This was spun off my "Einstein-Maxwell Conservation Law" worksheet,
which started as this but became what it is.
Combining Maxwell and fluids
primitives:
$U^I = \left[\matrix{
\rho \\ \rho v^i \\ E^i \\ B^i \\ E_{hydro}
}\right]$
conservatives:
$U^I = \downarrow I(i) \left[\matrix{
\rho \\ \rho v^i \\ E_{hydro} \\ \epsilon E^i \\ B^i
}\right]$
$E_{hydro} = E_{kin} + E_{int}$
$= \frac{1}{2} \rho v^2 + \frac{P}{\gamma - 1} =$ total hydrodynamic energy
$E_{EM} = \frac{1}{2} ( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 ) = $ electomagnetic energy, but I'm not incorporating it into the conservative energy just yet...
Conservative PDEs:
$\rho_{,t} + (\rho v^i)_{,i} = 0$
$(\rho v^i)_{,t} + (\rho v^i v^j + \delta^{ij} P)_{,j} = 0$
$(E_{hydro})_{,t} + (v^j H_{hydro})_{,j} = 0$
$(\epsilon_0 E^i)_{,t} - (\frac{1}{\mu_0} {\epsilon^{ij}}_k B^k)_{,j} = -J^i$
${B^i}_{,t} + ({\epsilon^{ij}}_k E^k)_{,j} = 0$
Election vs Ion
using Abgrall, Kumar "Robust Finite Volume Schemes for Two-Fluid Plasma Equations
variables:
$q_\alpha = $ charge of species $\alpha$
$m_\alpha = $ mass of species $\alpha$
$r_\alpha = q_\alpha / m_\alpha$ = ratio of charge to mass
$\rho_\alpha = n_\alpha m_\alpha$
$P_\alpha = n_\alpha k_B T_\alpha$
Flux derivative wrt conservative variables:
$\left[\matrix{
\rho_{ion} \\ \rho_{ion} v_{ion}^i \\ E_{hydro,ion} \\
\rho_{elec} \\ \rho_{elec} v_{elec}^i \\ E_{hydro,elec} \\
\epsilon_0 E^i \\ B^i
}\right]_{,t} + \left[\matrix{
0 & \delta^{jk} & 0 & 0 & 0 & 0 & 0 & 0 \\
-v_{ion}^i v_{ion}^j + \frac{1}{2} \delta^{ij} \tilde\gamma v_{ion}^2 &
\delta^i_k v_{ion}^j + \delta^j_k v_{ion}^i - \delta^{ij} \tilde\gamma v_{ion,k} &
\delta^{ij} \tilde\gamma &
0 & 0 & 0 & 0 & 0 \\
v_{ion}^j (\frac{1}{2} \tilde\gamma v_{ion}^2 - h_{hydro,ion}) &
-\tilde\gamma v_{ion}^j v_{ion,k} + \delta^j_k h_{hydro,ion} &
\gamma v_{ion}^j &
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 &0 & \delta^{jk} & 0 & 0 & 0 \\
0 & 0 & 0 & -v_{elec}^i v_{elec}^j + \frac{1}{2} \delta^{ij} \tilde\gamma v_{elec}^2 &
\delta^i_k v_{elec}^j + \delta^j_k v_{elec}^i - \delta^{ij} \tilde\gamma v_{elec,k} &
\delta^{ij} \tilde\gamma &
0 & 0 \\
0 & 0 & 0 & v_{elec}^j (\frac{1}{2} \tilde\gamma v_{elec}^2 - h_{hydro,elec}) &
-\tilde\gamma v_{elec}^j v_{elec,k} + \delta^j_k h_{hydro,elec} &
\gamma v_{elec}^j &
0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{1}{\mu_0} {\epsilon^{ij}}_k \\
0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\epsilon_0} {\epsilon^{ij}}_k & 0 \\
}\right] \left[\matrix{
\rho_{ion} \\ \rho_{ion} v_{ion}^k \\ E_{hydro,ion} \\
\rho_{elec} \\ \rho_{elec} v_{elec}^k \\ E_{hydro,elec} \\
\epsilon_0 E^k \\ B^k
}\right]_{,j} = \left[\matrix{
0 \\
r_{ion} \rho_{ion} (E^i + {\epsilon^i}_{jk} B^k v_{ion}^j) \\
r_{ion} \rho_{ion} (E_j v_{ion}^j) \\
0 \\
r_{elec} \rho_{elec} (E^i + {\epsilon^i}_{jk} B^k v_{elec}^j) \\
r_{elec} \rho_{elec} (E_j v_{elec}^j) \\
- \Sigma_\alpha r_\alpha \rho_\alpha v^i_\alpha
0 \\
}\right]$
with constraints:
${\epsilon_0 E^i}_{,i} = \Sigma_\alpha r_\alpha \rho_\alpha$
${B^i}_{,i} = 0$