%\documentstyle[11pt]{article} 
%\begin{document}
%\setcounter{page}{60}

\newpage

\begin{center}
\Large{\bf IX. RADIATION DIFFUSION}
\addcontentsline{toc}{section}{\protect\numberline{IX.}{\bf ~Radiation Diffusion}}
\end{center}   

\vspace*{5mm}
\noindent \rm For both nonequilibrium and equilibrium diffusion we set all Eddington factors $f_{k} \equiv \frac{1}{3}$, and set the surface flux Eddington factor $g_{N} \equiv \frac{1}{2}$ (consistent with isotropic radiation). Further we drop the time derivative and velocity-dependent terms.

\begin{flushleft}
\large {\bf A. Nonequilibrium Diffusion} \rm (Flux-Limited)\\
\addcontentsline{toc}{subsection}{\protect\numberline{IX.A.}{ ~Nonequilibrium Diffusion}}

\vspace*{5mm}
\rm Replace (RM2) with the flux-limited diffusion diffusion equation\\

\vspace*{3mm}
$F^{n+1}_{k} - \frac{c}{3}(r^{n+1}_{k})^{\mu}(E^{n+1}_{k-1}-E^{n+1}_{k})/D^{n+1}_{k} = 0 \hfill \rm (RD1) $\\

\vspace*{3mm}
\rm where\\

\vspace*{3mm}
$ D^{n+1}_{k} \equiv <\chi>^{n+1}_{k}\frac{1}{2}[\rho^{n+1}_{k-1}$\tt dvoln(k-1)$ + \rho^{n+1}_{k}$\tt dvoln(k)]\\
\hfill $+ \frac{2 \lambda}{3}(r^{n+1}_{k})^{\mu}|E^{n+1}_{k-1} - E^{n+1}_{k}|/(E^{n+1}_{k-1} + E^{n+1}_{k})$ \rm \hspace{5mm}(RD2)\\
\rm and\\
$$\frac{1}{<\chi>^{n+1}_{k}} \equiv \frac{1}{2}\left( \frac{1}{\chi^{n+1}_{k-1}} + \frac{1}{\chi^{n+1}_{k}} \right) \hspace{58mm} \rm (RD3) $$\\

\rm For convenience write (RD2) as\\
\vspace*{2mm}
$ D^{n+1}_{k} = \alpha^{n+1}_{k} + \frac{2}{3}\lambda\beta^{n+1}_{k}$\hfill \rm (RD4)\\
\end{flushleft}

\noindent \rm For no flux-limiting set $\lambda = 0$, and for flux limiting
set $\lambda = 1$. Boundary conditions (BC22) - (BC25), and (BC35) - (BC36)
remain unchanged.

\vspace*{3mm}
{\flushleft\large{\bf B. Equilibrium Diffusion}}\\
\addcontentsline{toc}{subsection}{\protect\numberline{IX.B.}{ ~Equilibrium Diffusion}}

\vspace*{2mm}
\noindent \rm To implement equilibrium diffusion, in addition to all replacements in equation (RD2) described above, replace (RE2) with\\

\vspace*{2mm}
{\flushleft $E^{n+1}_{k} = a_{R}(T^{n+1}_{k})^{4}$ \hfill \rm (RD5)}\\

\newpage
\vspace*{5mm}
{\flushleft \large{\bf C. Linearization}}\\
\addcontentsline{toc}{subsection}{\protect\numberline{IX.C.}{ ~Linearization}}

\vspace*{2mm}
\begin{center}
\it(1) Nonequilibrium Diffusion
\addcontentsline{toc}{subsubsection}{\protect\numberline{IX.C.1.}{\sl ~Nonequilibrium Diffusion}}
\end{center}   

\noindent \rm Define \\
{\flushleft \tt chdvol(k)$\equiv \alpha^{n+1}_{k}=\frac{1}{2}<\chi>^{n+1}_{k}[\rho^{n+1}_{k-1}$\tt dvoln(k-1)$+\rho^{n+1}_{k}$\tt dvoln(k)]\,\rm(RD6) }\\

\vspace*{2mm}
{\flushleft \tt chdvolrm(k)$\equiv \frac{\partial\alpha^{n+1}_{k}}{\partial lnr^{n+1}_{k-1}} = - \frac{1}{2} <\chi>^{n+1}_{k}$\tt rmup1n(k-1)$\rho^{n+1}_{k-1}$\hfill \rm (RD7) }\\

\vspace*{2mm}
{\flushleft \tt chdvolr0(k)$\equiv \frac{\partial\alpha^{n+1}_{k}}{\partial lnr^{n+1}_{k}} = \hspace{3mm} \frac{1}{2} <\chi>^{n+1}_{k}$\tt rmup1n(k\hspace{2mm})$(\rho^{n+1}_{k-1}-\rho^{n+1}_{k})$\hfill \rm (RD8) }\\

\vspace*{2mm}
{\flushleft \tt chdvolrp(k)$\equiv \frac{\partial\alpha^{n+1}_{k}}{\partial lnr^{n+1}_{k+1}} = \hspace{3mm} \frac{1}{2} <\chi>^{n+1}_{k}$\tt rmup1n(k+1)$\rho^{n+1}_{k}$\hfill \rm (RD9) }\\
 
\vspace*{2mm}
{\flushleft $ {\tt chdvoldm(k)} \equiv \frac{\partial\alpha^{n+1}_{k}}{\partial ln\rho^{n+1}_{k-1}} = $ \hfill}\\
{\flushright \hfill $\frac{1}{2}<\chi>^{n+1}_{k}{\tt dvoln(k)}\rho^{n+1}_{k-1} + \frac{1}{2}{\tt chdvol(k)} \frac{<\chi>^{n+1}_{k}}{\chi^{n+1}_{k-1}}\left(\frac{\partial ln\chi}{\partial ln\rho}\right)^{n+1}_{k-1} \hspace{5mm} \rm (RD10) $}\\
 
\vspace*{2mm}
{\flushleft $ {\tt chdvold0(k)} \equiv \frac{\partial\alpha^{n+1}_{k}}{\partial ln\rho^{n+1}_{k}} = $ \hfill}\\
{\flushright \hfill $\frac{1}{2}<\chi>^{n+1}_{k}{\tt dvoln(k)}\rho^{n+1}_{k} + \frac{1}{2}{\tt chdvol(k)} \frac{<\chi>^{n+1}_{k}}{\chi^{n+1}_{k}}\left(\frac{\partial ln\chi}{\partial ln\rho}\right)^{n+1}_{k} \hspace{5mm} \rm (RD11) $}\\

\vspace*{2mm}
{\flushleft $ {\tt chdvoltm(k)} \equiv \frac{\partial\alpha^{n+1}_{k}}{\partial lnT^{n+1}_{k-1}} = \frac{1}{2} {\tt chdvol(k)} \frac{<\chi>^{n+1}_{k}}{\chi^{n+1}_{k-1}}\left(\frac{\partial ln\chi}{\partial lnT}\right)^{n+1}_{k-1} \hfill \rm (RD12) $}\\

\vspace*{2mm}
{\flushleft $ {\tt chdvolt0(k)} \equiv \frac{\partial\alpha^{n+1}_{k}}{\partial lnT^{n+1}_{k}} = \frac{1}{2} {\tt chdvol(k)} \frac{<\chi>^{n+1}_{k}}{\chi^{n+1}_{k}}\left(\frac{\partial ln\chi}{\partial lnT}\right)^{n+1}_{k} \hfill \rm (RD13) $}\\
 
\vspace*{2mm}
\noindent \rm Further define \\

\vspace*{2mm}
{\flushleft \tt elim(k)$ = \frac{2}{3}\lambda\beta^{n+1}_{k} = \frac{2}{3}\lambda\,${\tt rmun(k)}$|E^{n+1}_{k-1} - E^{n+1}_{k}| / (E^{n+1}_{k-1} + E^{n+1}_{k})$ \hfill \rm (RD14)} \\

\vspace*{2mm}
{\flushleft \tt elimr0(k)$ = \frac{2}{3}\lambda \partial\beta^{n+1}_{k} /\partial r^{n+1}_{k} = \mu$ \tt elim(k) \hfill \rm(RD15)} \\

\vspace*{2mm}
{\flushleft \tt elimem(k)$ = \frac{2}{3}\lambda \partial \beta^{n+1}_{k} /\partial E^{n+1}_{k-1} $}\\
$$\hspace{22mm} = - \frac{[{\tt elim(k)} E^{n+1}_{k-1} + \frac{2}{3}\lambda\,{\tt rmun(k)}{\rm sgn}(E^{n+1}_{k-1} - E^{n+1}_{k})]}{(E^{n+1}_{k-1} + E^{n+1}_{k})} \hspace{5mm} \rm (RD16) $$\\

\vspace*{2mm}
{\flushleft \tt elime0(k)$ = \frac{2}{3}\lambda \partial \beta^{n+1}_{k} /\partial E^{n+1}_{k} $}\\
$$\hspace{22mm} = - \frac{[{\tt elim(k)} E^{n+1}_{k} + \frac{2}{3}\lambda\,{\tt rmun(k)}{\rm sgn}(E^{n+1}_{k-1} - E^{n+1}_{k})]}{(E^{n+1}_{k-1} + E^{n+1}_{k})} \hspace{5mm} \rm (RD17) $$\\
 
\vspace*{2mm}
\noindent \rm and \\
{\flushleft \tt fd(k)\hspace{8mm}$\equiv F^{n+1}_{k}D^{n+1}_{k} \hspace{11mm}= \hspace{3mm}(c/3)\ $\tt rmun(k)$(E^{n+1}_{k-1} - E^{n+1}_{k}) \hfill \rm (RD18)$}\\
 
\vspace*{2mm}
{\flushleft \tt dfddem(k)$\equiv \partial$\tt fd(k)$/\partial ln E^{n+1}_{k-1} = \hspace{3mm}(c/3)\ $\tt rmun(k)$E^{n+1}_{k-1} \hfill \rm (RD19)$}\\
 
\vspace*{2mm}
{\flushleft \tt dfdde0(k)$\equiv \partial$\tt fd(k)$/\partial ln E^{n+1}_{k} = -(c/3)\ $\tt rmun(k)$E^{n+1}_{k} \hfill \rm (RD20)$}\\
 
\vspace*{2mm}
{\flushleft \tt dfddr0(k)$\equiv \partial$\tt fd(k)$/\partial lnr^{n+1}_{k} \hspace{2mm} =  \hspace{3mm} \mu\ $\tt fd(k)\hfill \rm (RD21)}\\
 
\vspace*{2mm}
{\flushleft \tt denom(k)$\hspace{1mm}\equiv 1 / D^{n+1}_{k} = 1 / ($\tt chdvol(k)$+$elim(k)$+10^{-30})$ \hfill \rm (RD22)}\\

\vspace*{2mm}
\noindent \rm Then \\
 
\vspace*{2mm}
{\flushleft \tt dif(k)$\hspace{8mm}= F^{n+1}_{k} \hspace{18mm} = \hspace{3mm}$\tt denom(k)fd(k) \hfill \rm (RD23)}\\
 
\vspace*{2mm}
{\flushleft \tt ddifdrm(k)$= \partial F^{n+1}_{k}/\partial lnr^{n+1}_{k-1} = -$\tt denom(k)dif(k)chdvolrm(k) \hfill \rm (RD24)}\\
 
\vspace*{2mm}
{\flushleft \tt ddifdr0(k)$= \partial F^{n+1}_{k}/\partial lnr^{n+1}_{k}$}\\
{\flushright $=\ $\tt denom(k)\{dfddr0(k)$-$\tt dif(k)[chdvolr0(k)$+$\tt elimr0(k)]\} \hspace{9mm}\rm (RD25)}\\
 
\vspace*{2mm}
{\flushleft \tt ddifdrp(k)$= \partial F^{n+1}_{k}/\partial lnr^{n+1}_{k+1} = -$\tt denom(k)dif(k)chdvolrp(k) \hfill \rm (RD26)}\\
 
\vspace*{2mm}
{\flushleft \tt ddifddm(k)$= \partial F^{n+1}_{k}/\partial ln\rho^{n+1}_{k-1} = -$\tt denom(k)dif(k)chdvoldm(k) \hfill \rm (RD27)}\\
 
\vspace*{2mm}
{\flushleft \tt ddifdd0(k)$= \partial F^{n+1}_{k}/\partial ln\rho^{n+1}_{k} = -$\tt denom(k)dif(k)chdvold0(k) \hfill \rm (RD28)}\\

\vspace*{2mm}
{\flushleft \tt ddifdtm(k)$= \frac{\partial F^{n+1}_{k}}{\partial lnT^{n+1}_{k-1}} \hfill = -$\tt denom(k)dif(k)chdvoltm(k) \rm (RD29) }\\

\vspace*{2mm}
{\flushleft \tt ddifdt0(k)$= \frac{\partial F^{n+1}_{k}}{\partial lnT^{n+1}_{k}} \hfill = -$\tt denom(k)dif(k)chdvolt0(k) \rm (RD30) }\\

\vspace*{2mm}
{\flushleft \tt ddifdem(k)$= \frac{\partial F^{n+1}_{k}}{\partial lnE^{n+1}_{k-1}} = $\tt denom(k)[dfddem(k)$-$\tt dif(k)elimem(k)]\hspace{2mm}\rm (RD31)}\\

\vspace*{2mm}
{\flushleft \tt ddifde0(k)$= \frac{\partial F^{n+1}_{k}}{\partial lnE^{n+1}_{k}} = $\tt denom(k)[dfdde0(k)$-$\tt dif(k)elime0(k)]\hspace{2mm}\rm (RD32)}\\
 
\vspace*{2mm}
\noindent \rm Then\\
 
\vspace*{2mm}
{\flushleft \tt em1(if,jr):$-$\tt ddifdrm(k) \hfill \rm (RD33)}\\
 
\vspace*{2mm}
{\flushleft \tt e00(if,jr):$-$\tt ddifdr0(k) \hfill \rm (RD34)}\\
 
\vspace*{2mm}
{\flushleft \tt ep1(if,jr):$-$\tt ddifdrp(k) \hfill \rm (RD35)}\\
 
\vspace*{2mm}
{\flushleft \tt em1(if,jd):$-$\tt ddifddm(k) \hfill \rm (RD36)}\\
 
\vspace*{2mm}
{\flushleft \tt e00(if,jd):$-$\tt ddifdd0(k) \hfill \rm (RD37)}\\
 
\vspace*{2mm}
{\flushleft \tt em1(if,jt):$-$\tt ddifdtm(k) \hfill \rm (RD38)}\\
 
\vspace*{2mm}
{\flushleft \tt e00(if,jt):$-$\tt ddifdt0(k) \hfill \rm (RD39)}\\
 
\vspace*{2mm}
{\flushleft \tt em1(if,je):$-$\tt ddifdem(k) \hfill \rm (RD40)}\\
 
\vspace*{2mm}
{\flushleft \tt e00(if,je):$-$\tt ddifde0(k) \hfill \rm (RD41)}\\
 
\vspace*{2mm}
{\flushleft \tt e00(if,jf):\hspace{3mm}frnom(k) \hfill \rm (RD42)}\\

\vspace*{2mm}
\noindent \rm Finally

\vspace*{2mm}
{\flushleft $-$\ \tt rhs(if)\ $= F^{n+1}_{k} -$\tt dif(k) \hfill \rm (RD43) }\\

\vspace*{2mm}
\begin{center}
\it(2) Equilibrium Diffusion
\addcontentsline{toc}{subsubsection}{\protect\numberline{IX.C.2.}{\sl ~Equilibrium Diffusion}}
\end{center}   

\vspace*{2mm}
\tt e00(if,jt):$ - 4a_{R}(T^{n+1}_{k})^{4}$ \hfill \rm (RD44)\\

\vspace*{2mm}
\tt e00(if,je):$\ E^{n+1}_{k} $ \hfill \rm (RD45)\\

\vspace*{2mm}
{\flushleft $-$\ \tt rhs(ie)\,$= E^{n+1}_{k} - 4a_{R}(T^{n+1}_{k})^{4}$ \hfill \rm (RD46)}\\
 
%\end{document}   
.