{{c_s}}}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ v} _y}} {{\frac{1}{{|n_1|}}}} {{\frac{1}{{{c_s}}^{2}}}}}& {{{\frac{1}{2}}} {{{ {(n_1)}} _z}} {{\frac{1}{{c_s}}}} {{\frac{1}{{{|n_1|}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{{ v} _z}} {{\frac{1}{{|n_1|}}}} {{\frac{1}{{{c_s}}^{2}}}}}& {{\frac{1}{2}}} {{{(\gamma_{-1})}}} \cdot {{\frac{1}{{|n_1|}}}} {{\frac{1}{{{c_s}}^{2}}}}\end{array}\right]}$
${{{{R_F}}} \cdot {{X}}} = {\left[\begin{array}{c} {{X^2}} + {{{{X^1}}} \cdot {{{|n_1|}}}} + {{{{X^5}}} \cdot {{{|n_1|}}}}\\ {{{{X^2}}} \cdot {{{ v} ^x}}}{-{{{{X^3}}} \cdot {{{ {(n_1)}} _y}}}}{-{{{{X^4}}} \cdot {{{ {(n_1)}} _z}}}}{-{{{{X^1}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} ^x}}}} + {{{{X^1}}} \cdot {{{|n_1|}}} \cdot {{{ v} ^x}}} + {{{{X^5}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} ^x}}} + {{{{X^5}}} \cdot {{{|n_1|}}} \cdot {{{ v} ^x}}}\\ {{{{X^2}}} \cdot {{{ v} ^y}}} + {{{{X^3}}} \cdot {{{ {(n_1)}} _x}}}{-{{{{X^1}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} ^y}}}} + {{{{X^1}}} \cdot {{{|n_1|}}} \cdot {{{ v} ^y}}} + {{{{X^5}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} ^y}}} + {{{{X^5}}} \cdot {{{|n_1|}}} \cdot {{{ v} ^y}}}\\ {{{{X^2}}} \cdot {{{ v} ^z}}} + {{{{X^4}}} \cdot {{{ {(n_1)}} _x}}}{-{{{{X^1}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} ^z}}}} + {{{{X^1}}} \cdot {{{|n_1|}}} \cdot {{{ v} ^z}}} + {{{{X^5}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} ^z}}} + {{{{X^5}}} \cdot {{{|n_1|}}} \cdot {{{ v} ^z}}}\\ {\frac{1}{2}}{\left({{{{{(v)^2}}} \cdot {{{X^2}}}} + {{{2}} {{{X^1}}} \cdot {{{h_{total}}}} \cdot {{{|n_1|}}}} + {{{2}} {{{X^3}}} \cdot {{{ {(n_1)}} _x}} {{{ v} _y}}}{-{{{2}} {{{X^3}}} \cdot {{{ {(n_1)}} _y}} {{{ v} _x}}}}{-{{{2}} {{{X^4}}} \cdot {{{ {(n_1)}} _z}} {{{ v} _x}}}} + {{{2}} {{{X^4}}} \cdot {{{ {(n_1)}} _x}} {{{ v} _z}}} + {{{2}} {{{X^5}}} \cdot {{{h_{total}}}} \cdot {{{|n_1|}}}}{-{{{2}} {{{X^1}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}}}} + {{{2}} {{{X^5}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}}}}\right)}\end{array}\right]}$
${{{{L_F}}} \cdot {{X}}} = {\left[\begin{array}{c} \frac{{{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^5}}} \cdot {{{|n_1|}}}}{-{{{2}} {{{X^2}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _x}}}}{-{{{2}} {{{X^3}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _y}}}}{-{{{2}} {{{X^4}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _z}}}} + {{{{(\gamma_{-1})}}} \cdot {{{(v)^2}}} \cdot {{{X^1}}} \cdot {{{|n_1|}}}}{-{{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^2}}} \cdot {{{|n_1|}}} \cdot {{{ v} _x}}}}{-{{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^3}}} \cdot {{{|n_1|}}} \cdot {{{ v} _y}}}}{-{{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^4}}} \cdot {{{|n_1|}}} \cdot {{{ v} _z}}}} + {{{2}} {{{X^1}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}}}}{{{4}} {{{{c_s}}^{2}}} {{{{|n_1|}}^{2}}}}\\ \frac{{-{{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^5}}}}} + {{{2}} {{{X^1}}} \cdot {{{{c_s}}^{2}}}}{-{{{{(\gamma_{-1})}}} \cdot {{{(v)^2}}} \cdot {{{X^1}}}}} + {{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^2}}} \cdot {{{ v} _x}}} + {{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^3}}} \cdot {{{ v} _y}}} + {{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^4}}} \cdot {{{ v} _z}}}}{{{2}} {{{{c_s}}^{2}}}}\\ \frac{{{{{X^3}}} \cdot {{{{|n_1|}}^{2}}}}{-{{{{X^1}}} \cdot {{{ v} ^y}} {{{{|n_1|}}^{2}}}}}{-{{{{X^2}}} \cdot {{{ {(n_1)}} _x}} {{{ {(n_1)}} ^y}}}}{-{{{{X^3}}} \cdot {{{ {(n_1)}} ^y}} {{{ {(n_1)}} _y}}}}{-{{{{X^4}}} \cdot {{{ {(n_1)}} ^y}} {{{ {(n_1)}} _z}}}} + {{{{X^1}}} \cdot {{{ {(n_1)}} _k}} {{{ {(n_1)}} ^y}} {{{ v} ^k}}}}{{{{ {(n_1)}} _x}} {{{{|n_1|}}^{2}}}}\\ \frac{{{{{X^4}}} \cdot {{{{|n_1|}}^{2}}}}{-{{{{X^1}}} \cdot {{{ v} ^z}} {{{{|n_1|}}^{2}}}}}{-{{{{X^2}}} \cdot {{{ {(n_1)}} _x}} {{{ {(n_1)}} ^z}}}}{-{{{{X^3}}} \cdot {{{ {(n_1)}} _y}} {{{ {(n_1)}} ^z}}}}{-{{{{X^4}}} \cdot {{{ {(n_1)}} ^z}} {{{ {(n_1)}} _z}}}} + {{{{X^1}}} \cdot {{{ {(n_1)}} _k}} {{{ {(n_1)}} ^z}} {{{ v} ^k}}}}{{{{ {(n_1)}} _x}} {{{{|n_1|}}^{2}}}}\\ \frac{{{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^5}}} \cdot {{{|n_1|}}}} + {{{2}} {{{X^2}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _x}}} + {{{2}} {{{X^3}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _y}}} + {{{2}} {{{X^4}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _z}}} + {{{{(\gamma_{-1})}}} \cdot {{{(v)^2}}} \cdot {{{X^1}}} \cdot {{{|n_1|}}}}{-{{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^2}}} \cdot {{{|n_1|}}} \cdot {{{ v} _x}}}}{-{{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^3}}} \cdot {{{|n_1|}}} \cdot {{{ v} _y}}}}{-{{{2}} {{{(\gamma_{-1})}}} \cdot {{{X^4}}} \cdot {{{|n_1|}}} \cdot {{{ v} _z}}}}{-{{{2}} {{{X^1}}} \cdot {{{c_s}}} \cdot {{{ {(n_1)}} _k}} {{{ v} ^k}}}}}{{{4}} {{{{c_s}}^{2}}} {{{{|n_1|}}^{2}}}}\end{array}\right]}$
code:
real const nLen = normal_len(n);
real const gamma = solver->heatCapacityRatio;
real const gamma_1 = solver->heatCapacityRatio - 1.;
real3 const v_n = normal_vecDotNs(n, (eig)->v);
primFromCons:
real const tmp1 = 1. / (U)->rho;
(result)->rho = (U)->rho;
(result)->v.x = (U)->m.x * tmp1;
(result)->v.y = (U)->m.y * tmp1;
(result)->v.z = (U)->m.z * tmp1;
(result)->P = gamma_1 * (U)->ETotal + -1. * gamma_1 * 1. / 2. * coordLenSq((U)->m) * tmp1;
consFromPrim
(result)->rho = (W)->rho;
(result)->m.x = (W)->rho * (W)->v.x;
(result)->m.y = (W)->rho * (W)->v.y;
(result)->m.z = (W)->rho * (W)->v.z;
(result)->ETotal = coordLenSq((W)->v) * (W)->rho * 1. / 2. + P * 1. / gamma_1;
eigen_leftTransform
real const tmp1 = (X)->ptr[4] * nLen;
real const tmp2 = gamma_1 * tmp1;
real const tmp3 = (eig)->Cs * normal_l1x(n);
real const tmp4 = (X)->ptr[1] * tmp3;
real const tmp5 = 2. * tmp4;
real const tmp6 = (eig)->Cs * normal_l1y(n);
real const tmp7 = (X)->ptr[2] * tmp6;
real const tmp8 = 2. * tmp7;
real const tmp9 = (eig)->Cs * normal_l1z(n);
real const tmp10 = (X)->ptr[3] * tmp9;
real const tmp11 = 2. * tmp10;
real const tmp12 = (X)->ptr[0] * nLen;
real const tmp13 = coordLenSq((eig)->v) * tmp12;
real const tmp14 = nLen * (eig)->vL.x;
real const tmp15 = (X)->ptr[1] * tmp14;
real const tmp16 = gamma_1 * tmp15;
real const tmp17 = 2. * tmp16;
real const tmp18 = nLen * (eig)->vL.y;
real const tmp19 = (X)->ptr[2] * tmp18;
real const tmp20 = gamma_1 * tmp19;
real const tmp21 = 2. * tmp20;
real const tmp22 = nLen * (eig)->vL.z;
real const tmp23 = (X)->ptr[3] * tmp22;
real const tmp24 = gamma_1 * tmp23;
real const tmp25 = 2. * tmp24;
real const tmp26 = (eig)->Cs * v_n.x;
real const tmp27 = (X)->ptr[0] * tmp26;
real const tmp28 = -tmp25;
real const tmp29 = 2. * tmp27;
real const tmp30 = -tmp21;
real const tmp31 = -tmp17;
real const tmp32 = gamma_1 * tmp13;
real const tmp33 = 2. * tmp2;
real const tmp34 = (eig)->Cs * (eig)->Cs;
real const tmp35 = nLen * nLen;
real const tmp36 = tmp34 * tmp35;
real const tmp37 = 4. * tmp36;
real const tmp38 = 1. / tmp37;
real const tmp39 = normal_l1x(n) * tmp35;
real const tmp40 = 1. / tmp39;
(result)->ptr[0] = (-tmp5 + -tmp8 + -tmp11 + tmp28 + tmp29 + tmp30 + tmp31 + tmp32 + tmp33) * tmp38;
(result)->ptr[1] = 1. / (2. * tmp34) * (-2. * gamma_1 * (X)->ptr[4] + 2. * (X)->ptr[0] * tmp34 + -gamma_1 * coordLenSq((eig)->v) * (X)->ptr[0] + 2. * gamma_1 * (X)->ptr[1] * (eig)->vL.x + 2. * gamma_1 * (X)->ptr[3] * (eig)->vL.z + 2. * gamma_1 * (X)->ptr[2] * (eig)->vL.y);
(result)->ptr[2] = ((X)->ptr[2] * tmp35 + -(X)->ptr[0] * v_Uy * tmp35 + -(X)->ptr[1] * normal_l1x(n) * normal_u1y(n) + -(X)->ptr[2] * normal_l1y(n) * normal_u1y(n) + (X)->ptr[0] * v_n.x * normal_u1y(n) + -(X)->ptr[3] * normal_l1z(n) * normal_u1y(n)) * tmp40;
(result)->ptr[3] = ((X)->ptr[3] * tmp35 + -(X)->ptr[0] * v_Uz * tmp35 + -(X)->ptr[1] * normal_l1x(n) * normal_u1z(n) + -(X)->ptr[2] * normal_l1y(n) * normal_u1z(n) + (X)->ptr[0] * v_n.x * normal_u1z(n) + -(X)->ptr[3] * normal_l1z(n) * normal_u1z(n)) * tmp40;
(result)->ptr[4] = (-tmp29 + tmp5 + tmp8 + tmp11 + tmp28 + tmp30 + tmp31 + tmp32 + tmp33) * tmp38;
eigen_rightTransform
real const tmp1 = v_Ux;
real const tmp2 = (eig)->Cs * normal_u1x(n);
real const tmp3 = nLen * tmp1;
real const tmp4 = v_Uy;
real const tmp5 = (eig)->Cs * normal_u1y(n);
real const tmp6 = nLen * tmp4;
real const tmp7 = v_Uz;
real const tmp8 = (eig)->Cs * normal_u1z(n);
real const tmp9 = nLen * tmp7;
real const tmp10 = (eig)->hTotal * nLen;
real const tmp11 = (eig)->Cs * v_n.x;
(result)->ptr[0] = (X)->ptr[1] + (X)->ptr[4] * nLen + (X)->ptr[0] * nLen;
(result)->ptr[1] = (X)->ptr[1] * tmp1 + -(X)->ptr[2] * normal_l1y(n) + -(X)->ptr[3] * normal_l1z(n) + -(X)->ptr[0] * tmp2 + (X)->ptr[0] * tmp3 + (X)->ptr[4] * tmp3 + (X)->ptr[4] * tmp2;
(result)->ptr[2] = (X)->ptr[1] * tmp4 + (X)->ptr[2] * normal_l1x(n) + -(X)->ptr[0] * tmp5 + (X)->ptr[0] * tmp6 + (X)->ptr[4] * tmp6 + (X)->ptr[4] * tmp5;
(result)->ptr[3] = (X)->ptr[1] * tmp7 + (X)->ptr[3] * normal_l1x(n) + -(X)->ptr[0] * tmp8 + (X)->ptr[0] * tmp9 + (X)->ptr[4] * tmp9 + (X)->ptr[4] * tmp8;
(result)->ptr[4] = 1. / 2. * (coordLenSq((eig)->v) * (X)->ptr[1] + 2. * (X)->ptr[0] * tmp10 + 2. * (X)->ptr[2] * normal_l1x(n) * (eig)->vL.y + -2. * (X)->ptr[2] * normal_l1y(n) * (eig)->vL.x + -2. * (X)->ptr[3] * normal_l1z(n) * (eig)->vL.x + 2. * (X)->ptr[3] * normal_l1x(n) * (eig)->vL.z + 2. * (X)->ptr[4] * tmp10 + 2. * (X)->ptr[4] * tmp11 + -2. * (X)->ptr[0] * tmp11);