relation between eigenmodes and left eigenvector matrix:
$U_{,t} + F_{,x} = 0$
apply chain rule
$U_{,t} + \frac{\partial F}{\partial U} U_{,x} = 0$
let $A = \frac{\partial F}{\partial U}$
$U_{,t} + A U_{,x} = 0$
Eigenvector decomposition: let $Q \Lambda Q^{-1} = A$
$U_{,t} + Q \Lambda Q^{-1} U_{,x} = 0$
Left-multiply by left eigenvector matrix:
$Q^{-1} U_{,t} + \Lambda Q^{-1} U_{,x} = 0$
Let $w_{,a} = Q^{-1} U_{,a}$
$w_{,t} + \lambda w_{,x} = 0$
eigenfunctions are solutions of the form $w_{,t} + \lambda w_{,x} = 0$
So the derivative $w_{,a}$ is a linear combination of the derivatives of $u_{,a}$ and the row of the left eigenvector matrix
quasilinear form / relation between conservative and primitive PDEs:
$U_{,t} + F_{,x} = 0$
$U_{,t} + A U_{,x} = 0$
$\frac{\partial U}{\partial W} W_{,t} + A \frac{\partial U}{\partial W} W_{,x} = 0$
Let $P = \frac{\partial U}{\partial W}$
$P W_{,t} + A P W_{,x} = 0$
$W_{,t} + P^{-1} A P W_{,x} = 0$
Euler Fluid Equations
variables:
$\rho =$ density
$v_i =$ velocity
$m_i = \rho v_i = $ momentum
$P =$ pressure
$\gamma =$ heat capacity ratio
$\tilde\gamma = \gamma - 1$
$e_{int} = \frac{P}{\rho \tilde\gamma} =$ internal specific energy
$e_{kin} = \frac{1}{2} v^2 =$ kinetic specific energy
$e_{total} = e_{kin} + e_{int} = \frac{1}{2} v^2 + \frac{P}{\rho \tilde\gamma} =$ total specific energy
$E_{int} = \frac{P}{\tilde\gamma} =$ internal densitized energy
$E_{kin} = \frac{1}{2} \rho v^2 =$ kinetic densitized energy
$E_{total} = E_{kin} + E_{int} = \rho e_{total} = \frac{1}{2} \rho v^2 + \frac{P}{\tilde\gamma} =$ total densitized energy
$h_{int} = e_{int} + \frac{P}{\rho} = \gamma e_{int} = $ internal specific enthalpy
$h_{total} = h_{int} + e_{kin} = e_{total} + \frac{P}{\rho} =$ total specific enthalpy
$H_{int} = \rho h_{int} = E_{int} + P = $ internal densitized enthalpy
$H_{total} = \rho h_{total} = E_{total} + P =$ total densitized enthalpy
$C_s = \sqrt{\frac{\gamma P}{\rho}} =$ speed of sound
conservative and primitive variables:
$W_i = \left[\matrix{ \rho & v_i & P }\right]^T$
$U_i = \left[\matrix{ \rho & \rho v_i & E_{total} }\right]^T$
$P = \frac{\partial U_i}{\partial W_j} = \left[\matrix{
1 & 0 & 0 \\
v_i & \rho \delta_{ij} & 0 \\
\frac{1}{2} v^2 & \rho v_j & 1/\tilde\gamma
}\right]$
$P^{-1} = \frac{\partial W_i}{\partial U_j} = \left[\matrix{
1 & 0 & 0 \\
-v_i / \rho & \delta_{ij} / \rho & 0 \\
\frac{1}{2} \tilde\gamma v^2 & -\tilde\gamma v_j & \tilde\gamma
}\right]$
Euler fluid equations in flux-form:
$U_{,t} + F_{j,j} = 0$
$\rho_{,t} + (\rho v_j)_{,j} = 0$
$(\rho v_i)_{,t} + (\rho v_i v_j + \delta_{ij} P)_{,j} = 0$
$E_{total,t} + (v_j H_{total})_{,j} = 0$
$\rho_{,t} + \rho_{,j} v_j + \rho v_{j,j} = 0$
$(\rho v_i)_{,t} + \rho_{,j} v_i v_j + \rho v_{i,j} v_j + \rho v_i v_{j,j} + P_{,i} = 0$
$E_{total,t} + v_{j,j} H_{total} + v_j (\frac{1}{2} \rho_{,j} v^2 + \rho v_k v_{k,j} + \frac{\gamma}{\tilde\gamma} P_{,j}) = 0$
$\left[\matrix{
\rho \\ \rho v_i \\ E_{total}
}\right]_{,t}
+ \left[\matrix{
v_j &
\rho \delta_{jm} &
0 \\
v_i v_j &
\rho ( v_j \delta_{im} + v_i \delta_{jm} ) &
\delta_{ij} \\
\frac{1}{2} v_j v^2 &
\delta_{jm} H_{total} + \rho v_j v_m &
\frac{\gamma}{\tilde\gamma} v_j
}\right]
\left[\matrix{
\rho \\ v_m \\ P
}\right]_{,j} = 0$
$\left[\matrix{
\rho \\ \rho v_i \\ E_{total}
}\right]_{,t}
+ \left[\matrix{
v_j &
\rho \delta_{jm} &
0 \\
v_i v_j &
\rho ( v_j \delta_{im} + v_i \delta_{jm} ) &
\delta_{ij} \\
\frac{1}{2} v_j v^2 &
\delta_{jm} H_{total} + \rho v_j v_m &
\frac{\gamma}{\tilde\gamma} v_j
}\right]
\left[\matrix{
1 & 0 & 0 \\
-v_m / \rho & \delta_{mk} / \rho & 0 \\
\frac{1}{2} \tilde\gamma v^2 & -\tilde\gamma v_k & \tilde\gamma
}\right]
\left[\matrix{
\rho \\ \rho v_k \\ E_{total}
}\right]_{,j} = 0$
matrix conservative form:
$\left[\matrix{
\rho \\ \rho v_i \\ E_{total}
}\right]_{,t}
+ \left[\matrix{
0 &
\delta_{jk} &
0 \\
-v_i v_j + \frac{1}{2} \delta_{ij} \tilde\gamma v^2 &
\delta_{ik} v_j + \delta_{jk} v_i - \delta_{ij} \tilde\gamma v_k &
\delta_{ij} \tilde\gamma \\
v_j (\frac{1}{2} \tilde\gamma v^2 - h_{total}) &
-\tilde\gamma v_j v_k + \delta_{jk} h_{total} &
\gamma v_j
}\right]
\left[\matrix{
\rho \\ \rho v_k \\ E_{total}
}\right]_{,j} = 0$
$U_{,t} + A_j U_{,j} = 0$
matrix primitive form:
$W_{,t} + P^{-1} A_j P W_{,j} = 0$
$\left[\matrix{
\rho \\ v_m \\ P
}\right]_{,t} + \left[\matrix{
1 & 0 & 0 \\
-v_m / \rho & \delta_{mi} / \rho & 0 \\
\frac{1}{2} \tilde\gamma v^2 & -\tilde\gamma v_i & \tilde\gamma
}\right] \left[\matrix{
0 &
\delta_{jk} &
0 \\
-v_i v_j + \frac{1}{2} \delta_{ij} \tilde\gamma v^2 &
\delta_{ik} v_j + \delta_{jk} v_i - \delta_{ij} \tilde\gamma v_k &
\delta_{ij} \tilde\gamma \\
v_j (\frac{1}{2} \tilde\gamma v^2 - h_{total}) &
-\tilde\gamma v_j v_k + \delta_{jk} h_{total} &
\gamma v_j
}\right] \left[\matrix{
1 & 0 & 0 \\
v_k & \rho \delta_{kn} & 0 \\
\frac{1}{2} v^2 & \rho v_n & 1/\tilde\gamma
}\right] \left[\matrix{
\rho \\ v_n \\ P
}\right]_{,j} = 0$
$\left[\matrix{
\rho \\ v_m \\ P
}\right]_{,t} + \left[\matrix{
0 & \delta_{jk} & 0 \\
\frac{1}{\rho} (-v_m v_j + \frac{1}{2} \delta_{mj} \tilde\gamma v^2) &
\frac{1}{\rho} (\delta_{mk} v_j - \delta_{mj} \tilde\gamma v_k) &
\frac{1}{\rho} \delta_{mj} \tilde\gamma \\
\tilde\gamma v_j (v^2 - h_{total}) &
\tilde\gamma (
- v_j v_k
- \frac{1}{2} \delta_{jk} v^2
+ \delta_{jk} h_{total}) &
\tilde\gamma v_j
}\right] \left[\matrix{
1 & 0 & 0 \\
v_k & \rho \delta_{kn} & 0 \\
\frac{1}{2} v^2 & \rho v_n & 1/\tilde\gamma
}\right] \left[\matrix{
\rho \\ v_n \\ P
}\right]_{,j} = 0$
$\left[\matrix{
\rho \\ v_m \\ P
}\right]_{,t} + \left[\matrix{
v_j & \rho \delta_{jn} & 0 \\
0 & \delta_{mn} v_j & \frac{1}{\rho} \delta_{mj} \\
0 & \delta_{jn} \gamma P & v_j
}\right] \left[\matrix{
\rho \\ v_n \\ P
}\right]_{,j} = 0$
$\rho_{,t} + v_j \rho_{,j} + \rho v_{j,j} = 0$
$v_{i,t} + v_j v_{i,j} + \frac{1}{\rho} \delta_{ij} P_{,j} = 0$
$P_{,t} + v_j P_{,j} + \gamma P v_{j,j} = 0$
for the $x$-direction alone, assuming no flux along $y$ or $z$:
$\rho_{,t} + v_x \rho_{,x} = -\rho v_{x,x}$
$v_{x,t} + v_x v_{x,x} = -\frac{1}{\rho} P_{,x}$
$v_{p,t} + v_x v_{p,x} = 0$ for $p \ne x$
$P_{,t} + v_x P_{,x} = -\gamma P v_{x,x}$
conservative form eigenvector decomposition in x-direction with all indexes expanded:
$\left[\matrix{
\rho \\ \rho v_i \\ E_{total}
}\right]_{,t}
+ \left[\matrix{
1 & 1 & 0 & 0 & 1 \\
v_x - C_s & v_x & 0 & 0 & v_x + C_s \\
v_y & v_y & 1 & 0 & v_y \\
v_z & v_z & 0 & 1 & v_z \\
h_{total} - v_j C_s & \frac{1}{2} v^2 & v_y & v_z & h_{total} + v_j C_s
}\right]
\left[\matrix{
v_j - C_s & 0 & 0 & 0 & 0 \\
0 & v_j & 0 & 0 & 0 \\
0 & 0 & v_j & 0 & 0 \\
0 & 0 & 0 & v_j & 0 \\
0 & 0 & 0 & 0 & v_j + C_s
}\right]
\left(
\frac{1}{2 {C_s}^2}
\left[\matrix{
\frac{1}{2} \tilde\gamma v^2 + C_s v_x &
-C_s - \tilde\gamma v_x &
-\tilde\gamma v_y &
-\tilde\gamma v_z &
\tilde\gamma \\
2 {C_s}^2 - \tilde\gamma v^2 &
2 \tilde\gamma v_x &
2 \tilde\gamma v_y &
2 \tilde\gamma v_z &
-2 \tilde\gamma \\
-2 {C_s}^2 v_y &
0 &
2 {C_s}^2 &
0 &
0 \\
-2 {C_s}^2 v_z &
0 &
0 &
2 {C_s}^2 &
0 \\
\frac{1}{2} \tilde\gamma v^2 - C_s v_x &
C_s - \tilde\gamma v_x &
-\tilde\gamma v_y &
-\tilde\gamma v_z &
\tilde\gamma
}\right]
\right)
\left[\matrix{
\rho \\ \rho v_k \\ E_{total}
}\right]_{,x}
$
conservative form eigenvector decomposition in j-direction:
$\left[\matrix{
\rho \\ \rho v_i \\ E_{total}
}\right]_{,t}
+ \left[\matrix{
1 & 1 & 0 & 0 & 1 \\
v_i - \delta_{ij} C_s & v_i & n^2_i & n^3_i & v_i + \delta_{ij} C_s \\
h_{total} - v_j C_s & \frac{1}{2} v^2 & n^2_m v_m & n^3_m v_m & h_{total} + v_j C_s
}\right]
\left[\matrix{
v_j - C_s & 0 & 0 & 0 & 0 \\
0 & v_j & 0 & 0 & 0 \\
0 & 0 & v_j & 0 & 0 \\
0 & 0 & 0 & v_j & 0 \\
0 & 0 & 0 & 0 & v_j + C_s
}\right]
\left(
\frac{1}{2 {C_s}^2}
\left[\matrix{
\frac{1}{2} \tilde\gamma v^2 + C_s v_x &
-\delta_{jk} C_s - \tilde\gamma v_k &
\tilde\gamma \\
-2 {C_s}^2 (v_n - \delta_{jn} (v_j + 1)) - \delta_{jn} \tilde\gamma v^2 &
2 {C_s}^2 \delta_{nk} + \delta_{nj} (2 \tilde\gamma v_k - 2 \delta_{jk} {C_s}^2) &
-2 \delta_{jn} \tilde\gamma \\
\frac{1}{2} \tilde\gamma v^2 - C_s v_x &
\delta_{jk} C_s - \tilde\gamma v_k &
\tilde\gamma
}\right]
\right)
\left[\matrix{
\rho \\ \rho v_k \\ E_{total}
}\right]_{,j}
= 0$
The idea is that, for an eigenmode of the Euler fluid equations, it should satisfy $w_{,t} + \lambda w_{,x} = 0$
From here on out I'm choosing the $x$-direction flux, so $j = x$
The eigenmode becomes
$w_{i,a} =
\frac{1}{2 {C_s}^2}
\left[\matrix{
\frac{1}{2} \tilde\gamma v^2 + C_s v_x &
-\delta_{xk} C_s - \tilde\gamma v_k &
\tilde\gamma \\
-2 {C_s}^2 (v_i - \delta_{xi} (v_x + 1)) - \delta_{xi} \tilde\gamma v^2 &
2 {C_s}^2 \delta_{ik} + \delta_{xi} (2 \tilde\gamma v_k - 2 \delta_{xk} {C_s}^2) &
-2 \delta_{xi} \tilde\gamma \\
\frac{1}{2} \tilde\gamma v^2 - C_s v_x &
\delta_{xk} C_s - \tilde\gamma v_k &
\tilde\gamma
}\right]
\left[\matrix{
\rho \\ \rho v_k \\ E_{total}
}\right]_{,a}$
for $\lambda_i = \left[\matrix{
v_x - C_s \\ v_x \\ v_x + C_s
}\right]$
for $p \ne x$
$w_{i,a} =
\frac{1}{2 {C_s}^2}
\left[\matrix{
(\frac{1}{2} \tilde\gamma v^2 + C_s v_x) \rho_{,a}
+ (\pm\delta_{xk} C_s - \tilde\gamma v_k) (\rho v_k)_{,a}
+ \tilde\gamma E_{total,a} \\
(2 {C_s}^2 - \tilde\gamma v^2) \rho_{,a}
+ (2 {C_s}^2 \delta_{xk} + 2 \tilde\gamma v_k - 2 \delta_{xk} {C_s}^2) (\rho v_k)_{,a}
- 2 \tilde\gamma E_{total,a} \\
-2 {C_s}^2 v_p \rho_{,a}
+ 2 {C_s}^2 \delta_{pk} (\rho v_k)_{,a}
}\right]$
for $\lambda_i = \left[\matrix{
v_x \pm C_s \\ v_x \\ v_x
}\right]$
$w_{i,a} =
\left[\matrix{
\frac{1}{2 {C_s}^2} P_{,a} \pm \frac{1}{2 {C_s}} \rho v_{x,a} \\
\rho_{,a} - \frac{1}{{C_s}^2} P_{,a} \\
\rho v_{p,a}
}\right]$
for $\lambda_i = \left[\matrix{
v_x \pm C_s \\ v_x \\ v_x
}\right]$
so $w_{i,t} + \lambda_i w_{i,x}$ should equal zero ...
$\frac{1}{2 {C_s}^2} P_{,t} \pm \frac{1}{2 {C_s}} \rho v_{x,t} + (v_x \pm C_s) (\frac{1}{2 {C_s}^2} P_{,x} \pm \frac{1}{2 {C_s}} \rho v_{x,x})$
$\frac{1}{2 {C_s}^2} P_{,t} \pm \frac{1}{2 {C_s}} \rho v_{x,t} + (v_x \pm C_s) \frac{1}{2 {C_s}^2} P_{,x} \pm (v_x \pm C_s) \frac{1}{2 {C_s}} \rho v_{x,x}$
$\frac{1}{2 {C_s}^2} P_{,t}
\pm \frac{1}{2 {C_s}} \rho v_{x,t}
+ v_x \frac{1}{2 {C_s}^2} P_{,x}
\pm \frac{1}{2 C_s} P_{,x}
\pm v_x \frac{1}{2 {C_s}} \rho v_{x,x}
+ \frac{1}{2} \rho v_{x,x}$
$\frac{1}{2 {C_s}^2} (P_{,t} + v_x P_{,x})
\mp \frac{1}{2 C_s} P_{,x}
\pm \frac{1}{2 C_s} P_{,x}
+ \frac{1}{2} \rho v_{x,x}$
$\frac{1}{2 {C_s}^2} (-\gamma P v_{x,x})
+ \frac{1}{2} \rho v_{x,x}$
$-\frac{1}{2} \rho v_{x,x} + \frac{1}{2} \rho v_{x,x}$
$0$
$\rho_{,t} - \frac{1}{{C_s}^2} P_{,t} + v_x (\rho_{,x} - \frac{1}{{C_s}^2} P_{,x})$
$\rho_{,t} + \rho_{,x} v_x - \frac{1}{{C_s}^2} (P_{,t} + v_x P_{,x})$
$-\rho v_{x,x} - \frac{1}{{C_s}^2} (P_{,t} + v_x P_{,x})$
$-\rho v_{x,x} - \frac{1}{{C_s}^2} (-\gamma P v_{x,x})$
$-\rho v_{x,x} + \rho v_{x,x}$
$0$
$\rho v_{p,t} + v_x \rho v_{p,x}$ for $p \ne x$
$\rho (v_{p,t} + v_x v_{p,x})$
$0$