Simplified Spherical Harmonics (SPN ) Even-Parity Equation Set

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

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

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

$\displaystyle \xi_{{m,g}}^{}$	=	Even-parity pseudo-angular energy intensity,
$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$ _{m, g}	=	Even-parity pseudo-angular energy current,
$\displaystyle \mathcal {C}$ ^s_g	=	$\displaystyle \left(\vphantom{ \sigma_g^a - \sigma_g^s }\right.$ $\displaystyle \sigma_{g}^{a}$ - $\displaystyle \sigma_{g}^{s}$ $\displaystyle \left.\vphantom{ \sigma_g^a - \sigma_g^s }\right)$ $\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$ ⁽⁰⁾_g^. $\displaystyle {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$ ,
$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\mathcal{C}}$}$ ^v_{m, g}	=	3 $\displaystyle \mu^{2}_{m}$ $\displaystyle \sigma_{g}^{t}$ $\displaystyle \left(\vphantom{ P_g + \phi_g }\right.$ P_g + $\displaystyle \phi_{g}^{}$ $\displaystyle \left.\vphantom{ P_g + \phi_g }\right)$ $\displaystyle {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$ ,
$\displaystyle \phi_{g}^{}$	=	$\displaystyle \sum_{{m=1}}^{M}$ $\displaystyle \xi_{{m,g}}^{}$ w_m ,
P_g	=	$\displaystyle \sum_{{m=1}}^{M}$ $\displaystyle \xi_{{m,g}}^{}$ $\displaystyle \mu^{2}_{m}$ w_m ,
$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$ _g	=	$\displaystyle \sum_{{m=1}}^{M}$ $\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$ _{m, g} w_m ,
$\displaystyle \phi^{{(0)}}_{g}$	=	$\displaystyle \phi_{g}^{}$ -2 $\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$ ⁽⁰⁾_g^. $\displaystyle {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$ ,
$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$ ⁽⁰⁾_g	=	$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$ _g - $\displaystyle \left(\vphantom{ P_g + \phi_g }\right.$ P_g + $\displaystyle \phi_{g}^{}$ $\displaystyle \left.\vphantom{ P_g + \phi_g }\right)$ $\displaystyle {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$ ,
M	=	$\displaystyle \left(\vphantom{ N + 1 }\right.$ N + 1 $\displaystyle \left.\vphantom{ N + 1 }\right)$ /2 .