Hydrodynamics
Mass
Total mass in volume:
$m = \int_\Omega \rho dV$ = mass, in $[kg]$
$\rho$ = density, in $[\frac{kg}{m^3}]$
$\Omega$ = spatial domain, within $\textbf{R}^3$
Rate-of-change of mass:
$\partial_t m = \int_\Omega \partial_t \rho dV$, in $[\frac{kg}{s}]$
Rate-of-change only considering boundary:
$\partial_t m = \int_{\partial\Omega} \rho v^j dS_j$, in $[\frac{kg}{s}]$
$v^i$ = velocity, in $[\frac{m}{s}]$
Apply Green's theorem to rate-of-change of boundary:
$\partial_t m = -\int_\Omega \nabla_j (\rho v^j) dV$ in $[\frac{kg}{s}]$
Consider rate-of-change at any point:
Conservation of mass:
$\partial_t \rho + \nabla_j (\rho v^j) = 0$ in $[\frac{kg}{m^3 \cdot s}]$
Momentum
Conservation of momentum:
$\int_\Omega \Pi_C^i dV = \int_\Omega \rho v^i dV$ in $[\frac{kg \cdot m}{s}]$
$\partial_t \int_\Omega \Pi_C^i dV$ = change in momentum due to conservation, in $[\frac{kg \cdot m}{s^2}]$
Rate-of-change of momentum
$\partial_t \int_\Omega \Pi_C^i dV = \int_\Omega \partial_t (\rho v^i) dV$, in $[\frac{kg \cdot m}{s^2}]$
Rate-of-change of only considering boundary (neglecting forces like gravity, pressure, viscosity, etc)
$\partial_t \int_\Omega \Pi_C^i dV = \int_{\partial\Omega} \rho v^i v^j dS_j$, in $[\frac{kg \cdot m}{s^2}]$
Apply Green's theorem:
$\partial_t \int_\Omega \Pi_C^i dV = -\int_\Omega \nabla_j (\rho v^i v^j) dV$, in $[\frac{kg \cdot m}{s^2}]$
Consider at any point:
(TODO this should either be a different symbol, (as $\rho$ relates to $m$), or annotation of like $\Pi$ vs $\tilde\Pi$).
$\partial_t \Pi_C^i = -\nabla_j (\rho v^i v^j)$, in $[\frac{kg}{m^2 \cdot s^2}]$
Hydro pressure from internal energy (ideal gas law):
$P = (\gamma - 1) \rho e_{int} = (\gamma - 1) E_{int}$, in $[\frac{kg}{m \cdot s^2}]$
$\gamma = $ heat capacity ratio, in $[1]$
$P$ = hydro pressure, in $[\frac{kg}{m \cdot s^2}]$
$e_{int}$ = specific internal energy per density, per volume, in $[\frac{m^2}{s^2}]$
$E_{int}$ = internal energy per volume, in $[\frac{kg}{m \cdot s^2}]$
internal energy, per volume:
$E_{int} = \rho e_{int} = \frac{P}{\gamma - 1}$, in $[\frac{kg}{m \cdot s^2}]$
Force due to pressure:
$\partial_t \int_\Omega \Pi_P^i dV = \int_{\partial\Omega} P dS_i$, in $[\frac{kg \cdot m}{s^2}]$
$\partial_t \int_\Omega \Pi_P^i dV = -\int_\Omega \nabla_i P dV$ via Green's theorem, in $[\frac{kg \cdot m}{s^2}]$
$\partial_t \Pi_P^i = -\nabla_i P$ at any point, in $[\frac{kg}{m^2 \cdot s^2}]$
Total change in momemtum:
$\partial_t \Pi^i = \partial_t \Pi_C^i + \partial_t \Pi_P^i$, in $[\frac{kg}{m^2 \cdot s^2}]$
$\partial_t \Pi^i = -\nabla_j (\rho v^i v^j) - \nabla_i P$, in $[\frac{kg}{m^2 \cdot s^2}]$
$\partial_t \Pi^i + \nabla_j (\rho v^i v^j + g^{ij} P) = 0$, in $[\frac{kg}{m^2 \cdot s^2}]$
... where $g^{ij}$ is the inverse metric tensor associated with our manifold tangent-space basis.
$\partial_t \Pi^i + \nabla_j (\rho v^i v^j + g^{ij} P) = 0$, in $[\frac{kg}{m^2 \cdot s^2}]$
For $\Pi^i = $ hydro momentum $ = \rho v^i$ we find
$\partial_t (\rho v^i) + \nabla_j (\rho v^i v^j + g^{ij} P) = 0$, in $[\frac{kg}{m^2 \cdot s^2}]$
Energy
specific kinetic energy, per density, per volume:
$e_{kin} = \frac{1}{2} v^2$, in $[\frac{m^2}{s^2}]$
... where $v^2 = v^i v^j g_{ij}$.
kinetic energy, per volume:
$E_{kin} = \rho e_{kin} = \frac{1}{2} \rho v^2$, in $[\frac{kg}{m \cdot s^2}]$
Total hydro energy:
$\int_\Omega E_{hydro} dV = \int_\Omega (E_{int} + E_{kin}) dV$, in $[\frac{kg \cdot m^2}{s^2}]$
$\int_\Omega E_{hydro} dV = \int_\Omega (\frac{1}{2}\rho v^2 + \frac{P}{\gamma - 1}) dV$, in $[\frac{kg \cdot m^2}{s^2}]$
change in total energy due to conservation of hydro energy:
$\partial_t \int_\Omega E^C_{total} dV = \int_\Omega \partial_t E_{hydro} dV$, in $[\frac{kg \cdot m^2}{s^3}]$
consider velocity moving through boundaries:
$\partial_t \int_\Omega E^C_{total} dV = \int_{\partial\Omega} E_{hydro} v^i dS_i$, in $[\frac{kg \cdot m^2}{s^3}]$
apply Green's theorem:
$\partial_t \int_\Omega E^C_{total} dV = -\int_\Omega \nabla_j (E_{hydro} v^j) dV$, in $[\frac{kg \cdot m^2}{s^3}]$
for any point:
$\partial_t E^C_{total} = -\nabla_j (E_{hydro} v^j)$, in $[\frac{kg}{m \cdot s^3}]$
Change in energy due to hydro pressure force:
$\partial_t \int_\Omega E^P_{total} dV = \int_{\partial\Omega} P v^j dS_j$, in $[\frac{kg \cdot m^2}{s^3}]$
$\partial_t \int_\Omega E^P_{total} dV = -\int_\Omega \nabla_j (P v^j) dV$ via Green's theorem, in $[\frac{kg \cdot m^2}{s^3}]$
$\partial_t E^P_{total} = -\nabla_j (P v^j)$ at any point, in $[\frac{kg}{m \cdot s^3}]$
Total change in total energy:
$\partial_t E_{total} = \partial_t E^C_{total} + \partial_t E^P_{total}$, in $[\frac{kg}{m \cdot s^3}]$
$\partial_t E_{total} = -\nabla_j (E_{hydro} v^j) - \nabla_j (P v^j)$, in $[\frac{kg}{m \cdot s^3}]$
$\partial_t E_{total} + \nabla_j ((E_{hydro} + P) v^j) = 0$, in $[\frac{kg}{m \cdot s^3}]$
$\partial_t E_{total} + \nabla_j (H_{hydro} v^j) = 0$, in $[\frac{kg}{m \cdot s^3}]$
for $H_{hydro} = E_{hydro} + P$, in $[\frac{kg}{m \cdot s^2}]$
Conservative form of Euler:
$\partial_t \rho + \nabla_j (\rho v^j) = 0$, in $[\frac{kg}{m^3 \cdot s}]$
$\partial_t (\rho v^i) + \nabla_j (\rho v^i v^j + g^{ij} P) = 0$, in $[\frac{kg}{m^2 \cdot s^2}]$
$\partial_t E_{hydro} + \nabla_j (H_{hydro} v^j) = 0$, in $[\frac{kg}{m \cdot s^3}]$
the hyperbolic conservation law equation:
$\partial_t U^I + \nabla_j F^{Ij} = 0$
Euler fluid PDE conserved variables:
$U^I = \downarrow I \left[\begin{matrix}
\rho \\ \rho v^i \\ E_{hydro}
\end{matrix}\right]$
Euler fluid PDE flux vector:
$F^{Ij} = \downarrow I \left[\begin{matrix}
\rho v^j \\
\rho v^i v^j + g^{ij} P \\
H_{hydro} v^j
\end{matrix}\right]$
Euler fluid PDE primitive variables:
$W^K = \downarrow K \left[\begin{matrix}
\rho \\ v^k \\ P
\end{matrix}\right]$
$\frac{\partial E_{hydro}}{\partial \rho}$
$= \frac{\partial}{\partial \rho} ( \frac{1}{2} \rho v^2 + \frac{P}{\gamma-1})$
$= \frac{1}{2} v^2$, in $[\frac{m^2}{s^2}]$
$\frac{\partial E_{hydro}}{\partial v^k}$
$= \frac{\partial}{\partial v^k} ( \frac{1}{2} \rho v^2 + \frac{P}{\gamma-1})$
$= \rho v_k$, in $[\frac{kg}{m^2 \cdot s}]$
$\frac{\partial E_{hydro}}{\partial P}$
$= \frac{\partial}{\partial P} ( \frac{1}{2} \rho v^2 + \frac{P}{\gamma-1})$
$= \frac{1}{\gamma-1}$, in $[1]$
Derivative of conservatives wrt primitives:
$\frac{\partial U^I}{\partial W^J} = \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix}
1 & 0 & 0 \\
v^i & \rho \delta^i_j & 0 \\
\frac{1}{2} v^2 & \rho v_j & \frac{1}{\gamma-1}
\end{matrix}\right] }$
Derivative of primitives wrt conservatives:
$\frac{\partial W^J}{\partial U^I} = \downarrow J \overset{\rightarrow I}{ \left[\begin{matrix}
1 & 0 & 0 \\
-\frac{1}{\rho} v^j & \frac{1}{\rho} \delta_i^j & 0 \\
\frac{1}{2} (\gamma-1) v^2 & -(\gamma-1) v_i & \gamma-1
\end{matrix}\right] }$
Derivative of flux with respect to primitive variables:
$\frac{\partial F^{Ij}}{\partial W^K} = \downarrow I \overset{\rightarrow K}{ \left[\begin{matrix}
v^j & \rho \delta^j_k & 0 \\
v^i v^j & \rho (\delta^i_k v^j + v^i \delta^j_k) & g^{ij} \\
\frac{1}{2} v^2 v^j & \rho v^j v_k + H_{hydro} \delta^j_k & \frac{\gamma}{\gamma-1} v^j
\end{matrix}\right] }$
Derivative of flux with respect to conservative variables:
$\frac{\partial F^{Ij}}{\partial U^K} = \frac{\partial F^{Ij}}{\partial W^L} \frac{\partial W^L}{\partial U^K}
= \downarrow I \overset{\rightarrow L}{ \left[\begin{matrix}
v^j & \rho \delta^j_l & 0 \\
v^i v^j & \rho (\delta^i_l v^j + v^i \delta^j_l) & g^{ij} \\
\frac{1}{2} v^2 v^j & \rho v^j v_l + (E_{hydro} + P) \delta^j_l & \frac{\gamma}{\gamma-1} v^j
\end{matrix}\right] } \cdot \downarrow L \overset{\rightarrow K}{ \left[\begin{matrix}
1 & 0 & 0 \\
-\frac{1}{\rho} v^l & \frac{1}{\rho} \delta^l_k & 0 \\
\frac{1}{2} (\gamma-1) v^2 & -(\gamma-1) v_k & \gamma-1
\end{matrix}\right] }$
$= \downarrow I \overset{\rightarrow K}{ \left[\begin{matrix}
0 & \delta^j_k & 0 \\
-v^i v^j + \frac{1}{2} g^{ij} (\gamma-1) v^2 &
\delta^i_k v^j + \delta^j_k v^i - g^{ij} (\gamma-1) v_k &
g^{ij} (\gamma - 1) \\
\frac{1}{2} v^j v^2 (\gamma - 1) - v^j h_{hydro} &
-(\gamma-1) v^j v_k + \delta^j_k h_{hydro} &
\gamma v^j
\end{matrix}\right] }$
for $h_{hydro} = \frac{1}{\rho} H_{hydro}$, in $[\frac{m^2}{s^2}]$
Magnetohydrodynamics
$\mu_0 = $ permeability of free space, in $[\frac{kg \cdot m}{C^2}]$
$\epsilon_0 = $ permittivity of free space, in $[\frac{C^2 \cdot s^2}{kg \cdot m^3}]$
$E = $ electrical field, in $[\frac{kg \cdot m}{C \cdot s^2}]$
$B = $ magnetic field, in $[\frac{kg}{C \cdot s}]$
$J = $ electrical current, per volume, in $[\frac{C}{m^2 \cdot s}]$
Ampere's law:
$\vec{\nabla} \times \vec{B} = \mu_0 (\vec{J} + \epsilon_0 \partial_t \vec{E})$, in $[\frac{kg}{C \cdot m \cdot s}]$
${\epsilon^{ij}}_k \nabla_j B^k = \mu_0 (J^i + \epsilon_0 \partial_t E^i)$, in $[\frac{kg}{C \cdot m \cdot s}]$
Assume that our electric field is a steady-state, i.e. $\partial_t E^i = 0$
$J^i = \frac{1}{\mu_0} {\epsilon^{ij}}_k \nabla_j B^k$, in $[\frac{C}{m^2 \cdot s}]$
$q = $ electric charge, in $[C]$
$\sigma$ = electrical conductivity, in $[\frac{C^2 \cdot s^2}{kg \cdot m^3 \cdot s}]$
Lorentz force law:
$\vec{F} = q (\vec{E} + \vec{v} \times \vec{B})$, in $[\frac{kg \cdot m}{s^2}]$
$F^i = q (E^i + {\epsilon^i}_{jk} v^j B^k)$, in $[\frac{kg \cdot m}{s^2}]$
$E^i = \frac{1}{q} F^i - {\epsilon^i}_{jk} v^j B^k$, in $[\frac{kg \cdot m}{C \cdot s^2}]$
Let $J^i / \sigma = F^i / q$, in $[\frac{kg \cdot m}{C \cdot s^2}]$
$E^i = J^i / \sigma - {\epsilon^i}_{jk} v^j B^k$, in $[\frac{kg \cdot m}{C \cdot s^2}]$
Maxwell-Faraday equation:
$\partial_t \vec{B} = -\vec{\nabla} \times \vec{E}$
$\partial_t B^i = -{\epsilon^{ij}}_k \nabla_j E^k$
(TODO flip valence of everything below here, to make it match my other EMHD flux eigensystem worksheets ... and double check or just use those).
Magnetic Field
substitute Ampere's law into Lorentz force law:
$E^i = \frac{1}{\sigma\mu_0} {\epsilon^{ij}}_k \nabla_j B^k - {\epsilon^i}_{jk} v^j B^k$
$\eta = \frac{1}{\sigma\mu_0} = $ magnetic diffusivity
$E^i = \eta {\epsilon^{ij}}_k \nabla_j B^k - {\epsilon^i}_{jk} v^j B^k$
substitute this into Maxwell-Faraday equation:
$ \partial_t B_i = -{\epsilon_i}^{jk} \nabla_j (\eta {\epsilon_k}^{lm} \partial_l B_m - {\epsilon_k}^{lm} v_l B_m)$
$ \partial_t B_i = {\epsilon_i}^{jk} {\epsilon_k}^{lm} \nabla_j (v_l B_m) - {\epsilon_i}^{jk} {\epsilon_k}^{lm} \nabla_j (\eta \partial_l B_m)$
Levi-Civita identity: ${\epsilon_i}^{jk} {\epsilon_k}^{lm} = \delta^{ij}_{kl} = \delta^i_k \delta^j_l - \delta^i_l \delta^j_k$
$ \partial_t B_i = (\delta^i_l \delta^j_m - \delta^i_m \delta^j_l) \nabla_j (v_l B_m) - (\delta^i_l \delta^j_m - \delta^i_m \delta^j_l) \nabla_j (\eta \partial_l B_m)$
assume $\eta$ is constant
$ \partial_t B_i = \nabla_j (v_i B_j - v_j B_i) - \eta (\nabla_i \nabla_j B_j - \nabla_j \nabla_j B_i)$
No magnetic monopoles: $\nabla_i B_i = 0$
$ \partial_t B_i = \nabla_j (v_i B_j - v_j B_i) + \eta \nabla_j \nabla_j B_i$
assume $\eta = 0$
$ \partial_t B_i = \nabla_j (v_i B_j - v_j B_i)$
$ \partial_t B_i + \nabla_j (B_i v_j - B_j v_i) = 0$
Momentum
Lorentz force law:
$F_i = q (E_i + {\epsilon_i}^{jk} v_j B_k)$
assume $E_i = 0$
$\Pi^M_i = F_i =$ Magnetic force on fluid
$\Pi^M_i = {\epsilon_i}^{jk} q v_j B_k$
let $J_i = q v_i$
$\Pi^M_i = {\epsilon_i}^{jk} J_j B_k$
substitute Ampere's law approximation: $J_i = \frac{1}{\mu_0} {\epsilon_i}^{jk} \nabla_j B_k$
$\Pi^M_i = {\epsilon_i}^{jk} (\frac{1}{\mu_0} {\epsilon_j}^{lm} \partial_l B_m) B_k$
$\Pi^M_i = -\frac{1}{\mu_0} {\epsilon_i}^{kj} {\epsilon^{lm}}_j B_k \partial_l B_m$
$\Pi^M_i = \frac{1}{\mu_0} (\delta^i_m \delta^k_l - \delta^i_l \delta^k_m) B_k \partial_l B_m$
$\Pi^M_i = \frac{1}{\mu_0} (B_j \nabla_j B_i - B_j \nabla_i B_j)$
insert additional $\nabla_k B_k = 0$ terms, split $B_j \nabla_i B_j$ into two halves
$\Pi^M_i = \frac{1}{\mu_0} ((\nabla_j B_i) B_j + B_i (\nabla_j B_j) - B_i - \frac{1}{2} (\nabla_i B_j) B_j - \frac{1}{2} B_j (\nabla_i B_j))$
un-distribute gradient:
$\Pi^M_i = \frac{1}{\mu_0} (\nabla_j (B_i B_j) - \frac{1}{2} \nabla_i (B^2))$
$\Pi^M_i = \nabla_j (\frac{1}{\mu_0} (B_i B_j - \frac{1}{2} g^{ij} B_i B_j))$
$\Pi^M_i = \nabla_j \sigma_{ij}$
$\sigma_{ij} = \frac{1}{\mu_0} (B_i B_j - \frac{1}{2} g^{ij} B_i B_j) =$ Maxwell stress tensor for $E_i = 0$
Change in momentum due to magnetic field:
$\Pi^M_i = -\frac{1}{2} \frac{1}{\mu_0} \nabla_j B^2 + \frac{1}{\mu_0} \nabla_j (B_i B_j)$
Total change in momemtum:
$\partial_t \Pi_i = \partial_t \Pi^C_i + \partial_t \Pi^P_i + \partial_t \Pi^M_i$
$\partial_t \Pi_i = -\nabla_j (\rho v_i v_j) - \nabla_i P - \frac{1}{\mu_0} \frac{1}{2} \nabla_i B^2 + \frac{1}{\mu_0} \nabla_j (B_i B_j)$
$\partial_t \Pi_i + \nabla_j (\rho v_i v_j + g^{ij} P + \frac{1}{\mu_0} \frac{1}{2} g^{ij} B^2 - \frac{1}{\mu_0} B_i B_j) = 0$
Let $P_{mag} = \frac{1}{\mu_0} \frac{1}{2} B^2$
$\partial_t \Pi_i + \nabla_j (\rho v_i v_j + g^{ij} (P + P_{mag}) - \frac{1}{\mu_0} B_i B_j) = 0$
Let $P_{total} = P + P_{mag}$
$\partial_t \Pi_i + \nabla_j (\rho v_i v_j + g^{ij} P_{total} - \frac{1}{\mu_0} B_i B_j) = 0$
For $\Pi_i = $ total momentum $ = \rho v_i$ we find
$\partial_t (\rho v_i) + \nabla_j (\rho v_i v_j + g^{ij} P_{total} - \frac{1}{\mu_0} B_i B_j) = 0$
Energy
Change in energy due to magnetic field:
$\partial_t E_{total}^M = -\int_{\partial\Omega} B_k (v_k B_j - v_j B_k) dS_j$ TODO where does this come from? I'm guessing $\partial_t B_i$
$\partial_t E_{total}^M = \int_\Omega \nabla_j (B_k (v_k B_j - v_j B_k)) dV$ via Green's theorem
$\partial_t E_{total}^M = \nabla_j (B_k (v_k B_j - v_j B_k))$ at any point
Total change in total energy:
$\partial_t E_{total} = \partial_t E_{total}^C + \partial_t E_{total}^P + \partial_t E_{total}^M$
$\partial_t E_{total} = -\nabla_j (E_{hydro} v_j) - \nabla_j (P v_j) - \nabla_j \frac{1}{\mu_0} (v_j B^2 - B_k v_k B_j)$
$\partial_t E_{total} + \nabla_j ((E_{hydro} + P) v_j + \frac{1}{\mu_0} (v_j B^2 - B_k v_k B_j) = 0$
$\partial_t E_{total} + \nabla_j ((E_{hydro} + P) v_j + \frac{1}{\mu_0} B_k (v_j B_k - v_k B_j)) = 0$
$\partial_t E_{total} + \nabla_j ((E_{hydro} + P) v_j + \frac{1}{\mu_0} B^2 v_j - \frac{1}{\mu_0} B_k v_k B_j) = 0$
$\partial_t E_{total} + \nabla_j ((E_{hydro} + \frac{1}{2} \frac{1}{\mu_0} B^2 + P + \frac{1}{2} \frac{1}{\mu_0} B^2) v_j - \frac{1}{\mu_0} B_k v_k B_j) = 0$
Let $E_{mag} = \frac{1}{2} \frac{1}{\mu_0} B^2$
$\partial_t E_{total} + \nabla_j ((E_{hydro} + E_{mag} + P + P_{mag}) v_j - \frac{1}{\mu_0} B_k v_k B_j) = 0$
Let $E_{total} = E_{hydro} + E_{mag}$
Let $H_{total} = E_{total} + P_{total}$
$\partial_t E_{total} + \nabla_j (H_{total} v_j - \frac{1}{\mu_0} B_k v_k B_j) = 0$
Conservative Form of MHD:
$\partial_t \rho + \nabla_j (\rho v_j) = 0$
$\partial_t (\rho v_i) + \nabla_j (\rho v_i v_j + g^{ij} P_{total} - \frac{1}{\mu_0} B_i B_j) = 0$
$\partial_t B_i + \nabla_j (B_i v_j - B_j v_i) = 0$
$\partial_t E_{total} + \nabla_j (H_{total} v_j - \frac{1}{\mu_0} B_k v_k B_j) = 0$
$\partial_t U_i + \partial_{x_j} F_{ij} = 0$
Primitive variables:
$W_i \left[\begin{matrix}
\rho \\ v_i \\ B_i \\ P
\end{matrix}\right]$
Conservative variables:
$U_i = \left[\begin{matrix}
\rho \\ \rho v_i \\ B_i \\ E_{total}
\end{matrix}\right]$
$\frac{\partial E_{total}}{\partial \rho}$
$= \frac{\partial E_{hydro}}{\partial \rho} + \frac{\partial E_{mag}}{\partial \rho}$
$= \frac{1}{2} v^2$
$\frac{\partial E_{total}}{\partial v_k}$
$= \frac{\partial E_{hydro}}{\partial v_k} + \frac{\partial E_{mag}}{\partial v_k}$
$= \rho v_k$
$\frac{\partial E_{total}}{\partial B_k}$
$= \frac{\partial E_{hydro}}{\partial B_k} + \frac{\partial E_{mag}}{\partial B_k}$
$= \frac{1}{\mu_0} B_k$
$\frac{\partial E_{total}}{\partial P}$
$= \frac{\partial E_{hydro}}{\partial P} + \frac{\partial E_{mag}}{\partial P}$
$= \frac{1}{\gamma-1} $
Derivative of conservative with respect to primitive:
$\frac{\partial U_i}{\partial W_j} = \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix}
1 & 0 & 0 & 0 \\
v_i & \rho g^{ij} & 0 & 0 \\
0 & 0 & g^{ij} & 0 \\
\frac{1}{2}v^2 & \rho v_j & \frac{1}{\mu_0} B_j & \frac{1}{\gamma-1}
\end{matrix}\right] }$
Derivative of primitive with respect to conservative:
$\frac{\partial W_j}{\partial U_i} = \downarrow J \overset{\rightarrow L}{ \left[\begin{matrix}
1 & 0 & 0 & 0 \\
-\frac{1}{\rho} v_j & \frac{1}{\rho} g^{ij} & 0 & 0 \\
0 & 0 & g^{ij} & 0 \\
\frac{1}{2} (\gamma-1) v^2 & -(\gamma-1) v_i & -(\gamma-1) \frac{1}{\mu_0} B_i & \gamma-1
\end{matrix}\right] }$
Flux vector:
$F_{ij} = \left[\begin{matrix}
\rho v_j \\
\rho v_i v_j + g^{ij} P_{total} - \frac{1}{\mu_0} B_i B_j \\
B_i v_j - B_j v_i \\
H_{total} v_j - \frac{1}{\mu_0} B_k v_k B_j
\end{matrix}\right]$
Derivative of flux vector with respect to primitive variables:
$\frac{\partial F_{ij}}{\partial W_k} = \downarrow I \overset{\rightarrow K}{ \left[\begin{matrix}
v_j &
\rho g^{jk} &
0 &
0 \\
v_i v_j &
\rho (g^{ik} v_j + v_i g^{jk}) &
\frac{1}{\mu_0} (g^{ij} B_k - g^{ik} B_j - g^{jk} B_i) &
g^{ij} \\
0 &
B_i g^{jk} - B_j g^{ik} &
g^{ik} v_j - g^{jk} v_i &
0 \\
\frac{1}{2} v^2 v_j &
\rho v_j v_k + H_{total} g^{jk} - \frac{1}{\mu_0} B_j B_k &
\frac{1}{\mu_0} (2 B_k v_j - v_k B_j - g^{jk} v_l B_l) &
\frac{\gamma}{\gamma-1} v_j
\end{matrix}\right] }$
Derivative of flux vector with respect to conservative variables:
$\frac{\partial F_{ij}}{\partial U_k}
= \frac{\partial F_{ij}}{\partial W_l} \frac{\partial W_l}{\partial U_k}$
$= \downarrow I \overset{\rightarrow L}{ \left[\begin{matrix}
v_j &
\rho g^{jl} &
0 &
0 \\
v_i v_j &
\rho (g^{il} v_j + v_i g^{jl}) &
\frac{1}{\mu_0} (g^{ij} B_l - g^{il} B_j - g^{jl} B_i) &
g^{ij} \\
0 &
B_i g^{jl} - B_j g^{il} &
g^{il} v_j - g^{jl} v_i &
0 \\
\frac{1}{2} v^2 v_j &
\rho v_j v_l + (E_{hydro} + \frac{1}{\mu_0} B^2 + P) g^{jl} - \frac{1}{\mu_0} B_j B_l &
\frac{1}{\mu_0} (2 B_l v_j - v_l B_j - g^{jl} v_m B_m) &
\frac{\gamma}{\gamma-1} v_j
\end{matrix}\right] } \cdot \downarrow L \overset{\rightarrow K}{ \left[\begin{matrix}
1 & 0 & 0 & 0 \\
-\frac{1}{\rho} v_l & \frac{1}{\rho} g^{kl} & 0 & 0 \\
0 & 0 & g^{kl} & 0 \\
\frac{1}{2} (\gamma-1) v^2 & -(\gamma-1) v_k & -(\gamma-1) \frac{1}{\mu_0} B_k & \gamma-1
\end{matrix}\right] }$
$= \downarrow I \overset{\rightarrow K}{ \left[\begin{matrix}
0 &
g^{jk} &
0 &
0 \\
-v_i v_j + \frac{1}{2} g^{ij} (\gamma-1) v^2 &
g^{ik} v_j + g^{jk} v_i - g^{ij} (\gamma-1) v_k &
\frac{1}{\mu_0}(-g^{ij} (\gamma-2) B_k - g^{ik} B_j - g^{jk} B_i) &
g^{ij} (\gamma-1) \\
-\frac{1}{\rho} (B_i v_j - B_j v_i) &
\frac{1}{\rho} (B_i g^{jk} - B_j g^{ik}) &
g^{ik} v_j - g^{jk} v_i &
0 \\
v_j (\frac{1}{2} (\gamma-1) v^2 - h_{total}) + \frac{1}{\mu_0\rho} B_j B_m v_m &
-(\gamma-1) v_j v_k + (g^{jk} h_{total} - \frac{1}{\mu_0\rho} B_j B_k) &
\frac{1}{\mu_0} (-(\gamma - 2) v_j B_k - v_k B_j - g^{jk} v_m B_m) &
\gamma v_j
\end{matrix}\right] }$
for $h_{total} = (E_{total} + P_{total}) / \rho$
Similarity Transforms:
$\partial_t U_i(x,t) + \partial_{x_j} F_{ij}(U(x,t)) = 0$
substitute $\tilde{F}_{ij}$ and $\tilde{U}_i$ into conservative equations
$\partial_t \tilde{U}_i + \partial_{x_j} (\tilde{F}_{ij} + \xi_j \tilde{U}_i) = 0$
time derivative of state (in self-similar form)
$\partial_t \tilde{U}_i = {\partial\tilde{U}_i\over\partial\xi_j} {\partial\xi_j\over\partial t} = -{x_j\over t^2} \partial_{\xi_j} \tilde{U}_i$
substitute $\xi_j = {x_j\over t}$
$\partial_t \tilde{U}_i = -{1\over t} \xi_j \partial_{\xi_j} \tilde{U}_i$
spatial derivative of flux (in self-similar form)
$\partial_{x_j} F_{ij} = \partial_{x_j} (\tilde{F}_{ij} + \xi_j \tilde{U}_i)$
$= {{\partial \xi_j}\over{\partial x_j}} \partial_{\xi_j} (\tilde{F}_{ij} + \xi_j \tilde{U}_i)$
$= {1\over t} (\partial_{\xi_j} \tilde{F}_{ij} + (\partial_{\xi_j} \xi_j) \tilde{U}_i + \xi_j \partial_{\xi_j} \tilde{U}_i)$
$= {1\over t} (\partial_{\xi_j} \tilde{F}_{ij} + n \tilde{U}_i) + {1\over t} \xi_j \partial_{\xi_j} \tilde{U}_i$ for $n$ dimensions
substitute to find:
$ -{1\over t} \xi_j \partial_{\xi_j} \tilde{U}_i
+ {1\over t} (\partial_{\xi_j} \tilde{F}_{ij} + n \tilde{U}_i) + {1\over t} \xi_j \partial_{\xi_j} \tilde{U}_i = 0$
simplifies to
$n \tilde{U}_i + \partial_{\xi_j} \tilde{F}_{ij} = 0$
finite volume form:
$\int_\Omega (n \tilde{U}_i + \partial_{\xi_j} \tilde{F}_{ij}) dx = 0$
$\int_\Omega dx n \tilde{U}_i + \int_\Omega \partial_{\xi_j} \tilde{F}_{ij} dx = 0$
$|\Omega| n \tilde{U}_i + \int_{\partial\Omega} \tilde{F}_{ij} n_j d\sigma = 0$
discretize:
$\Delta \xi n \tilde{U}_i + \tilde{F}_{ij}(x_R) - \tilde{F}_{ij}(x_L) = 0$
$n \tilde{U}_i + {1\over\Delta \xi}(\tilde{F}_{ij}(x_R) - \tilde{F}_{ij}(x_L)) = 0$
...for $\Delta \xi = \Pi_{k=1}^n \Delta \xi_k$
Similarity Transform of MHD Equations:
$U_i = \tilde{U}_i = \left[\begin{matrix}
\rho \\ \rho v_i \\ B_i \\ E_{total}
\end{matrix}\right]$
$F_{ij} = \left[\begin{matrix}
\rho v_j \\
\rho v_i v_j + g^{ij} P_{total} - \frac{1}{\mu_0} B_i B_j \\
B_i v_j - B_j v_i \\
(E_{total} + P_{total}) v_j - \frac{1}{\mu_0} B_k v_k B_j
\end{matrix}\right]$
$\tilde{F}_{ij} = F_{ij} - \xi_j \tilde{U}_i = \left[\begin{matrix}
\rho (v_j - \xi_j) \\
\rho v_i (v_j - \xi_j) + g^{ij} P_{total} - \frac{1}{\mu_0} B_i B_j \\
B_i (v_j - \xi_j) - B_j v_i \\
E_{total} (v_j - \xi_j) + P_{total} v_j - \frac{1}{\mu_0} B_k v_k B_j
\end{matrix}\right]$
Newton Method
Nonlinear function:
$G_i(\tilde{U}) = n \tilde{U}_i + \partial_{\xi_j} \tilde{F}_{ij}$
solve for $G_i = 0$ by minimizing $\tilde{U}_j$
Taylor expansion of function to minimize:
$G_i(\tilde{U}^{k+1}) = G_i(\tilde{U}^k) + {{\partial G_i(\tilde{U}^k)}\over{\partial \tilde{U}^k_j}} (\tilde{U}_j^{k+1} - \tilde{U}_j^k) + \mathscr(O)((\tilde{U})^2)$
$\tilde{U}_i^k$ the $k$th iteration of our state vector $\tilde{U}_i$
$G_i(\tilde{U}^{k+1}) = G_i(\tilde{U}^k) + {{\partial G_i(\tilde{U}^k)}\over{\partial \tilde{U}^k_j}} (\tilde{U}_j^{k+1} - \tilde{U}_j^k) + \mathscr(O)((\tilde{U})^2)$
Rearranged, and ignoring the higher-order terms:
$||{{\partial G_i(\tilde{U}^k)}\over{\partial \tilde{U}^k_j}}||^{-1}_{ij} (G_j(\tilde{U}^{k+1}) - G_j(\tilde{U}^k)) = \tilde{U}_i^{k+1} - \tilde{U}_i^k$
$\tilde{U}_i^{k+1} = \tilde{U}_i^k + ||{{\partial G_i(\tilde{U}^k)}\over{\partial \tilde{U}^k_j}}||^{-1}_{ij} (G_j(\tilde{U}^{k+1}) - G_j(\tilde{U}^k))$
Assume that we arrive at our destination, i.e. $G_i(\tilde{U}^{k+1}) = 0$:
$\tilde{U}_i^{k+1} = \tilde{U}_i^k - ||{{\partial G_i(\tilde{U}^k)}\over{\partial \tilde{U}^k_j}}||^{-1}_{ij} G_j(\tilde{U}^k)$
This formula is equivalent to Newton's method of an n-dimensions scalar function with value equal to $\frac{1}{2}||G||^2$
Jacobian Approximation:
${{\partial G_i(u)}\over{\partial \tilde{U}_j}} v_j \approx ( G_i (u + \epsilon v) - G_i(u) ) / \epsilon$
Evaluation of function to minimize
${{\partial G_i}\over{\partial \tilde{U}_j}} = \partial_{\tilde{U}_j} G_i$
$= \partial_{\tilde{U}_j} (n \tilde{U}_i + \partial_{\xi_k} \tilde{F}_{ik})$
$= n g^{ij} + \partial_{\tilde{U}_j} \partial_{\xi_k} \tilde{F}_{ik}$
Curved Space Partial of Conserved Quantities wrt Primitives
$U^I = \left[\begin{matrix}
\rho \\
m^i \\
B^i \\
E_{total}
\end{matrix}\right] = \left[\begin{matrix}
\rho \\
\rho v^i \\
B^i \\
\tilde\gamma^{-1} P + \frac{1}{2} \rho v^2 + \frac{1}{2} \frac{1}{\mu_0} B^2
\end{matrix}\right]$
$E_{total} = E_{int} + E_{kin} + E_{mag}$
$= \tilde\gamma^{-1} P + \frac{1}{2} \rho v^2 + \frac{1}{2} \frac{1}{\mu_0} B^2$
$P = \tilde\gamma (E_{total} - \frac{1}{2} \rho v^2 - \frac{1}{2} \frac{1}{\mu_0} B^2)$
$P_{total} = P + P_{mag} = \tilde\gamma (E_{total} - \frac{1}{2} \rho v^2 - \frac{1}{2} \frac{1}{\mu_0} B^2) + \frac{1}{2} \frac{1}{\mu_0} B^2$
$= \tilde\gamma E_{total}
- \frac{1}{2} \tilde\gamma \rho v^2
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2
$
$H_{total} = E_{total} + P_{total}$
$= \gamma E_{total}
- \frac{1}{2} \tilde\gamma \rho v^2
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2
$
$F^{Ij}$
$= \left[\begin{matrix}
\rho v^j \\
\rho v^i v^j + g^{ij} P_{total} - \frac{1}{\mu_0} B^i B^j \\
B^i v^j - B^j v^i \\
H_{total} v^j - \frac{1}{\mu_0} B^k v_k B^j
\end{matrix}\right]$
$= \left[\begin{matrix}
m^j \\
\frac{1}{\rho} m^i m^j + g^{ij} (\tilde\gamma (E_{total} - \frac{1}{2} \frac{1}{\rho} m^2 - \frac{1}{2} \frac{1}{\mu_0} B^2) + \frac{1}{2} \frac{1}{\mu_0} B^2) - \frac{1}{\mu_0} B^i B^j \\
\frac{1}{\rho} (B^i m^j - B^j m^i) \\
\frac{1}{\rho} (E_{total} + \tilde\gamma (E_{total} - \frac{1}{2} \frac{1}{\rho} m^2 - \frac{1}{2} \frac{1}{\mu_0} B^2) + \frac{1}{2} \frac{1}{\mu_0} B^2) m^j - \frac{1}{\mu_0} \frac{1}{\rho} B^k m_k B^j
\end{matrix}\right]$
$= \left[\begin{matrix}
m^j
\\
+ \frac{1}{\rho} m^i m^j
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{ij}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{ij}
- \frac{1}{\mu_0} B^i B^j
+ \tilde\gamma g^{ij} E_{total}
\\
+ \frac{1}{\rho} B^i m^j
- \frac{1}{\rho} B^j m^i
\\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^2
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^2
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l g_{kl} B^j
+ \frac{1}{\rho} \gamma m^j E_{total}
\end{matrix}\right]$
Flux partial derivative:
${F^{Ij}}_{,j}$
$= \left[\begin{matrix}
m^j
\\
+ \frac{1}{\rho} m^i m^j
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{ij}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{ij}
- \frac{1}{\mu_0} B^i B^j
+ \tilde\gamma g^{ij} E_{total}
\\
+ \frac{1}{\rho} B^i m^j
- \frac{1}{\rho} B^j m^i
\\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^2
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^2
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l g_{kl} B^j
+ \frac{1}{\rho} \gamma m^j E_{total}
\end{matrix}\right]_{,j}$
$= \left[\begin{matrix}
{m^j}_{,j}
\\
- \frac{1}{\rho^2} \rho_{,j} m^i m^j
+ \frac{1}{\rho} {m^i}_{,j} m^j
+ \frac{1}{\rho} m^i {m^j}_{,j}
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} \rho_{,j} m^2 g^{ij}
- \tilde\gamma \frac{1}{\rho} {m^k}_{,j} m_k g^{ij}
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^k m^l g_{kl,j} g^{ij}
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 {g^{ij}}_{,j}
+ (2 - \gamma) \frac{1}{\mu_0} {B^k}_{,j} B_k g^{ij}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^k B^l g_{kl,j} g^{ij}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 {g^{ij}}_{,j}
- \frac{1}{\mu_0} {B^i}_{,j} B^j
- \frac{1}{\mu_0} B^i {B^j}_{,j}
+ \tilde\gamma {g^{ij}}_{,j} E_{total}
+ \tilde\gamma g^{ij} E_{total,j}
\\
- \frac{1}{\rho^2} \rho_{,j} B^i m^j
+ \frac{1}{\rho} {B^i}_{,j} m^j
+ \frac{1}{\rho} B^i {m^j}_{,j}
+ \frac{1}{\rho^2} \rho_{,j} B^j m^i
- \frac{1}{\rho} {B^j}_{,j} m^i
- \frac{1}{\rho} B^j {m^i}_{,j}
\\
+ \tilde\gamma \frac{1}{\rho^3} \rho_{,j} m^j m^2
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} {m^j}_{,j} m^2
- \tilde\gamma \frac{1}{\rho^2} m^j {m^k}_{,j} m_k
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l g_{kl,j}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho^2} \rho_{,j} m^j B^2
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} {m^j}_{,j} B^2
+ (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j {B^k}_{,j} B_k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l g_{kl,j}
+ \frac{1}{\mu_0} \frac{1}{\rho^2} \rho_{,j} B^k m^l g_{kl} B^j
- \frac{1}{\mu_0} \frac{1}{\rho} {B^k}_{,j} m^l g_{kl} B^j
- \frac{1}{\mu_0} \frac{1}{\rho} B^k {m^l}_{,j} g_{kl} B^j
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l g_{kl,j} B^j
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l g_{kl} {B^j}_{,j}
- \gamma \frac{1}{\rho^2} \rho_{,j} m^j E_{total}
+ \gamma \frac{1}{\rho} {m^j}_{,j} E_{total}
+ \gamma \frac{1}{\rho} m^j E_{total,j}
\end{matrix}\right]$
$= \left[\begin{matrix}
0 &
\delta^j_k &
0 &
0 \\
\frac{1}{\rho^2} (
- m^i m^j
+ \frac{1}{2} \tilde\gamma m^2 g^{ij}
) &
\frac{1}{\rho} (
m^j \delta^i_k
+ m^i \delta^j_k
- \tilde\gamma m_k g^{ij}
) &
\frac{1}{\mu_0} (
(2 - \gamma) B_k g^{ij}
- B^j \delta^i_k
- B^i \delta^j_k
) &
\tilde\gamma g^{ij}
\\
\frac{1}{\rho^2} (
- B^i m^j
+ B^j m^i
) &
\frac{1}{\rho} (
B^i \delta^j_k
- B^j \delta^i_k
) &
\frac{1}{\rho} (
m^j \delta^i_k
- m^i \delta^j_k
) &
0
\\
\frac{1}{\rho^2} (
\tilde\gamma \frac{1}{\rho} m^j m^2
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} m^j B^2
+ \frac{1}{\mu_0} B^k m_k B^j
- \gamma m^j E_{total}
) &
\frac{1}{\rho} (
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 \delta^j_k
- \tilde\gamma \frac{1}{\rho} m^j m_k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 \delta^j_k
- \frac{1}{\mu_0} B_k B^j
+ \gamma E_{total} \delta^j_k
) &
\frac{1}{\mu_0} \frac{1}{\rho} (
(2 - \gamma) m^j B_k
- m_k B^j
- B^l m_l \delta^j_k
) &
\gamma \frac{1}{\rho} m^j
\end{matrix}\right] \left[\begin{matrix}
\rho \\
m^k \\
B^k \\
E_{total}
\end{matrix}\right]_{,j} + g_{kl,j}\left[\begin{matrix}
0 \\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^k m^l g^{ij}
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{ik} g^{lj}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^k B^l g^{ij}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{ik} g^{lj}
- \tilde\gamma E_{total} g^{ik} g^{lj}
\\
0 \\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l B^j
\end{matrix}\right]$
Flux covariant derivative:
${F^{Ij}}_{;j}$
$= \left[\begin{matrix}
0 &
\delta^j_k &
0 &
0 \\
\frac{1}{\rho^2} (
- m^i m^j
+ \frac{1}{2} \tilde\gamma m^2 g^{ij}
) &
\frac{1}{\rho} (
m^j \delta^i_k
+ m^i \delta^j_k
- \tilde\gamma m_k g^{ij}
) &
\frac{1}{\mu_0} (
(2 - \gamma) B_k g^{ij}
- B^j \delta^i_k
- B^i \delta^j_k
) &
\tilde\gamma g^{ij}
\\
\frac{1}{\rho^2} (
- B^i m^j
+ B^j m^i
) &
\frac{1}{\rho} (
B^i \delta^j_k
- B^j \delta^i_k
) &
\frac{1}{\rho} (
m^j \delta^i_k
- m^i \delta^j_k
) &
0
\\
\frac{1}{\rho^2} (
\tilde\gamma \frac{1}{\rho} m^j m^2
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} m^j B^2
+ \frac{1}{\mu_0} B^k m_k B^j
- \gamma m^j E_{total}
) &
\frac{1}{\rho} (
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 \delta^j_k
- \tilde\gamma \frac{1}{\rho} m^j m_k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 \delta^j_k
- \frac{1}{\mu_0} B_k B^j
+ \gamma E_{total} \delta^j_k
) &
\frac{1}{\mu_0} \frac{1}{\rho} (
(2 - \gamma) m^j B_k
- m_k B^j
- B^m m_m \delta^j_k
) &
\gamma \frac{1}{\rho} m^j
\end{matrix}\right] \left[\begin{matrix}
\rho_{,j} \\
{m^k}_{,j} + {\Gamma^k}_{lj} m^l \\
{B^k}_{,j} + {\Gamma^k}_{lj} B^l \\
E_{total,j}
\end{matrix}\right]$
difference:
${F^{Ij}}_{;j}$
$= {F^{Ij}}_{,j}
+ \left[\begin{matrix}
0 &
\delta^j_k &
0 &
0 \\
\frac{1}{\rho^2} (
- m^i m^j
+ \frac{1}{2} \tilde\gamma m^2 g^{ij}
) &
\frac{1}{\rho} (
m^j \delta^i_k
+ m^i \delta^j_k
- \tilde\gamma m_k g^{ij}
) &
\frac{1}{\mu_0} (
(2 - \gamma) B_k g^{ij}
- B^j \delta^i_k
- B^i \delta^j_k
) &
\tilde\gamma g^{ij}
\\
\frac{1}{\rho^2} (
- B^i m^j
+ B^j m^i
) &
\frac{1}{\rho} (
B^i \delta^j_k
- B^j \delta^i_k
) &
\frac{1}{\rho} (
m^j \delta^i_k
- m^i \delta^j_k
) &
0
\\
\frac{1}{\rho^2} (
\tilde\gamma \frac{1}{\rho} m^j m^2
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} m^j B^2
+ \frac{1}{\mu_0} B^k m_k B^j
- \gamma m^j E_{total}
) &
\frac{1}{\rho} (
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 \delta^j_k
- \tilde\gamma \frac{1}{\rho} m^j m_k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 \delta^j_k
- \frac{1}{\mu_0} B_k B^j
+ \gamma E_{total} \delta^j_k
) &
\frac{1}{\mu_0} \frac{1}{\rho} (
(2 - \gamma) m^j B_k
- m_k B^j
- B^m m_m \delta^j_k
) &
\gamma \frac{1}{\rho} m^j
\end{matrix}\right] \left[\begin{matrix}
0 \\
{\Gamma^k}_{lj} m^l \\
{\Gamma^k}_{lj} B^l \\
0
\end{matrix}\right]
- g_{kl,j}\left[\begin{matrix}
0 \\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^k m^l g^{ij}
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{ik} g^{lj}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^k B^l g^{ij}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{ik} g^{lj}
- \tilde\gamma E_{total} g^{ik} g^{lj}
\\
0 \\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l B^j
\end{matrix}\right]
$
$= {F^{Ij}}_{,j}
+ \left[\begin{matrix}
\delta^j_k {\Gamma^k}_{lj} m^l \\
\frac{1}{\rho} (
m^j \delta^i_k
+ m^i \delta^j_k
- \tilde\gamma m_k g^{ij}
) {\Gamma^k}_{lj} m^l
+ \frac{1}{\mu_0} (
(2 - \gamma) B_k g^{ij}
- B^j \delta^i_k
- B^i \delta^j_k
) {\Gamma^k}_{lj} B^l
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^k m^l g^{ij} (\Gamma_{klj} + \Gamma_{lkj})
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{ik} g^{lj} (\Gamma_{klj} + \Gamma_{lkj})
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^k B^l g^{ij} (\Gamma_{klj} + \Gamma_{lkj})
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{ik} g^{lj} (\Gamma_{klj} + \Gamma_{lkj})
+ \tilde\gamma E_{total} g^{ik} g^{lj} (\Gamma_{klj} + \Gamma_{lkj})
\\
\frac{1}{\rho} (
B^i \delta^j_k
- B^j \delta^i_k
) {\Gamma^k}_{lj} m^l
+ \frac{1}{\rho} (
m^j \delta^i_k
- m^i \delta^j_k
) {\Gamma^k}_{lj} B^l
\\
\frac{1}{\rho} (
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 \delta^j_k
- \tilde\gamma \frac{1}{\rho} m^j m_k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 \delta^j_k
- \frac{1}{\mu_0} B_k B^j
+ \gamma E_{total} \delta^j_k
) {\Gamma^k}_{lj} m^l
+ \frac{1}{\mu_0} \frac{1}{\rho} (
(2 - \gamma) m^j B_k
- m_k B^j
- B^m m_m \delta^j_k
) {\Gamma^k}_{lj} B^l
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l (\Gamma_{klj} + \Gamma_{lkj})
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l (\Gamma_{klj} + \Gamma_{lkj})
+ \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l B^j (\Gamma_{klj} + \Gamma_{lkj})
\end{matrix}\right]
$
$= {F^{Ij}}_{,j}
+ {\Gamma^k}_{jk} F^{Ij}
+ \left[\begin{matrix}
0 \\
\frac{1}{\rho} (
{\Gamma^i}_{jk} m^j m^k
- \tilde\gamma {\Gamma_{jk}}^i m^j m^k
)
+ \frac{1}{\mu_0} (
(2 - \gamma) {\Gamma_{jk}}^i B^j B^k
- {\Gamma^i}_{jk} B^j B^k
)
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho} {\Gamma_{jk}}^i m^j m^k
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{jk} {\Gamma^i}_{jk}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} {\Gamma_{jk}}^i B^j B^k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{jk} {\Gamma^i}_{jk}
+ \tilde\gamma E_{total} {\Gamma^i}_{jk} g^{jk}
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^j m^k {\Gamma_{jk}}^i
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^j B^k {\Gamma_{jk}}^i
\\
0 \\
\frac{1}{\rho} (
- \tilde\gamma \frac{1}{\rho} m^j m_k {\Gamma^k}_{lj} m^l
- \frac{1}{\mu_0} B_k B^j {\Gamma^k}_{lj} m^l
)
+ \frac{1}{\mu_0} \frac{1}{\rho} (
(2 - \gamma) m^j B_k {\Gamma^k}_{lj} B^l
- m_k B^j {\Gamma^k}_{lj} B^l
)
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l \Gamma_{klj}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l \Gamma_{klj}
+ \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l B^j \Gamma_{klj}
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l \Gamma_{lkj}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l \Gamma_{lkj}
+ \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l B^j \Gamma_{lkj}
\end{matrix}\right]$
$= \frac{1}{\sqrt{g}} ( \sqrt{g} F^{Ij} )_{,j}
+ \left[\begin{matrix}
0 \\
{\Gamma^i}_{jk} (
\frac{1}{\rho} m^j m^k
+ P_{total} g^{jk}
- \frac{1}{\mu_0} B^j B^k
)
\\
0 \\
0
\end{matrix}\right]$
[1]: http://www.av8n.com/physics/euler-flow.htm:
[2]: https://en.wikipedia.org/wiki/Maxwell%27s_equations
[3]: https://en.wikipedia.org/wiki/Lorentz_force
[4]: https://en.wikipedia.org/wiki/Magnetic_pressure
[5]: http://www.mpia.de/homes/dullemon/lectures/fluiddynamics/
[6]: Samtaney (1997). Computational Methods for Self-similar Solutions of the Compressible Euler Equations. Journal of Computational Physics, 132, 327-345
[7]: Toro (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd Ed. 115-
[8]: Keyes, JFNK