Changes between Version 195 and Version 196 of FluxLimitedDiffusion

Timestamp:

05/30/13 17:20:39 (11 years ago)

Author:

Baowei Liu

Comment:

—

FluxLimitedDiffusion

@@ -5 +5 @@
 [[CollapsibleStart(Spectral Intensity)]]
 == Spectral Intensity ==
-Typically when we discuss the radiation field we use the spectral intensity [[latex(I \left ( \nu, \mathbf{x}, \Omega \right ) )]] which is a function of frequency, position, and direction.  This is very similar to the phase space density used in deriving the fluid equations [[latex(f \left ( \mathbf{x}, \mathbf{v} \right ) )]] except that 
- * light always travels at \(c\), so the velocity dependence is just a direction dependence.  
+Typically when we discuss the radiation field we use the spectral intensity [[latex($I \left ( \nu, \mathbf{x}, \Omega \right ) $)]] which is a function of frequency, position, and direction.  This is very similar to the phase space density used in deriving the fluid equations [[latex($f \left ( \mathbf{x}, \mathbf{v} \right ) $)]] except that 
+ * light always travels at  [[latex($c$)]], so the velocity dependence is just a direction dependence.  
  * Furthermore, photons can have different frequencies, so there is an extra dimension to the phase space.  
  * Instead of storing the phase space density of photons, the spectral intensity is the phase space density of energy flux...  
 
-Going between photon number and energy just involves a factor of \( h \nu \) and going from energy density to energy flux density just involves a factor of \(c\) so we have
-
- $$ I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) $$
+Going between photon number and energy just involves a factor of [[latex($h\nu$)]] and going from energy density to energy flux density just involves a factor of [[latex($c$)]] so we have
+
+ [[latex($I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) $)]]
 
 This can also be seen by considering the differential energy 
 
- [[latex(dE = I \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA dt = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dV)]]
+ [[latex($dE = I \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA dt = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dV$)]]
 
 where the number of photons traveling normal to the surface dA that cross the surface dA in time dt is just the number of photons in the volume dV = dA c dt (assuming the photons are headed normal to dA)...
 
 so we also have
- [[latex(dE = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA c dt)]]
+ [[latex($dE = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA c dt$)]]
 
 which gives 
 
- [[latex(I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) )]]
+ [[latex($I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) $)]]
 
 
@@ -33 +33 @@
 == Deriving the Transport Equation ==
 
-If we consider the Boltzmann transport equation for photons of a specific frequency \(f_\nu\) we have
-
- [[latex(\frac{\partial}{\partial t} f_\nu + \mathbf{v} \cdot  \nabla f + \mathbf{F} \cdot \nabla_p f = \left ( \frac{\partial f}{\partial t} \right )_{coll} )]] 
+If we consider the Boltzmann transport equation for photons of a specific frequency [[latex($f_\nu$)]] we have
+
+ [[latex($\frac{\partial}{\partial t} f_\nu + \mathbf{v} \cdot  \nabla f + \mathbf{F} \cdot \nabla_p f = \left (\frac{\partial f}{\partial t} \right )_{coll} $)]] 
 
 Now photons don't experience body forces, always travel at the speed of light,  and in general the "collision term" consists of photon emission and absorptions... so we have
 
- [[latex(\frac{\partial}{\partial t} f_\nu + c \mathbf{n} \cdot  \nabla f_\nu  = A_{\nu} - \chi_\nu f_\nu c  )]] 
-
-where \(A_{\nu} \) is the emission rate of photons of frequency \(\nu\) and the mean free path length is given by \(\chi_\nu = \sigma_\nu n\) where \(\sigma_\nu\) is the particle scattering cross section and \(n\) is the number density of particles.
-
-Now if we multiply through by \( h\nu\) we have
- [[latex(\frac{\partial}{c \partial t} I_\nu + \mathbf{n} \cdot  \nabla I_\nu  = \eta_\nu - \sigma_\nu I_\nu  )]] 
-
-where \(\eta_\nu = h\nu A_{\nu}\) is the radiative power
+ [[latex($\frac{\partial}{\partial t} f_\nu + c \mathbf{n} \cdot  \nabla f_\nu  = A_{\nu} - \chi_\nu f_\nu c  $)]] 
+
+where [[latex($A_{\nu} $)]] is the emission rate of photons of frequency [[latex($\nu$)]] and the mean free path length is given by [[latex($\chi_\nu = \sigma_\nu n$)]] where [[latex($\sigma_\nu$)]] is the particle scattering cross section and [[latex($n$)]] is the number density of particles.
+
+Now if we multiply through by [[latex($ h\nu$)]] we have
+ [[latex($\frac{\partial}{c \partial t} I_\nu + \mathbf{n} \cdot  \nabla I_\nu  = \eta_\nu - \sigma_\nu I_\nu  $)]] 
+
+where [[latex($\eta_\nu = h\nu A_{\nu}$)]] is the radiative power
 
 If we solve the transport equation along a characteristic 
 
- [[latex(\left [ \mathbf{x} \left ( s \right ) , t \left ( s \right ) \right ] = \left [ \mathbf{x0} + \mathbf{n} s, \frac{s}{c} \right ] )]]
+ [[latex($\left [ \mathbf{x} \left ( s \right ) , t \left ( s \right ) \right ] = \left [ \mathbf{x0} + \mathbf{n} s, \frac{s}{c} \right ] $)]]
 
 we have
 
- [[latex(\frac{d I_\nu}{d s} = \frac{\partial I_\nu}{\partial x^i} \frac{\partial{x^i}}{\partial s} + \frac{\partial I_\nu}{\partial t}\frac{\partial t}{\partial s} = \mathbf{n} \cdot \nabla I_\nu + \frac{1}{c}\frac{\partial I_\nu}{\partial t} = \eta_\nu(s) - \chi_\nu(s) I_\nu (s)  )]]
-
-where \(f(s) = f(\mathbf{x}(s), t(s)) = f \left ( \mathbf{x_0}+\mathbf{n} s, \frac{s}{c} \right ) \) and then we can divide through by \(\chi_\nu(s)\) we get
-
- [[latex(\frac{dI_\nu}{\chi_\nu(s) ds} = \frac{\eta_\nu(s)}{\chi_\nu(s)} - I_\nu(s) = S_\nu(s) - I_\nu(s))]]
-
-Now if we define \(d\tau_\nu = \chi_\nu(s) ds \) which gives 
-
- [[latex(\tau_\nu(s) = \int\limits_0^s \chi_\nu(s') ds')]]
-and
- [[latex(s(\tau_\nu) = \int\limits_0^\tau_\nu \frac{1}{\chi_\nu} d\tau'_\nu )]] 
+ [[latex($\frac{d I_\nu}{d s} = \frac{\partial I_\nu}{\partial x^i} \frac{\partial{x^i}}{\partial s} + \frac{\partial I_\nu}{\partial t}\frac{\partial t}{\partial s} = \mathbf{n} \cdot \nabla I_\nu + \frac{1}{c}\frac{\partial I_\nu}{\partial t} = \eta_\nu(s) - \chi_\nu(s) I_\nu (s)  $)]]
+
+where [[latex($f(s) = f(\mathbf{x}(s), t(s)) = f \left ( \mathbf{x_0}+\mathbf{n} s, \frac{s}{c} \right ) $)]] and then we can divide through by [[latex($\chi_\nu(s)$)]] we get
+
+ [[latex($\frac{dI_\nu}{\chi_\nu(s) ds} = \frac{\eta_\nu(s)}{\chi_\nu(s)} - I_\nu(s) = S_\nu(s) - I_\nu(s)$)]]
+
+Now if we define [[latex($d\tau_\nu = \chi_\nu(s) ds $)]] which gives 
+
+ [[latex($\tau_\nu(s) = \int\limits_0^s \chi_\nu(s') ds'$)]]
+and
+
+ [[latex($s(\tau_\nu) = \int\limits_0^\tau_\nu \frac{1}{\chi_\nu} d\tau'_\nu$)]] 
 
 we can write the transport equation in the simplest form
 
- [[latex(\frac{dI_\nu}{d\tau_\nu} = S_\nu(\tau_\nu) - I_\nu(\tau_\nu))]]
+ [[latex($\frac{dI_\nu}{d\tau_\nu} = S_\nu(\tau_\nu) - I_\nu(\tau_\nu)$)]]
 
 although the RHS is now more difficult to evaluate as
 
- [[latex( f \left ( \tau_\nu \right ) = f \left ( s \left ( \tau_\nu \right ) \right ) = f \left ( \mathbf{x} \left (s \left ( \tau_\nu \right ) \right ), t \left ( s \left ( \tau_\nu \right ) \right ) \right ) )]]
-
-Also if we include scattering then the source function can depend on the mean radiative flux \(\frac{cE}{4 \pi}\) and the transport equation becomes an integro-differential equation that must be solved iteratively...
+ [[latex($ f \left ( \tau_\nu \right ) = f \left ( s \left ( \tau_\nu \right ) \right ) = f \left ( \mathbf{x} \left (s \left ( \tau_\nu \right ) \right ), t \left ( s \left ( \tau_\nu \right ) \right ) \right ) $)]]
+
+Also if we include scattering then the source function can depend on the mean radiative flux [[latex($\frac{cE}{4 \pi}$)]] and the transport equation becomes an integro-differential equation that must be solved iteratively...
 
 There are also a few important dimensionless numbers to consider:
 
-|| [[latex(\tau = l \kappa=\frac{l}{\lambda_p})]] || [[latex(\beta = \frac{u}{c})]] ||
-|| [[latex(\tau << 1 )]] || streaming limit ||
-|| [[latex(\tau >> 1 \mbox{, } \beta \tau << 1)]] || static diffusion limit ||
-|| [[latex(\tau >> 1 \mbox{, } \beta \tau >> 1)]] || dynamic diffusion limit || 
+|| [[latex($\tau = l \kappa=\frac{l}{\lambda_p}$)]] || [[latex($\beta = \frac{u}{c}$)]] ||
+|| [[latex($\tau << 1 $)]] || streaming limit ||
+|| [[latex($\tau >> 1 \mbox{, } \beta \tau << 1$)]] || static diffusion limit ||
+|| [[latex($\tau >> 1 \mbox{, } \beta \tau >> 1$)]] || dynamic diffusion limit || 
 
 [[CollapsibleEnd()]]
@@ -90 +91 @@
 Some of what follows is taken from [http://adsabs.harvard.edu/abs/2007ApJ...667..626K Krumholz et al. 2007]
 
-||  [[latex(\frac{\partial \rho}{\partial t} + \nabla \cdot \left ( \rho \mathbf{v} \right ) = 0)]]  ||
-||  [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P+\mathbf{G})]]  ||
-||  [[latex(\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = c G^0)]]  ||
-||  [[latex(\frac{\partial E}{\partial t} + \nabla \cdot \mathbf{F} = -c G_0 )]]  ||
-||  [[latex(\frac{1}{c^2} \frac{\partial \mathbf{F}}{\partial t} + \nabla \cdot \mathbf{P} = -\mathbf{G})]]  ||
+||  [[latex($\frac{\partial \rho}{\partial t} + \nabla \cdot \left ( \rho \mathbf{v} \right ) = 0$)]]  ||
+||  [[latex($\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P+\mathbf{G}$)]]  ||
+||  [[latex($\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = c G^0$)]]  ||
+||  [[latex($\frac{\partial E}{\partial t} + \nabla \cdot \mathbf{F} = -c G_0 $)]]  ||
+||  [[latex($\frac{1}{c^2} \frac{\partial \mathbf{F}}{\partial t} + \nabla \cdot \mathbf{P} = -\mathbf{G}$)]]  ||
 
 where the moments of the specific intensity are defined as
 ||  Radiation Energy moments  ||   Corresponding fluid moments   ||
-||  [[latex(cE =\int_0^\infty d \nu \int d \Omega I(\mathbf{n}, \nu))]]  ||  [[latex(\rho=\int d \mathbf{v} f(\mathbf{v}))]]  ||  
-||  [[latex(\mathbf{F}=\int_0^\infty d \nu \int d \Omega \mathbf{n} I(\mathbf{n}, \nu))]]  || [[latex(\rho \mathbf{v} =\int d \mathbf{v} \mathbf{v} f(\mathbf{v}))]]  ||   
-||  [[latex(c\mathbf{P}=\int_0^\infty d \nu \int d \Omega \mathbf{nn} I(\mathbf{n}, \nu))]]   ||  [[latex(\mathbf{P} =\int d \mathbf{v} \mathbf{vv} f(\mathbf{v}))]]  ||    
+||  [[latex($cE =\int_0^\infty d \nu \int d \Omega I(\mathbf{n}, \nu)$)]]  ||  [[latex($\rho=\int d \mathbf{v} f(\mathbf{v})$)]]  ||  
+||  [[latex($\mathbf{F}=\int_0^\infty d \nu \int d \Omega \mathbf{n} I(\mathbf{n}, \nu)$)]]  || [[latex($\rho \mathbf{v} =\int d \mathbf{v} \mathbf{v} f(\mathbf{v})$)]]  ||   
+||  [[latex($c\mathbf{P}=\int_0^\infty d \nu \int d \Omega \mathbf{nn} I(\mathbf{n}, \nu)$)]]   ||  [[latex($\mathbf{P} =\int d \mathbf{v} \mathbf{vv} f(\mathbf{v})$)]]  ||    
 
 and the radiation 4-force density is given by
 
-||   [[latex(cG^0 = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)-\eta(\mathbf{n},v) \right ])]]  ||
-||   [[latex(c\mathbf{G} = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)-\eta(\mathbf{n},v) \right ] \mathbf{n})]]  ||
-
-If we had a closure relation for the radiation pressure then we could solve this system.  For gas particles, collisions tend to produce a Boltzmann Distribution which is isotropic and gives a pressure tensor that is a multiple of the identity tensor.  Photons do not "collide" with each other and they all have the same velocity 'c' but in various directions.  If the field were isotropic than \(P^{ij}=\delta^{ij} 1/3 E\) but in general \(P^{ij}=f^{ij} E\) where 'f' is the Eddington Tensor.
+||   [[latex($cG^0 = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)-\eta(\mathbf{n},v) \right ]$)]]  ||
+||   [[latex($c\mathbf{G} = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)-\eta(\mathbf{n},v) \right ] \mathbf{n}$)]]  ||
+
+If we had a closure relation for the radiation pressure then we could solve this system.  For gas particles, collisions tend to produce a Boltzmann Distribution which is isotropic and gives a pressure tensor that is a multiple of the identity tensor.  Photons do not "collide" with each other and they all have the same velocity 'c' but in various directions.  If the field were isotropic than [[latex($P^{ij}=\delta^{ij} 1/3 E$)]] but in general [[latex($P^{ij}=f^{ij} E$)]] where 'f' is the Eddington Tensor.
 
 [[CollapsibleEnd()]]
@@ -113 +114 @@
 == Simplifying assumptions ==
 * If the flux spectrum of the radiation is direction-independent then we can write the radiation four-force density in terms of the moments of the radiation field
-||   [[latex(G^0=\gamma [ \gamma^2 \kappa_{0E} + (1-\gamma^2)\kappa_{0F}]E-\gamma \kappa_{0P}a_RT_0^4-\gamma(\mathbf{v} \cdot \mathbf{F}/c^2)[\kappa_{0F}-2\gamma^2(\kappa_{0F}-\kappa_{0E})] - \gamma^3(\kappa_{0F}-\kappa+{0E})(\mathbf{vv}) : \mathbf{P}/c^2)]]   ||
-||   [[latex(\mathbf{G} = \gamma \kappa_{0F}(\mathbf{F}/c)-\gamma \kappa_{0P}a_RT_0^4 (\mathbf{v}/c)  - \left [ \gamma^3( \kappa_{0F} - \kappa_{0E})( \mathbf{v}/c)E + \gamma \kappa_{0F}(\mathbf{v}/c) \cdot \mathbf{P} \right ] + \gamma^3 ( \kappa_{0F} - \kappa_{0E}) \left [ 2 \mathbf{v} \cdot \mathbf{F}/c^3 - ( \mathbf{vv}) : \mathbf{P}/c^3 \right ] \mathbf{v} )]]   ||
+||   [[latex($G^0=\gamma [ \gamma^2 \kappa_{0E} + (1-\gamma^2)\kappa_{0F}]E-\gamma \kappa_{0P}a_RT_0^4-\gamma(\mathbf{v} \cdot \mathbf{F}/c^2)[\kappa_{0F}-2\gamma^2(\kappa_{0F}-\kappa_{0E})] - \gamma^3(\kappa_{0F}-\kappa+{0E})(\mathbf{vv}) : \mathbf{P}/c^2$)]]   ||
+||   [[latex($\mathbf{G} = \gamma \kappa_{0F}(\mathbf{F}/c)-\gamma \kappa_{0P}a_RT_0^4 (\mathbf{v}/c)  - \left [ \gamma^3( \kappa_{0F} - \kappa_{0E})( \mathbf{v}/c)E + \gamma \kappa_{0F}(\mathbf{v}/c) \cdot \mathbf{P} \right ] + \gamma^3 ( \kappa_{0F} - \kappa_{0E}) \left [ 2 \mathbf{v} \cdot \mathbf{F}/c^3 - ( \mathbf{vv}) : \mathbf{P}/c^3 \right ] \mathbf{v} $)]]   ||
 
  where 
 
-||  [[latex(\kappa_{0P})]]  ||  comoving-frame Planck function weighted opacity ||
-||  [[latex(\kappa_{0E})]]  ||  comoving-frame radiation energy weighted opacity ||
-||  [[latex(\kappa_{0F})]]  ||  comoving-frame radiation flux weighted opacity ||
-
-* If the radiation has a blackbody spectrum then \(\kappa_{0E}=\kappa_{0P}\)
+||  [[latex($\kappa_{0P}$)]]  ||  comoving-frame Planck function weighted opacity ||
+||  [[latex($\kappa_{0E}$)]]  ||  comoving-frame radiation energy weighted opacity ||
+||  [[latex($\kappa_{0F}$)]]  ||  comoving-frame radiation flux weighted opacity ||
+
+* If the radiation has a blackbody spectrum then [[latex($\kappa_{0E}=\kappa_{0P}$)]]
 * If the radiation is optically thick, then 
-[[latex(\mathbf{F_0}(\nu_0) \propto -\nabla E_0(\nu_0)/\kappa_0(\nu_0) \propto -[\partial B(\nu_0, T_0) / \partial T_0](\nabla T_0)/\kappa_0(\nu_0))]] 
- which implies that  [[latex(\kappa_{0F}^{-1}=\kappa_{0R}^{-1}=\frac{\int_0^\infty d \nu_0 \kappa_0(\nu_0)^{-1}[\partial B(\nu_0,T_0)/\partial T_0]}{\int_0^\infty d \nu_0 [\partial B(\nu_0, T_0)/\partial T_0]})]] 
-* In the optically thin regime, \(|\mathbf{F}_0(\nu_0)| \rightarrow cE_0(\nu_0)\) so we would have \(\kappa_{0F}=\kappa_{0E}\) however assuming a blackbody temperature in the optically thin limit may be any more accurate than assuming that \(\kappa_{0F}=\kappa_{0R}\)
+[[latex($\mathbf{F_0}(\nu_0) \propto -\nabla E_0(\nu_0)/\kappa_0(\nu_0) \propto -[\partial B(\nu_0, T_0) / \partial T_0](\nabla T_0)/\kappa_0(\nu_0)$)]] 
+ which implies that  [[latex($\kappa_{0F}^{-1}=\kappa_{0R}^{-1}=\frac{\int_0^\infty d \nu_0 \kappa_0(\nu_0)^{-1}[\partial B(\nu_0,T_0)/\partial T_0]}{\int_0^\infty d \nu_0 [\partial B(\nu_0, T_0)/\partial T_0]}$)]] 
+* In the optically thin regime, [[latex($|\mathbf{F}_0(\nu_0)| \rightarrow cE_0(\nu_0)$)]] so we would have [[latex($\kappa_{0F}=\kappa_{0E}$)]] however assuming a blackbody temperature in the optically thin limit may be any more accurate than assuming that [[latex($\kappa_{0F}=\kappa_{0R}$)]]
 [[CollapsibleEnd()]]
 
@@ -133 +134 @@
 
 The flux limited diffusion approximation drops the radiation momentum equation in favor of 
-\(\mathbf{F}_0=-\frac{c\lambda}{\kappa_{0R}}\nabla E_0\) where \(\lambda\) is the flux-limiter and is given by
-
-  [[latex(\lambda = \frac{1}{R} \left ( \coth R - \frac{1}{R} \right ) )]]  
-and
-  [[latex(R=\frac{|\nabla E_0|}{\kappa_{0R}E_0})]]  
+[[latex($\mathbf{F}_0=-\frac{c\lambda}{\kappa_{0R}}\nabla E_0$)]] where [[latex($\lambda$)]] is the flux-limiter and is given by
+
+  [[latex($\lambda = \frac{1}{R} \left ( \coth R - \frac{1}{R} \right ) $)]]  
+and
+  [[latex($R=\frac{|\nabla E_0|}{\kappa_{0R}E_0}$)]]  
 
 which corresponds to a pressure tensor
-   [[latex(\mathbf{P}_0=\frac{E_0}{2}[(1-R_2)\mathbf{I}+(3R_2-1)\mathbf{n}_0\mathbf{n}_0])]]   
+   [[latex($\mathbf{P}_0=\frac{E_0}{2}[(1-R_2)\mathbf{I}+(3R_2-1)\mathbf{n}_0\mathbf{n}_0]$)]]   
 where
-   [[latex(R_2=\lambda+\lambda^2R^2)]]   
-
-Here are \(\lambda(R)\) and \(R_2(R)\) plotted
-  [[Image(fld.png, width=400)]]
+   [[latex($R_2=\lambda+\lambda^2R^2$)]]   
+
+Here are [[latex($\lambda(R)$)]] and [[latex($R_2(R)$)]] plotted
+  [[Image(fld.png, width=400$)]]
 
 
 If we Lorentz boost the comoving terms into the lab frame and keep terms necessary to maintain accuracy we get:
 
-   [[latex(G^0=\kappa_{0P} \left ( E-\frac{4 \pi B}{c} \right ) + \left ( \frac{\lambda}{c} \right ) \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \mathbf{v} \cdot \nabla E - \frac{\kappa_{0P}}{c^2} E \left [ \frac{3-R_2}{2}v^2 + \frac{3R_2-1}{2}(\mathbf{v} \cdot \mathbf{n})^2 \right ] + \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \kappa_{0P} \left ( E - \frac{4 \pi B}{c} \right ) )]]   
-   [[latex(\mathbf{G} = -\lambda \nabla E + \kappa_{0P} \frac{\mathbf{v}}{c} \left ( E - \frac{4 \pi B}{c} \right ) - \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \lambda \nabla E + 2 \lambda \left ( \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \frac{(\mathbf{v} \cdot \nabla E )\mathbf{v}}{c^2})]]   
+   [[latex($G^0=\kappa_{0P} \left ( E-\frac{4 \pi B}{c} \right ) + \left ( \frac{\lambda}{c} \right ) \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \mathbf{v} \cdot \nabla E - \frac{\kappa_{0P}}{c^2} E \left [ \frac{3-R_2}{2}v^2 + \frac{3R_2-1}{2}(\mathbf{v} \cdot \mathbf{n})^2 \right ] + \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \kappa_{0P} \left ( E - \frac{4 \pi B}{c} \right ) $)]]   
+   [[latex($\mathbf{G} = -\lambda \nabla E + \kappa_{0P} \frac{\mathbf{v}}{c} \left ( E - \frac{4 \pi B}{c} \right ) - \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \lambda \nabla E + 2 \lambda \left ( \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \frac{(\mathbf{v} \cdot \nabla E )\mathbf{v}}{c^2}$)]]   
 
 [[CollapsibleEnd()]]
@@ -159 +160 @@
 If we plug the expressions for the radiation 4-momentum back into the gas equations and keep terms necessary to maintain accuracy we get:
 
-  [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P\color{green}{-\lambda \nabla E})]]   
-  [[latex(\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{-\kappa_{0P}(4 \pi B-cE)} \color{green}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E} \color{blue}{-\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E})]]  
-  [[latex(\frac{\partial E}{\partial t}  \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} = \color{red}{\kappa_{0P} (4 \pi B-cE)} \color{green}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E} \color{green}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )} \color{blue}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E}  )]]  
-
-
-For static diffusion, the terms in blue with \(\frac{v^2}{c}\) can be dropped and the system can be split into the usual hydro update (black), radiative source terms (green), and a coupled implicit solve (red) for the radiation energy density and thermal energy density (ie temperature). 
+  [[latex($\frac{\partial }{\partial t} \left(\rho\mathbf{v}\right)+\nabla\cdot\left(\rho\mathbf{vv}\right)=\nabla P\color{green}{-\lambda \nabla E}$)]]
+   
+  [[latex($\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] =\color{red{\kappa_{0P}(4 \pi B-cE)} \color{green}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right)\mathbf{v}\cdot \nabla E} \color{blue}{-\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E}$)]]  
+  [[latex($\frac{\partial E}{\partial t}  \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E}=\color{red}{\kappa_{0P} (4 \pi B-cE)} \color{green}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1\right)\mathbf{v}\cdot \nabla E} \color{green}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E\right )}\color{blue}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E}  $)]]  
+
+
+For static diffusion, the terms in blue with [[latex($\frac{v^2}{c}$)]] can be dropped and the system can be split into the usual hydro update (black), radiative source terms (green), and a coupled implicit solve (red) for the radiation energy density and thermal energy density (ie temperature). 
 
 [[CollapsibleStart(Operator Splitting 1)]]
@@ -172 +174 @@
 In AstroBEAR this would look like:
 * Initialization
- * Prolongate, \(\rho\), \(\rho\mathbf{v}\), \(e\), \(E\), and \(\dot{E}^I\)
+ * Prolongate, [[latex($\rho$)]], [[latex($\rho\mathbf{v}$)]], [[latex($e$)]], [[latex($E$)]], and [[latex($\dot{E}^I$)]]
 * Step 1
- * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\) and do physical BC's
- * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}^I_1\)
- * Update \(E_{2\mbox{mbc}}\) using \(\dot{E}^I_{2\mbox{mbc}}\)
- * Update \(e_{2\mbox{mbc}}\) using \(E_{2\mbox{mbc}}\), \(\dot{E}^I_{2\mbox{mbc}}\), and \(e_{2\mbox{mbc}}\)
- * Update \(\dot{E}^I_0\) using pre IR and post IR \(E_0\)
- * Ghost \(e_{2\mbox{mbc}}\), \(E_{\mbox{mbc}+1}\), \(\dot{E}^I_{\mbox{mbc}+1}\)
+ * Overlap [[latex($\rho$)]], [[latex($\rho \mathbf{v} $)]] , [[latex($e$)]], [[latex($E$)]] and do physical BC's
+ * Do IR which updates [[latex($e_0$)]] and [[latex($E_0$)]] using [[latex($\rho_1$)]], [[latex($e_1$)]], [[latex($E_1$)]], and [[latex($\dot{E}^I_1$)]]
+ * Update [[latex($E_{2\mbox{mbc}}$)]] using [[latex($\dot{E}^I_{2\mbox{mbc}}$)]]
+ * Update [[latex($e_{2\mbox{mbc}}$)]] using [[latex($E_{2\mbox{mbc}}$)]], [[latex($\dot{E}^I_{2\mbox{mbc}}$)]], and [[latex($e_{2\mbox{mbc}}$)]]
+ * Update [[latex($\dot{E}^I_0$)]] using pre IR and post IR [[latex($E_0$)]]
+ * Ghost [[latex($e_{2\mbox{mbc}}$)]], [[latex($E_{\mbox{mbc}+1}$)]], [[latex($\dot{E}^I_{\mbox{mbc}+1}$)]]
  * Do first EH,,mbc,,
- * Do ER,,mbc,, --- Terms with \(\nabla E\) can be done without ghosting since EH did not change \(E\).  The \(\nabla \cdot \mathbf{v}E\) term needs time centered face centered velocities which can be stored during the hydro update.
- * Store \(\dot{E}\) in child arrays to be prolongated
+ * Do ER,,mbc,, --- Terms with [[latex($\nabla E$)]] can be done without ghosting since EH did not change [[latex($E$)]].  The [[latex($\nabla \cdot \mathbf{v}E$)]] term needs time centered face centered velocities which can be stored during the hydro update.
+ * Store [[latex($\dot{E}$)]] in child arrays to be prolongated
 * Step 2
- * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\) and do physical BC's
- * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}^I_1\)
- * Update \(\dot{E}^I_0\) using pre IR and post IR \(E_0\)
- * Update \(E_{1}\) using \(\dot{E}^I_{1}\)
- * Ghost \(e_{mbc}\), \(E_{1}\), \(\dot{E}^I_{1}\)
+ * Overlap [[latex($\rho$)]], [[latex($\rho \mathbf{v} $)]] , [[latex($e$)]], [[latex($E$)]] and do physical BC's
+ * Do IR which updates [[latex($e_0$)]] and [[latex($E_0$)]] using [[latex($\rho_1$)]], [[latex($e_1$)]], [[latex($E_1$)]], and [[latex($\dot{E}^I_1$)]]
+ * Update [[latex($\dot{E}^I_0$)]] using pre IR and post IR [[latex($E_0$)]]
+ * Update [[latex($E_{1}$)]] using [[latex($\dot{E}^I_{1}$)]]
+ * Ghost [[latex($e_{mbc}$)]], [[latex($E_{1}$)]], [[latex($\dot{E}^I_{1}$)]]
  * Do second EH,,0,,
- * Do ER,,mbc,, --- Terms with \(\nabla E\) can be done without ghosting since EH did not 
- * Do ER,,0,, --- Terms with grad E can be done without ghosting since EH did not change \(E\).  The \(\nabla \cdot \mathbf{v}E\) term needs time centered face centered velocities which can be stored during the hydro update.
+ * Do ER,,mbc,, --- Terms with [[latex($\nabla E$)]] can be done without ghosting since EH did not 
+ * Do ER,,0,, --- Terms with grad E can be done without ghosting since EH did not change [[latex($E$)]].  The [[latex($\nabla \cdot \mathbf{v}E$)]] term needs time centered face centered velocities which can be stored during the hydro update.
 
 [[CollapsibleEnd()]]
@@ -199 +201 @@
 The extra terms in the explicit update due to radiation energy are as follows:
 
- [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) =-\lambda \nabla E)]]   
- [[latex(\frac{\partial e}{\partial t}  = \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E )]]  
- [[latex(\frac{\partial E}{\partial t}  = -\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) )]]  
+ [[latex($\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) =-\lambda \nabla E$)]]   
+ [[latex($\frac{\partial e}{\partial t}  = \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E $)]]  
+ [[latex($\frac{\partial E}{\partial t}  = -\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) $)]]  
 
 These can be discretized as follows:
 
- [[latex(p^{n+1}_i=p^n_i - \frac{1}{2}\frac{\Delta t}{\Delta x} \lambda_{i} \left ( E^n_{i+1}-E^n_{i-1} \right ) )]]
-
- [[latex(e^{n+1}_i=e^n_i + \frac{1}{2}\frac{\Delta t}{\Delta x} \lambda_i \left ( 2 \frac{\kappa^n_{i,0P}}{\kappa^n_{i,0R}}-1 \right ) \left ( v^n_i \left ( E^n_{i+1}-E^n_{i-1} \right ) \right ) )]]  
-
- [[latex(E^{n+1}_i=E^n_i - \frac{\Delta t}{\Delta x} \left ( \frac{\lambda_i}{2} \left ( 2 \frac{\kappa^n_{i,0P}}{\kappa^n_{i,0R}}-1 \right ) \left ( v^n_i \left ( E^n_{i+1}-E^n_{i-1} \right ) \right ) + \left ( F_{i+1/2}-F_{i-1/2} \right ) \right ))]]  
+ [[latex($p^{n+1}_i=p^n_i - \frac{1}{2}\frac{\Delta t}{\Delta x} \lambda_{i} \left ( E^n_{i+1}-E^n_{i-1} \right ) $)]]
+
+ [[latex($e^{n+1}_i=e^n_i + \frac{1}{2}\frac{\Delta t}{\Delta x} \lambda_i \left ( 2 \frac{\kappa^n_{i,0P}}{\kappa^n_{i,0R}}-1 \right ) \left ( v^n_i \left ( E^n_{i+1}-E^n_{i-1} \right ) \right ) $)]]  
+
+ [[latex($E^{n+1}_i=E^n_i - \frac{\Delta t}{\Delta x} \left ( \frac{\lambda_i}{2} \left ( 2 \frac{\kappa^n_{i,0P}}{\kappa^n_{i,0R}}-1 \right ) \left ( v^n_i \left ( E^n_{i+1}-E^n_{i-1} \right ) \right ) + \left ( F_{i+1/2}-F_{i-1/2} \right ) \right )$)]]  
 
 where
-[[latex(F_{i+1/2}=\frac{3-R_{2,i+1/2}}{8} \left(v_{i}^n+v_{i+1}^n \right ) \left ( E^n_i+E^n_{i+1} \right ) )]]
+[[latex($F_{i+1/2}=\frac{3-R_{2,i+1/2}}{8} \left(v_{i}^n+v_{i+1}^n \right ) \left ( E^n_i+E^n_{i+1} \right ) $)]]
 
 where
 
-[[latex(R_{2,i+1/2} = \lambda_{i+1/2}+\lambda_{i+1/2}^2 R_{i+1/2}^2)]]
-and 
-[[latex(R_{i+1/2} = \frac{\left | E^n_{i+1}-E^n_{i} \right | }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )})]]
-
-and
-[[latex(\lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2} - \frac{1}{R_{i+1/2}} \right ) )]]
-
-and 
-[[latex(\kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,i+1}}{2} \mbox{ and } \lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2} - \frac{1}{R_{i+1/2}} \right ) )]]
-
-and 
-
-[[latex(\lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i} - \frac{1}{R_{i}} \right ) )]]
-
-and
-[[latex(R_{i} = \frac{\left | E^n_{i+1}-E^n_{i-1} \right | }{2 \kappa_{0R,i} E^n_{i}})]]
+[[latex($R_{2,i+1/2} = \lambda_{i+1/2}+\lambda_{i+1/2}^2 R_{i+1/2}^2$)]]
+and 
+[[latex($R_{i+1/2} = \frac{\left | E^n_{i+1}-E^n_{i} \right | }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )}$)]]
+
+and
+[[latex($\lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2} - \frac{1}{R_{i+1/2}} \right ) $)]]
+
+and 
+[[latex($\kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,i+1}}{2} \mbox{ and } \lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2} - \frac{1}{R_{i+1/2}} \right ) $)]]
+
+and 
+
+[[latex($\lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i} - \frac{1}{R_{i}} \right ) $)]]
+
+and
+[[latex($R_{i} = \frac{\left | E^n_{i+1}-E^n_{i-1} \right | }{2 \kappa_{0R,i} E^n_{i}}$)]]
 
 [[CollapsibleEnd()]]
@@ -238 +240 @@
 == Implicit Update 1 ==
 
-For now we will assume that \(\kappa_{0P}\) and \(\kappa_{0R}\) are constant over the implicit update and we will treat the energy as the total internal energy ignoring kinetic and magnetic contributions.  In this case we can solve the radiation energy equations:
-
-   [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)-cE) = \nabla \cdot \mathbf{F} + \kappa_{0P} (4 \pi B(T)-cE))]]   
-   [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} (4 \pi B(T)-cE))]]   
-
-where [[latex(\mathbf{F} = \frac{c\lambda}{\kappa_{0R}} \nabla E)]]
+For now we will assume that [[latex($\kappa_{0P}$)]] and [[latex($\kappa_{0R}$)]] are constant over the implicit update and we will treat the energy as the total internal energy ignoring kinetic and magnetic contributions.  In this case we can solve the radiation energy equations:
+
+   [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)-cE) = \nabla \cdot \mathbf{F} + \kappa_{0P} (4 \pi B(T)-cE)$)]]   
+   [[latex($\frac{\partial e}{\partial t} = - \kappa_{0P} (4 \pi B(T)-cE)$)]]   
+
+where [[latex($\mathbf{F} = \frac{c\lambda}{\kappa_{0R}} \nabla E$)]]
 
 === Expanding about e,,0,, ===
 
-Of course even if the opacity is independent of energy and radiation energy, the above combined system of equations is still non-linear due to the dependence on Temperature of the Planck Function \(B(T)\)
-
-However we can expand the Plank Function about \(e_0\) 
-
-[[latex(B(T) = B \left ( T_0+dT \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right | _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right ) )]]
+Of course even if the opacity is independent of energy and radiation energy, the above combined system of equations is still non-linear due to the dependence on Temperature of the Planck Function [[latex($B(T)$)]]
+
+However we can expand the Plank Function about [[latex($e_0$)]] 
+
+[[latex($B(T) = B \left ( T_0+dT \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right | _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right ) $)]]
 
 where
 
-[[latex(\Gamma = \frac{\partial T}{\partial e} = \frac{(\gamma-1)}{n k_B})]]
+[[latex($\Gamma = \frac{\partial T}{\partial e} = \frac{(\gamma-1)}{n k_B}$)]]
 
 Then the system of equations becomes
 
-   [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} \left [4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]]   
-   [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]]   
-
-which will be accurate as long as \(\left | 4\Gamma \frac{e-e_0}{T_0} \right | < \xi << 1\) or \(\Delta e = \left | e-e_0 \right | < \xi \frac{T_0}{4 \Gamma}\)
+   [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} \left [4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] $)]]   
+   [[latex($\frac{\partial e}{\partial t} = - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] $)]]   
+
+which will be accurate as long as [[latex($\left | 4\Gamma \frac{e-e_0}{T_0} \right | < \xi << 1$)]] or [[latex($\Delta e = \left | e-e_0 \right | < \xi \frac{T_0}{4 \Gamma}$)]]
 
 We can calculate the time scale for this to be true using the evolution equation for the energy density
 
-[[latex(\Delta e = \Delta t \kappa_{0P} \left | 4 \pi B_0 -cE \right | < \xi \frac{T_0}{4 \Gamma})]]
-
-which gives [[latex(\Delta t < \xi \frac{T_0}{4 \Gamma \kappa_{0P} \left | 4 \pi B_0 - cE \right | })]]
+[[latex($\Delta e = \Delta t \kappa_{0P} \left | 4 \pi B_0 -cE \right | < \xi \frac{T_0}{4 \Gamma}$)]]
+
+which gives [[latex($\Delta t < \xi \frac{T_0}{4 \Gamma \kappa_{0P} \left | 4 \pi B_0 - cE \right | }$)]]
 
 
@@ -275 +277 @@
 We can now discretize the equations
 
-   [[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha_{i+1/2} \left ( E^{*}_{i+1}-E^{*}_{i} \right ) - \alpha_{i-1/2} \left ( E^{*}_{i}-E^{*}_{i-1} \right ) \right ] - \epsilon E^{*}_i  + \phi e^{*}_i  + \theta_i - \phi_i e^n_i) )]]   
-   [[latex(e^{n+1}_i-e^{n}_i = \epsilon_i E^{*}_i  - \phi_i e^{*}_i  - \left(\theta_i-\phi_i e^n_i \right ) )]]   
+   [[latex($E^{n+1}_i-E^{n}_i = \left [ \alpha_{i+1/2} \left ( E^{*}_{i+1}-E^{*}_{i} \right ) - \alpha_{i-1/2} \left ( E^{*}_{i}-E^{*}_{i-1} \right ) \right ] - \epsilon E^{*}_i  + \phi e^{*}_i  + \theta_i - \phi_i e^n_i) $)]]   
+   [[latex($e^{n+1}_i-e^{n}_i = \epsilon_i E^{*}_i  - \phi_i e^{*}_i  - \left(\theta_i-\phi_i e^n_i \right ) $)]]   
 
 where the diffusion coefficient is given by 
 
-[[latex(\alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2}  \frac{c \lambda_{i+1/2}}{\kappa_{0R,i+1/2}} \mbox{ where } \kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,i+1}}{2} \mbox{ and } \lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2} - \frac{1}{R_{i+1/2}} \right ) )]]
+[[latex($\alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2}  \frac{c \lambda_{i+1/2}}{\kappa_{0R,i+1/2}} \mbox{ where } \kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,i+1}}{2} \mbox{ and } \lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2} - \frac{1}{R_{i+1/2}} \right ) $)]]
 and where
-[[latex(R_{i+1/2} = \frac{\left | E^n_{i+1}-E^n_{i} \right | }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )})]]
-
-
-and 
-
-[[latex(\epsilon_i=c\Delta t \kappa_{0P,i})]]
+[[latex($R_{i+1/2} = \frac{\left | E^n_{i+1}-E^n_{i} \right | }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )}$)]]
+
+
+and 
+
+[[latex($\epsilon_i=c\Delta t \kappa_{0P,i}$)]]
 
 represents the number of absorption/emissions during the time step
@@ -294 +296 @@
 and 
 
-[[latex(\phi_i = \epsilon_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) )]]
-
-[[latex(\theta_i = \epsilon_i \frac{4 \pi}{c} B \left ( T^n_i \right ) )]]
+[[latex($\phi_i = \epsilon_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) $)]]
+
+[[latex($\theta_i = \epsilon_i \frac{4 \pi}{c} B \left ( T^n_i \right ) $)]]
 
 
 and we can think of the radiative flux as
 
-[[latex(\frac{\Delta t}{\Delta x}\mathbf{F}_{i+1/2} = \alpha_{i+1/2} \left ( E^{*}_{i+1} - E^{*}_i \right ) )]]
+[[latex($\frac{\Delta t}{\Delta x}\mathbf{F}_{i+1/2} = \alpha_{i+1/2} \left ( E^{*}_{i+1} - E^{*}_i \right ) $)]]
 
 === Time Discretization ===
 
 Now all the terms on the right hand side that are linear in E or e have been written as E^*^ or e^*^ because there are different ways to approximate E^*^ (e^*^).  For Backward Euler we have
-[[latex(E^*_i = E^{n+1}_i)]]
+[[latex($E^*_i = E^{n+1}_i$)]]
 and for Crank Nicholson we have
-[[latex(E^*_i = \frac{1}{2} \left ( E^{n+1}_i + E^n_i \right ) )]]
+[[latex($E^*_i = \frac{1}{2} \left ( E^{n+1}_i + E^n_i \right ) $)]]
 or we can parameterize the solution
-[[latex(E^*_i = \psi E^{n+1}_i + \bar{\psi}E^n_i)]]
-where \(\bar{\psi} = 1-\psi\)
-
-Backward Euler has \(\psi=1\) and Crank Nicholson has \(\psi=1/2\)
-
-Forward Euler has \(\psi=0\)
+[[latex($E^*_i = \psi E^{n+1}_i + \bar{\psi}E^n_i$)]]
+where [[latex($\bar{\psi} = 1-\psi$)]]
+
+Backward Euler has [[latex($\psi=1$)]] and Crank Nicholson has [[latex($\psi=1/2$)]]
+
+Forward Euler has [[latex($\psi=0$)]]
 
 In any event in 1D we have the following matrix coefficients
 
-   [[latex(\left [ 1 + \psi \left( \alpha_{i+1/2} + \alpha_{i-1/2} + \epsilon_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha_{i-1/2} \right ) E^{n+1}_{i-1} - \left ( \psi \phi_i \right ) e^{n+1}_i=\left [ 1 - \bar{\psi} \left( \alpha_{i+1/2} + \alpha_{i-1/2} + \epsilon_i \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha_{i-1/2} \right ) E^{n}_{i-1} - \psi\phi_i e^n_i + \theta_i)]]   
-
-   [[latex(\left ( 1 +\psi \phi_i \right ) e^{n+1}_i - \left ( \psi \epsilon_i \right )E^{n+1}_i =\left ( 1 +\psi\phi_i \right ) e^n_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i-\theta_i )]]   
+   [[latex($\left [ 1 + \psi \left( \alpha_{i+1/2} + \alpha_{i-1/2} + \epsilon_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha_{i-1/2} \right ) E^{n+1}_{i-1} - \left ( \psi \phi_i \right ) e^{n+1}_i=\left [ 1 - \bar{\psi} \left( \alpha_{i+1/2} + \alpha_{i-1/2} + \epsilon_i \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha_{i-1/2} \right ) E^{n}_{i-1} - \psi\phi_i e^n_i + \theta_i$)]]   
+
+   [[latex($\left ( 1 +\psi \phi_i \right ) e^{n+1}_i - \left ( \psi \epsilon_i \right )E^{n+1}_i =\left ( 1 +\psi\phi_i \right ) e^n_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i-\theta_i $)]]   
 
 
 Now since the second equation has no spatial dependence, we can solve it for 
-   [[latex(\color{purple}{e^{n+1}_i = e^n_i + \frac{1}{ 1 +\psi \phi_i}\left \{ \left ( \psi \epsilon_i \right )E^{n+1}_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i-\theta_i \right \}} )]]   
+   [[latex($\color{purple}{e^{n+1}_i = e^n_i + \frac{1}{ 1 +\psi \phi_i}\left \{ \left ( \psi \epsilon_i \right)E^{n+1}_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i-\theta_i \right \}}$)]]   
 
 and plug the result into the first equation to get a matrix equation involving only one variable.
 
-   [[latex(\color{purple}{\left [ 1 + \psi \left( \alpha_{i+1/2} + \alpha_{i-1/2} + \frac{\epsilon_i}{ 1 +\psi \phi_i}\right ) \right ] E^{n+1}_i - \left ( \psi \alpha_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha_{i-1/2} \right ) E^{n+1}_{i-1} =\left [ 1 - \bar{\psi} \left( \alpha_{i+1/2} + \alpha_{i-1/2}  +\frac{\epsilon_i }{ 1 +\psi \phi_i} \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha_{i-1/2} \right ) E^{n}_{i-1} + \frac{\theta_i}{ 1 +\psi \phi_i}})]]   
+   [[latex($\color{purple}{\left [ 1 + \psi \left( \alpha_{i+1/2} + \alpha_{i-1/2} + \frac{\epsilon_i}{ 1 +\psi \phi_i}\right ) \right ] E^{n+1}_i - \left ( \psi \alpha_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha_{i-1/2} \right ) E^{n+1}_{i-1} =\left [ 1 - \bar{\psi} \left( \alpha_{i+1/2} + \alpha_{i-1/2}  +\frac{\epsilon_i }{ 1 +\psi \phi_i} \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha_{i-1/2} \right ) E^{n}_{i-1} + \frac{\theta_i}{ 1 +\psi \phi_i}}$)]]   
 
 
@@ -343 +345 @@
 For the initial solution vector, we can just use Edot from the parent update (or last time step if we are on the coarse grid) to guess E, and then we can solve for the new e given our guess at the new E using the same time stepping (Backward Euler, Crank Nicholson, etc...).
 
-[[latex(E^{n+1}_i = E^{n}_i+\dot{E}^{I}_i \Delta t)]]
-[[latex(e^{n+1}_i = e^n_i + \frac{1}{1+\psi \phi_i} \left \{ \left ( \psi \epsilon_i \right )E^{n+1}_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i-\theta_i \right \} )]]
+[[latex($E^{n+1}_i = E^{n}_i+\dot{E}^{I}_i \Delta t$)]]
+[[latex($e^{n+1}_i = e^n_i + \frac{1}{1+\psi \phi_i} \left \{ \left ( \psi \epsilon_i \right )E^{n+1}_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i-\theta_i \right \} $)]]
 
 == Coarse-Fine Boundaries ==
@@ -356 +358 @@
 === Open (Free streaming) boundaries ===
 We would like the radiation to leave at the free streaming limit.  So 
-[[latex(\frac{c \lambda}{\kappa_{0R}} \nabla E = \mathbf{F} = cE\mathbf{n} = \frac{c \lambda_g}{\kappa_g}\frac{ \left ( E-E_g \right ) }{\Delta x})]]
-
-Clearly if we set \(E_g = 0\) and \(\lambda_g =\kappa_g \Delta x\) we should get the correct flux.
+[[latex($\frac{c \lambda}{\kappa_{0R}} \nabla E = \mathbf{F} = cE\mathbf{n} = \frac{c \lambda_g}{\kappa_g}\frac{ \left ( E-E_g \right ) }{\Delta x}$)]]
+
+Clearly if we set [[latex($E_g = 0$)]] and [[latex($\lambda_g =\kappa_g \Delta x$)]] we should get the correct flux.
 
 This corresponds to an
-\(\alpha = c \frac{\Delta t}{\Delta x}\)
-
-So we would just modify \(\alpha\) and zero out the matrix coefficient to the ghost zone
+[[latex($\alpha = c \frac{\Delta t}{\Delta x}$)]]
+
+So we would just modify [[latex($\alpha$)]] and zero out the matrix coefficient to the ghost zone
 
 === Constant Slope Boundary ===
-Here we want the flux to be constant so energy does not pile up near the boundary.  If we cancel all derivative terms on both sides of the cell, this will effectively match the incoming flux with the outgoing flux.  This can also be done by setting \(\alpha_g = \alpha_i = 0\)
+Here we want the flux to be constant so energy does not pile up near the boundary.  If we cancel all derivative terms on both sides of the cell, this will effectively match the incoming flux with the outgoing flux.  This can also be done by setting [[latex($\alpha_g = \alpha_i = 0$)]]
 
 === Periodic Boundary ===
@@ -376 +378 @@
 
 === Reflecting/ZeroSlope Boundary ===
-Reflecting boundary should be fairly straightforward.  This an be achieved by setting \(\alpha_g = 0\) which zeros out any flux - and has the same effect as setting \(E^{*}_g=E^{*}_i\) or \(E^{n+1}_g=E^{n+1}_i \mbox{ and } E^{n}_g=E^{n}_i\)
+Reflecting boundary should be fairly straightforward.  This an be achieved by setting [[latex($\alpha_g = 0$)]] which zeros out any flux - and has the same effect as setting [[latex($E^{*}_g=E^{*}_i$)]] or [[latex($E^{n+1}_g=E^{n+1}_i \mbox{ and } E^{n}_g=E^{n}_i$)]]
 
 === Constant radiative flux ===
-To have a constant radiative flux we must zero out terms involving the gradient and just add \(F_0 \frac{\Delta t}{\Delta x}\) in the source vector
+To have a constant radiative flux we must zero out terms involving the gradient and just add [[latex($F_0 \frac{\Delta t}{\Delta x}$)]] in the source vector
 
 === Summary ===
 
-|| Numerical value  ||  Boundary                  ||  [[latex(E^{n+1}_{i+1})]]        ||  [[latex(E^{n+1}_i)]]                         ||   [[latex(E^{n+1}_{i-1})]]                       ||  [[latex(S)]] 
-||  0               ||  RAD_FREE_STREAMING        ||  [[latex(0)]]                    ||  [[latex(\psi c \frac{\Delta t}{\Delta x})]]  ||  [[latex(-\psi\alpha_{i-1/2})]]  ||  [[latex(-\bar{\psi}c \frac{\Delta t}{\Delta x} E^n_i + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right ))]]    || 
-||  1               ||  RAD_EXTRAPOLATE_FLUX     ||  [[latex(0)]]                    ||  [[latex(0)]]                                 ||  [[latex(0)]]  ||  [[latex(0)]]    ||
-||  2               ||  INTERNAL/PERIODIC         ||  [[latex(-\psi\alpha_{i+1/2})]]  ||  [[latex(\psi\alpha_{i+1/2})]]                ||  [[latex(-\psi\alpha_{i-1/2})]]  ||  [[latex(\bar{\psi}\alpha_{i+1/2} \left ( E^n_{i+1}-E^n_i \right ) + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right ))]]  ||
-||  3               ||  RAD_REFLECTING            ||  [[latex(0)]]                    ||  [[latex(0)]]                                 ||  [[latex(-\psi\alpha_{i-1/2})]]  ||  [[latex( \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right ))]]    ||
-||  4               ||  RAD_USERDEFINED_BOUNDARY/AMR_BOUNDARY  ||  [[latex(0)]]       ||  [[latex(\psi\alpha_{i+1/2})]]                ||  [[latex(-\psi\alpha_{i-1/2})]]  ||   [[latex(\bar{\psi}\alpha_{i+1/2} \left ( E^n_{i+1}-E^n_i \right ) - \psi \alpha_{i+1/2} E^{n+1}_{i+1} + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right ))]]   ||
-||  5               ||  RAD_USERDEFINED_FLUX      ||  [[latex(0)]]                    ||  [[latex(0)]]                                 ||   [[latex(-\psi\alpha_{i-1/2})]]  ||    [[latex(F_0 \frac{\Delta t}{\Delta x} + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right ))]]   ||
+|| Numerical value  ||  Boundary                  ||  [[latex($E^{n+1}_{i+1}$)]]        ||  [[latex($E^{n+1}_i$)]]                         ||   [[latex($E^{n+1}_{i-1}$)]]                       ||  [[latex($S$)]] 
+||  0               ||  RAD_FREE_STREAMING        ||  [[latex($0$)]]                    ||  [[latex($\psi c \frac{\Delta t}{\Delta x}$)]]  ||  [[latex($-\psi\alpha_{i-1/2}$)]]  ||  [[latex($-\bar{\psi}c \frac{\Delta t}{\Delta x} E^n_i + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right )$)]]    || 
+||  1               ||  RAD_EXTRAPOLATE_FLUX     ||  [[latex($0$)]]                    ||  [[latex($0$)]]                                 ||  [[latex($0$)]]  ||  [[latex($0$)]]    ||
+||  2               ||  INTERNAL/PERIODIC         ||  [[latex($-\psi\alpha_{i+1/2}$)]]  ||  [[latex($\psi\alpha_{i+1/2}$)]]                ||  [[latex($-\psi\alpha_{i-1/2}$)]]  ||  [[latex($\bar{\psi}\alpha_{i+1/2} \left ( E^n_{i+1}-E^n_i \right ) + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right )$)]]  ||
+||  3               ||  RAD_REFLECTING            ||  [[latex($0$)]]                    ||  [[latex($0$)]]                                 ||  [[latex($-\psi\alpha_{i-1/2}$)]]  ||  [[latex($ \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right )$)]]    ||
+||  4               ||  RAD_USERDEFINED_BOUNDARY/AMR_BOUNDARY  ||  [[latex($0$)]]       ||  [[latex($\psi\alpha_{i+1/2}$)]]                ||  [[latex($-\psi\alpha_{i-1/2}$)]]  ||   [[latex($\bar{\psi}\alpha_{i+1/2} \left ( E^n_{i+1}-E^n_i \right ) - \psi \alpha_{i+1/2} E^{n+1}_{i+1} + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right )$)]]   ||
+||  5               ||  RAD_USERDEFINED_FLUX      ||  [[latex($0$)]]                    ||  [[latex($0$)]]                                 ||   [[latex($-\psi\alpha_{i-1/2}$)]]  ||    [[latex($F_0 \frac{\Delta t}{\Delta x} + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right )$)]]   ||
 
 
@@ -401 +403 @@
 
 
-  [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P\color{green}{-\lambda \nabla E})]]   
-  [[latex(\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{-\kappa_{0P}(4 \pi B-cE)} \color{magenta}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E} \color{magenta}{-\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E})]]  
-  [[latex(\frac{\partial E}{\partial t}  \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} = \color{red}{\kappa_{0P} (4 \pi B-cE)} \color{magenta}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E} \color{magenta}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )} \color{magenta}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E}  )]]  
+  [[latex($\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P\color{green}{-\lambda \nabla E}$)]]   
+  [[latex($\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{-\kappa_{0P}(4 \pi B-cE)} \color{magenta}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E} \color{magenta}{-\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E}$)]]  
+  [[latex($\frac{\partial E}{\partial t}  \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} = \color{red}{\kappa_{0P} (4 \pi B-cE)} \color{magenta}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E} \color{magenta}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )} \color{magenta}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E}  $)]]  
 
 It is possible that with included the terms in magenta in a semi-implicit method, the dynamic diffusion regime may be stable...  In any event, it costs very little to add all of the terms in magenta to the implicit solve (using the old velocity).  Then the momentum update can be done explicitly - though using a time centered radiation field.
@@ -413 +415 @@
 In AstroBEAR this would look like:
 * Initialization
- * Prolongate, \(\rho\), \(\rho\mathbf{v}\), \(e\), \(E\), and \(\dot{E}\)
+ * Prolongate, [[latex($\rho$)]], [[latex($\rho\mathbf{v}$)]], [[latex($e$)]], [[latex($E$)]], and [[latex($\dot{E}$)]]
 * Step 1
- * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\) and do physical BC's
- * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}_1\)
- * Update \(E_{2\mbox{mbc}}\) using \(\dot{E}_{2\mbox{mbc}}\)
- * Update \(e_{2\mbox{mbc}}\) using \(E_{2\mbox{mbc}}\), \(\dot{E}_{2\mbox{mbc}}\), and \(e_{2\mbox{mbc}}\)
- * Update \(\dot{E}_0\) using pre IR and post IR \(E_0\)
- * Ghost \(e_{2\mbox{mbc}}\), \(E_{\mbox{mbc}+1}\), \(\dot{E}_{\mbox{mbc}+1}\)
+ * Overlap [[latex($\rho$)]], [[latex($\rho \mathbf{v} $)]] , [[latex($e$)]], [[latex($E$)]] and do physical BC's
+ * Do IR which updates [[latex($e_0$)]] and [[latex($E_0$)]] using [[latex($\rho_1$)]], [[latex($e_1$)]], [[latex($E_1$)]], and [[latex($\dot{E}_1$)]]
+ * Update [[latex($E_{2\mbox{mbc}}$)]] using [[latex($\dot{E}_{2\mbox{mbc}}$)]]
+ * Update [[latex($e_{2\mbox{mbc}}$)]] using [[latex($E_{2\mbox{mbc}}$)]], [[latex($\dot{E}_{2\mbox{mbc}}$)]], and [[latex($e_{2\mbox{mbc}}$)]]
+ * Update [[latex($\dot{E}_0$)]] using pre IR and post IR [[latex($E_0$)]]
+ * Ghost [[latex($e_{2\mbox{mbc}}$)]], [[latex($E_{\mbox{mbc}+1}$)]], [[latex($\dot{E}_{\mbox{mbc}+1}$)]]
  * Do first EH,,mbc,,
- * Do ER,,mbc,, --- Terms with \(\nabla E\) can be done without ghosting since EH did not change \(E\).  
- * Store \(\dot{E}\) in child arrays to be prolongated
+ * Do ER,,mbc,, --- Terms with [[latex($\nabla E$)]] can be done without ghosting since EH did not change [[latex($E$)]].  
+ * Store [[latex($\dot{E}$)]] in child arrays to be prolongated
 * Step 2
- * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\) and do physical BC's
- * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}_1\)
- * Update \(\dot{E}_0\) using pre IR and post IR \(E_0\)
- * Update \(E_{1}\) using \(\dot{E}_{1}\)
- * Ghost \(e_{mbc}\), \(E_{1}\), \(\dot{E}_{1}\)
+ * Overlap [[latex($\rho$)]], [[latex($\rho \mathbf{v} $)]] , [[latex($e$)]], [[latex($E$)]] and do physical BC's
+ * Do IR which updates [[latex($e_0$)]] and [[latex($E_0$)]] using [[latex($\rho_1$)]], [[latex($e_1$)]], [[latex($E_1$)]], and [[latex($\dot{E}_1$)]]
+ * Update [[latex($\dot{E}_0$)]] using pre IR and post IR [[latex($E_0$)]]
+ * Update [[latex($E_{1}$)]] using [[latex($\dot{E}_{1}$)]]
+ * Ghost [[latex($e_{mbc}$)]], [[latex($E_{1}$)]], [[latex($\dot{E}_{1}$)]]
  * Do second EH,,0,,
- * Do ER,,0,, --- Terms with \(\nabla E\) can be done without ghosting since EH did not change \(E\).  
+ * Do ER,,0,, --- Terms with [[latex($\nabla E$)]] can be done without ghosting since EH did not change [[latex($E$)]].  
 
 
@@ -440 +442 @@
 The extra terms in the explicit update due to radiation energy are as follows:
 
- [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) =-\lambda \nabla E)]]   
+ [[latex($\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) =-\lambda \nabla E$)]]   
 
 These can be discretized as follows:
 
- [[latex(p^{n+1}_i=p^n_i - \frac{1}{4}\frac{\Delta t}{\Delta x} \lambda_{i} \left ( \left ( E^{n}_{i+1}+E^{n+1}_{i+1} \right ) - \left ( E^n_{i-1}+E^{n+1}_{i-1} \right ) \right ) )]]
-
-[[latex(\lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i} - \frac{1}{R_{i}} \right ) )]]
-
-and
-[[latex(R_{i} = \frac{\left | E^n_{i+1}-E^n_{i-1} \right | }{2 \kappa_{0R,i} E^n_{i}})]]
+ [[latex($p^{n+1}_i=p^n_i - \frac{1}{4}\frac{\Delta t}{\Delta x} \lambda_{i} \left ( \left ( E^{n}_{i+1}+E^{n+1}_{i+1} \right ) - \left ( E^n_{i-1}+E^{n+1}_{i-1} \right ) \right ) $)]]
+
+[[latex($\lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i} - \frac{1}{R_{i}} \right ) $)]]
+
+and
+[[latex($R_{i} = \frac{\left | E^n_{i+1}-E^n_{i-1} \right | }{2 \kappa_{0R,i} E^n_{i}}$)]]
 
 [[CollapsibleEnd()]]
@@ -456 +458 @@
 == Implicit Update 2 ==
 
-For now we will assume that \(\kappa_{0P}\) and \(\kappa_{0R}\) are constant over the implicit update and we will treat the energy as the total internal energy ignoring kinetic and magnetic contributions.  In this case we can solve the radiation energy equations:
-
-  [[latex(\frac{\partial e}{\partial t}  = -\kappa_{0P}(4 \pi B-cE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E )]]  
-  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)-cE) -\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E + \frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) )]]  
+For now we will assume that [[latex($\kappa_{0P}$)]] and [[latex($\kappa_{0R}$)]] are constant over the implicit update and we will treat the energy as the total internal energy ignoring kinetic and magnetic contributions.  In this case we can solve the radiation energy equations:
+
+  [[latex($\frac{\partial e}{\partial t}  = -\kappa_{0P}(4 \pi B-cE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E $)]]  
+  [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)-cE) -\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E + \frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) $)]]  
 
 which we can also write as
 
-  [[latex(\frac{\partial e}{\partial t}  = f \left ( e \right ) + g \left ( E,\nabla E \right ) )]]
-  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E  \right ) - f \left ( e \right ) - g \left (e,E, \nabla E \right )  )]]  
+  [[latex($\frac{\partial e}{\partial t}  = f \left ( e \right ) + g \left ( E,\nabla E \right ) $)]]
+  [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E  \right ) - f \left ( e \right ) - g \left (e,E, \nabla E \right )  $)]]  
   where
-[[latex( f \left ( e \right ) = -4 \pi \kappa_{0P} B(T(e)) )]]
+[[latex($ f \left ( e \right ) = -4 \pi \kappa_{0P} B(T(e)) $)]]
  and
-  [[latex( g \left (e,E, \nabla E \right ) = \kappa_{0P}(cE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E )]]
+  [[latex($ g \left (e,E, \nabla E \right ) = \kappa_{0P}(cE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E $)]]
 
 Now we can linearize f about e,,0,,
-[[latex( f \left ( e \right ) = f \left ( e_0 \right ) + \frac{\partial f}{\partial e} \left ( e - e_0 \right ) )]]
+[[latex($ f \left ( e \right ) = f \left ( e_0 \right ) + \frac{\partial f}{\partial e} \left ( e - e_0 \right ) $)]]
 
 so that the first equation can be written as 
 
-  [[latex(\frac{\partial e}{\partial t}  =  f \left ( e_0 \right ) + g \left ( E,\nabla E \right )  +  \frac{\partial f}{\partial e} \left ( e - e_0 \right ) )]]
+  [[latex($\frac{\partial e}{\partial t}  =  f \left ( e_0 \right ) + g \left ( E,\nabla E \right )  +  \frac{\partial f}{\partial e} \left ( e - e_0 \right ) $)]]
 
 and then discretized as 
 
-  [[latex(e^{n+1}_i-e^{n}_i = \Delta t  \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) +  \psi \phi  e^n_i -  \psi \phi e^{n+1}_i )]]
+  [[latex($e^{n+1}_i-e^{n}_i = \Delta t  \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) +  \psi \phi  e^n_i -  \psi \phi e^{n+1}_i $)]]
 
 where
 
-[[latex(\phi = -\Delta t \frac{\partial f}{\partial e} = 4 \pi \kappa_{0P} \Delta t \frac{\partial B(T(e))}{\partial e})]]
+[[latex($\phi = -\Delta t \frac{\partial f}{\partial e} = 4 \pi \kappa_{0P} \Delta t \frac{\partial B(T(e))}{\partial e}$)]]
 
 which can be solved for 
-[[latex(e^{n+1}_i = e^{n}_i + \frac{\Delta t}{1 + \psi \phi} \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) )]]
+[[latex($e^{n+1}_i = e^{n}_i + \frac{\Delta t}{1 + \psi \phi} \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) $)]]
 
 or
 
-[[latex(e^{n+1}_i = e^{n}_i + \frac{1}{1 + \psi \phi_i} \left ( -\theta_i + \epsilon_i E^*_i + \omega_{x,i} v^n_{x,i} \left ( E^*_{i+1}-E^*_{i-1} \right ) - \xi_i E^* \right ) )]]
+[[latex($e^{n+1}_i = e^{n}_i + \frac{1}{1 + \psi \phi_i} \left ( -\theta_i + \epsilon_i E^*_i + \omega_{x,i} v^n_{x,i} \left ( E^*_{i+1}-E^*_{i-1} \right ) - \xi_i E^* \right ) $)]]
 
 
 Then if we take the semi-discretized equation for E 
 
-  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c \lambda}{\kappa_{0R}} \nabla E - \nabla \cdot \left ( \frac{3-R_2}{2} \mathbf{v} E \right ) - f \left ( e^n_i \right ) - g \left ( E, \nabla E \right ) - \frac{1}{\Delta t} \left ( \psi \phi_i  e^n_i -  \psi \phi_i e^{n+1}_i \right ) )]]  
+  [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c \lambda}{\kappa_{0R}} \nabla E - \nabla \cdot \left ( \frac{3-R_2}{2} \mathbf{v} E \right ) - f \left ( e^n_i \right ) - g \left ( E, \nabla E \right ) - \frac{1}{\Delta t} \left ( \psi \phi_i  e^n_i -  \psi \phi_i e^{n+1}_i \right ) $)]]  
 
 and then plugin the solution for e^n+1^,,i,, we get
 
-  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - f \left (e^n_i \right ) - g \left (E, \nabla E \right ) - \frac{1}{\Delta t} \left ( \psi \phi_i e^n_i - \psi \phi_i e^n_i - \frac{\psi \phi_i }{1+\psi \phi_i} \left ( f \left ( e^n_i \right ) + g \left (E,\nabla E \right ) \right ) \right ) )]]  
+  [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - f \left (e^n_i \right ) - g \left (E, \nabla E \right ) - \frac{1}{\Delta t} \left ( \psi \phi_i e^n_i - \psi \phi_i e^n_i - \frac{\psi \phi_i }{1+\psi \phi_i} \left ( f \left ( e^n_i \right ) + g \left (E,\nabla E \right ) \right ) \right ) $)]]  
 
 which simplifies to 
 
-  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - \frac{1}{1+\psi \phi_i} \left ( f \left ( e^n_i \right ) + g \left ( E,\nabla E \right ) \right ) )]]  
-
-Now we have 1 equation with 1 variable that we can solve implicitly using hypre, and then we can use \(E^{n+1}\) and \(E^n\) to construct \(E^*\) which we can plug into the equation for \(e^{n+1}\)
+  [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - \frac{1}{1+\psi \phi_i} \left ( f \left ( e^n_i \right ) + g \left ( E,\nabla E \right ) \right ) $)]]  
+
+Now we have 1 equation with 1 variable that we can solve implicitly using hypre, and then we can use [[latex($E^{n+1}$)]] and [[latex($E^n$)]] to construct [[latex($E^*$)]] which we can plug into the equation for [[latex($e^{n+1}$)]]
 
 
@@ -512 +514 @@
 Expanding 
 
-[[latex(B(T(e)) = B \left ( T_0+dT(e) \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right | _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right ) )]]
+[[latex($B(T(e)) = B \left ( T_0+dT(e) \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right | _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right ) $)]]
 
 where
 
-[[latex(\Gamma = \frac{\partial T}{\partial e} = \frac{(\gamma-1)}{n k_B})]]
-
-and we can identify [[latex(\phi = -\Delta t \frac{\partial f}{\partial e} = 4 \pi \kappa_{0P} \Delta t \frac{\partial B}{\partial e} = 16 \pi \kappa_{0P} \Delta t B_0 \frac{\Gamma}{T_0})]]
+[[latex($\Gamma = \frac{\partial T}{\partial e} = \frac{(\gamma-1)}{n k_B}$)]]
+
+and we can identify [[latex($\phi = -\Delta t \frac{\partial f}{\partial e} = 4 \pi \kappa_{0P} \Delta t \frac{\partial B}{\partial e} = 16 \pi \kappa_{0P} \Delta t B_0 \frac{\Gamma}{T_0}$)]]
 
 Then the equation for e becomes
 
-   [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E)]]   
-
-which will be accurate as long as \(4\Gamma \frac{\left | e-e_0 \right | }{T_0} < \xi << 1\) or \(\left | \Delta e \right | = \left | e-e_0 \right | < \xi \frac{T_0}{4 \Gamma}\)
+   [[latex($\frac{\partial e}{\partial t} = - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E$)]]   
+
+which will be accurate as long as [[latex($4\Gamma \frac{\left | e-e_0 \right | }{T_0} < \xi << 1$)]] or [[latex($\left | \Delta e \right | = \left | e-e_0 \right | < \xi \frac{T_0}{4 \Gamma}$)]]
 
 We can calculate the time scale for this to be true using the evolution equation for the energy density
 
-[[latex(\left | e-e_0 \right | = \left | \Delta e \right | = \Delta t \left | \frac{\partial e}{\partial t} \right | < \xi \frac{T_0}{4 \Gamma})]]
-
-which gives [[latex(\Delta t < \xi \frac{T_0}{4 \Gamma \left | \frac{\partial e}{\partial t} \right | } = \xi \frac{T_0}{4 \Gamma \left | - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E  \right | } )]]
+[[latex($\left | e-e_0 \right | = \left | \Delta e \right | = \Delta t \left | \frac{\partial e}{\partial t} \right | < \xi \frac{T_0}{4 \Gamma}$)]]
+
+which gives [[latex($\Delta t < \xi \frac{T_0}{4 \Gamma \left | \frac{\partial e}{\partial t} \right | } = \xi \frac{T_0}{4 \Gamma \left | - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E  \right | } $)]]
 
 or in discretized form
 
-[[latex(\Delta t < \xi \frac{\theta_i}{\phi_i \frac{1}{\Delta t} \left | \left ( -\theta_i + \epsilon_i E^*_i + \omega_{x,i} v^n_{x,i} \left ( E^*_{i+1}-E^*_{i-1} \right ) - \xi_i E^* \right ) \right |} )]]
+[[latex($\Delta t < \xi \frac{\theta_i}{\phi_i \frac{1}{\Delta t} \left | \left ( -\theta_i + \epsilon_i E^*_i + \omega_{x,i} v^n_{x,i} \left ( E^*_{i+1}-E^*_{i-1} \right ) - \xi_i E^* \right ) \right |} $)]]
 
 
 === Implicit Discretization 2 ===
 Now we can discretize 
-[[latex(g(E*, \nabla E*) =  \kappa_{0P}cE^* + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E^* -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E^*)]]
+[[latex($g(E*, \nabla E*) =  \kappa_{0P}cE^* + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E^* -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E^*$)]]
 
 as
 
-[[latex(g = \epsilon_i \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) + \omega_{x,i} \left ( \psi E^{n+1}_{i+1} - \psi E^{n+1}_{i-1} + \bar{\psi} E^n_{i+1}- \bar{\psi} E^n_{i-1} \right ) - \xi_i \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) )]]
+[[latex($g = \epsilon_i \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) + \omega_{x,i} \left ( \psi E^{n+1}_{i+1} - \psi E^{n+1}_{i-1} + \bar{\psi} E^n_{i+1}- \bar{\psi} E^n_{i-1} \right ) - \xi_i \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) $)]]
 
 which along with the other terms gives
-{{{
-#!latex
+[[latex($
 \begin{eqnarray}
 E^{n+1}_i-E^{n}_i & = & \left [ \alpha_{i+1/2} \left ( \psi E^{n+1}_{i+1} + \bar{\psi} E^{n}_{i+1}- \psi E^{n+1}_{i} - \bar{\psi} E^n_{i} \right ) - \alpha_{i-1/2} \left ( \psi  E^{n+1}_{i} + \bar{\psi} E^{n}_i - \psi E^{n+1}_{i-1} - \bar{\psi}E^{n}_{i-1} \right ) \right ] \\
@@ -553 +554 @@
  & - & \frac{1}{1+\psi \phi_i}  \left [ - \theta_i  + \epsilon_i \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) + \omega_{x,i} v^n_{x,i} \left ( \psi E^{n+1}_{i+1} + \bar{\psi} E^n_{i+1} - \psi E^{n+1}_{i-1} - \bar{\psi} E^n_{i-1} \right ) - \xi_i \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) \right ] \\
 \end{eqnarray}
-}}}
+$)]]
 
 where the diffusion coefficient is given by 
 
-[[latex(\alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2}  \frac{c \lambda_{i+1/2}}{\kappa_{0R,i+1/2}} )]]  
+[[latex($\alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2}  \frac{c \lambda_{i+1/2}}{\kappa_{0R,i+1/2}} $)]]  
 
 and where
 
-[[latex(\zeta_{i+1/2}= \frac{\Delta t}{\Delta x}\frac{3-R_{2,i+1/2}}{4} )]]
-
-and 
-
-[[latex(\epsilon_i=c\Delta t \kappa_{0P,i})]]
-
-and 
-
-[[latex(\phi_i = \Delta t 4 \pi \kappa_{0P,i} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) )]]
-
-and
-
-[[latex(\theta_i = - \Delta t f(e^n_i) =  \Delta t 4 \pi \kappa_{0P,i} B \left ( T^n_i \right ) )]]
-
-and
-
-[[latex(\omega_{x,i} = \frac{\lambda_{x,i} \Delta t}{\Delta x} \left ( \frac{\kappa_{0P,i}}{\kappa_{0R,i}}-\frac{1}{2} \right ))]]
-
-and
-
-[[latex(\xi_i = \Delta t \frac{3-R_{2,i}}{2}\kappa_{0P,i}\frac{v_i^2}{c} )]]
-and
-
-where [[latex( \kappa_{0R,i+1/2} = \frac{\kappa_{0R,i}+\kappa_{0R,i+1}}{2} )]]
-
-and 
-
-[[latex(R_{i+1/2} = \frac{\left | E^n_{i+1}-E^n_{i} \right | }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )})]]
-
-and
-
-[[latex(\lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2} - \frac{1}{R_{i+1/2}} \right ) )]]
-
-and
-
-[[latex(R_{2,i+1/2} = \lambda_{i+1/2}+\lambda_{i+1/2}^2 R_{i+1/2}^2)]]
-
-and
-
-
-[[latex(R_{i} = \frac{\left | E^n_{i+1}-E^n_{i-1} \right | }{2 \kappa_{0R,i} E^n_{i}})]]
-
-and
-
-[[latex(\lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i} - \frac{1}{R_{i}} \right ) )]]
-
-and
-
-[[latex(R_{2,i} = \lambda_{i}+\lambda_{i}^2 R_{i}^2)]]
+[[latex($\zeta_{i+1/2}= \frac{\Delta t}{\Delta x}\frac{3-R_{2,i+1/2}}{4} $)]]
+
+and 
+
+[[latex($\epsilon_i=c\Delta t \kappa_{0P,i}$)]]
+
+and 
+
+[[latex($\phi_i = \Delta t 4 \pi \kappa_{0P,i} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) $)]]
+
+and
+
+[[latex($\theta_i = - \Delta t f(e^n_i) =  \Delta t 4 \pi \kappa_{0P,i} B \left ( T^n_i \right ) $)]]
+
+and
+
+[[latex($\omega_{x,i} = \frac{\lambda_{x,i} \Delta t}{\Delta x} \left ( \frac{\kappa_{0P,i}}{\kappa_{0R,i}}-\frac{1}{2} \right )$)]]
+
+and
+
+[[latex($\xi_i = \Delta t \frac{3-R_{2,i}}{2}\kappa_{0P,i}\frac{v_i^2}{c} $)]]
+and
+
+where [[latex($ \kappa_{0R,i+1/2} = \frac{\kappa_{0R,i}+\kappa_{0R,i+1}}{2} $)]]
+
+and 
+
+[[latex($R_{i+1/2} = \frac{\left | E^n_{i+1}-E^n_{i} \right | }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )}$)]]
+
+and
+
+[[latex($\lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2} - \frac{1}{R_{i+1/2}} \right ) $)]]
+
+and
+
+[[latex($R_{2,i+1/2} = \lambda_{i+1/2}+\lambda_{i+1/2}^2 R_{i+1/2}^2$)]]
+
+and
+
+
+[[latex($R_{i} = \frac{\left | E^n_{i+1}-E^n_{i-1} \right | }{2 \kappa_{0R,i} E^n_{i}}$)]]
+
+and
+
+[[latex($\lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i} - \frac{1}{R_{i}} \right ) $)]]
+
+and
+
+[[latex($R_{2,i} = \lambda_{i}+\lambda_{i}^2 R_{i}^2$)]]
 
 Which we can arrange into the following form
 
-{{{
-#!latex
+[[latex($
 \begin{eqnarray}
  & \left ( 1 + \psi \left ( \alpha_{i+1/2} +  \alpha_{i-1/2} + \zeta_{i+1/2} v^n_{x,i+1/2} - \zeta_{i-1/2} v^n_{x,i-1/2} + \frac{\epsilon_i - \xi_i}{1+\psi \phi_i} \right ) \right ) E^{n+1}_i   \\
@@ -624 +624 @@
 + & \frac{\theta_i}{1+\psi \phi_i}   \\
 \end{eqnarray}
-}}}
+$)]]
 
 === 2D etc... ===
@@ -633 +633 @@
 === Initial solution vector ===
 For the initial solution vector, we can just use Edot from the parent update (or last time step if we are on the coarse grid) to guess E
-[[latex(E^{n+1}_i = E^{n}_i+\dot{E}^{n}_i \Delta t)]]
+[[latex($E^{n+1}_i = E^{n}_i+\dot{E}^{n}_i \Delta t$)]]
 
 == Coarse-Fine Boundaries ==
@@ -642 +642 @@
 
 == Physical Boundary Conditions ==
-For each boundary type we need to specify \(E^n_g, E^{n+1}_g, \kappa_g, \mbox { and } v_g\) in terms of other quantities (ie including E^n+1^_i).  The v,,g,, and kappa,,g,, terms should come from the hydro boundary conditions so we just need an equation for E^n^,,g,,
+For each boundary type we need to specify [[latex($E^n_g, E^{n+1}_g, \kappa_g, \mbox { and } v_g$)]] in terms of other quantities (ie including E^n+1^_i).  The v,,g,, and kappa,,g,, terms should come from the hydro boundary conditions so we just need an equation for E^n^,,g,,
 
 
@@ -648 +648 @@
 We would like the radiation to leave at the free streaming limit.  But we could have to modify the opacity in the ghost zone to keep the radiation energy positive
 
-[[latex(\frac{c \lambda}{\kappa_{0R}} \nabla E = \mathbf{F} = cE\mathbf{n} = \frac{c \lambda_g}{\kappa_g}\frac{ \left ( E-E_g \right ) }{\Delta x})]]
-
-Clearly if we set \(E_g = 0\) and \(\kappa_g << \frac{1}{\Delta x} \) we should get \(\lambda = \kappa_g \Delta x\) and a free streaming flux of \(c E \mathbf{n}\) 
+[[latex($\frac{c \lambda}{\kappa_{0R}} \nabla E = \mathbf{F} = cE\mathbf{n} = \frac{c \lambda_g}{\kappa_g}\frac{ \left ( E-E_g \right ) }{\Delta x}$)]]
+
+Clearly if we set [[latex($E_g = 0$)]] and [[latex($\kappa_g << \frac{1}{\Delta x} $)]] we should get [[latex($\lambda = \kappa_g \Delta x$)]] and a free streaming flux of [[latex($c E \mathbf{n}$)]] 
 
 which also corresponds to an
-\(\alpha = c \frac{\Delta t}{\Delta x}\)
+[[latex($\alpha = c \frac{\Delta t}{\Delta x}$)]]
 
 === Constant Slope Boundary ===
-[[latex(E_g = 2 E_i - E_{i+1})]]
+[[latex($E_g = 2 E_i - E_{i+1}$)]]
 
 === Periodic Boundary ===
@@ -667 +667 @@
 === Reflecting/ZeroSlope Boundary ===
 
-[[latex(E_g = E_i)]]
+[[latex($E_g = E_i$)]]
 
 === Constant radiative flux ===
 
-[[latex(E_g = E_i - F_0 \frac{\kappa_g \Delta x}{c \lambda_g})]]
-
-[[CollapsibleEnd()]]
+[[latex($E_g = E_i - F_0 \frac{\kappa_g \Delta x}{c \lambda_g}$)]]
+
+[[CollapsibleEnd()]]