SP_N Even-Parity Equation Set

Radiation transport equations:

$${\frac{{1}}{{c}}}$$$${\frac{{\partial}}{{\partial t}}}$$$$\xi_{{m,g}}^{}$$ + $$\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$} \cdot$}$$$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$$_{m, g} + $$\sigma_{g}^{t}$$$$\xi_{{m,g}}^{}$$ = $$\sigma_{g}^{s}$$$$\phi_{g}^{}$$ + $$\sigma_{g}^{e}$$B_g + $$\mathcal {C}$$^s_g  ,

$${\frac{{1}}{{c}}}$$$${\frac{{\partial}}{{\partial t}}}$$$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$$_{m, g} + $$\mu^{2}_{m}$$$$\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$}$$$$\xi_{{m,g}}^{}$$ + $$\sigma_{g}^{t}$$$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$$_{m, g} = $$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\mathcal{C}}$}$$^v_{m, g}

for m = 1, M , and g = 1, G .

Temperature equations:

$${\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}} {\frac{{\partial T_i}}{{\partial t}}} \alpha \left(\vphantom{ T_e-T_i }\right. \left.\vphantom{ T_e-T_i }\right)$$ =

$${\frac{{\partial T_e}}{{\partial t}}} \alpha \left(\vphantom{ T_i-T_e }\right. \left.\vphantom{ T_i-T_e }\right) \sum_{{g=1}}^{{G}} \left(\vphantom{ \sigma_g^a \phi^{(0)}_g- \sigma_g^e B_g }\right. \sigma_{g}^{a} \phi^{{(0)}}_{g} \sigma_{g}^{e} \left.\vphantom{ \sigma_g^a \phi^{(0)}_g- \sigma_g^e B_g }\right)$$ =

where

$$\xi_{{m,g}}^{}$$ = Even-parity pseudo-angular energy intensity,

$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$$ = Even-parity pseudo-angular energy current,

$$\mathcal {C} \left(\vphantom{ \sigma_g^a - \sigma_g^s }\right. \sigma_{g}^{a} \sigma_{g}^{s} \left.\vphantom{ \sigma_g^a - \sigma_g^s }\right) \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$} {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$$ =

$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\mathcal{C}}$} \mu^{2}_{m} \sigma_{g}^{t} \left(\vphantom{ P_g + \phi_g }\right. \phi_{g}^{} \left.\vphantom{ P_g + \phi_g }\right) {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$$ =

$$\phi_{g}^{} \sum_{{m=1}}^{M} \xi_{{m,g}}^{}$$ =

$$\sum_{{m=1}}^{M} \xi_{{m,g}}^{} \mu^{2}_{m}$$ P_g =

$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$} \sum_{{m=1}}^{M} \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$$ =

$$\phi^{{(0)}}_{g} \phi_{g}^{} \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$} {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$$ =

$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$} \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$} \left(\vphantom{ P_g + \phi_g }\right. \phi_{g}^{} \left.\vphantom{ P_g + \phi_g }\right) {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$$ =

$$\left(\vphantom{ N + 1 }\right. \left.\vphantom{ N + 1 }\right)$$ M =