SP_N Temporal Discretization

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:

$${\frac{{\partial T_i}}{{\partial t}}} \alpha \left(\vphantom{ T_e-T_i }\right. \left.\vphantom{ T_e-T_i }\right)$$ =

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

where

Blue = Implicit or backwards Euler terms,

Magenta = Explicit or extrapolated implicit terms,

Red = Implicit terms accelerated by DSA,

Green = Linearized implicit terms accelerated by LMFG.

This is not quite accurate -- it's actually more complicated than this -- but this captures the flavor of the temporal discretization.