Most of what follows is taken from ​Krumholz et al. 2007

Physics of Flux Limited Diffusion

Physics of Radiation Transfer

tau = l \kappa=\frac{l}{\lambda_p	beta = \frac{u}{c
tau << 1	streaming limit
tau >> 1 \mbox{, } \beta \tau <<	static diffusion limit
tau >> 1 \mbox{, } \beta \tau >>	dynamic diffusion limit

Equations of Radiation Hydrodynamics

frac{\partial \rho}{\partial t} + \nabla \cdot \left ( \rho \mathbf{v} \right ) =

frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P+\mathbf{G

frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = c G^

frac{\partial E}{\partial t} + \nabla \cdot \mathbf{F} = -c G_0

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

E=\int_0^\infty d \nu \int d \Omega I(\mathbf{n}, \nu

mathbf{F}=\int_0^\infty d \nu \int d \Omega \mathbf{n} I(\mathbf{n}, \nu

\mathbf{P}=\int_0^\infty d \nu \int d \Omega \mathbf{nn} I(\mathbf{n}, \nu

and the radiation 4-momentum is given by

G^0 = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)-\eta(\mathbf{n},v) \right

\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 ^{ij}=\delta^{ij} 1/3 but in general ^{ij}=f^{ij} where 'f' is the Eddington Tensor.

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

^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^

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

kappa_{0P	comoving-frame Planck function weighted opacity
kappa_{0E	comoving-frame radiation energy weighted opacity
kappa_{0F	comoving-frame radiation flux weighted opacity

• If the radiation has a blackbody spectrum then kappa_{0E}=\kappa_{0P
• If the radiation is optically thick, then

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 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

Flux limited diffusion

The flux limited diffusion approximation drops the radiation momentum equation in favor of mathbf{F}_0=-\frac{c\lambda}{\kappa_{0R}}\nabla E_

where lambd is the flux-limiter

lambda = \frac{1}{R} \left ( \coth R - \frac{1}{R} \right ) =\frac{|\nabla E_0|}{\kappa_{0R}E_0

which corresponds to a pressure tensor

mathbf{P}_0=\frac{E_0}{2}[(1-R_2)\mathbf{I}+(3R_2-1)\mathbf{n}_0\mathbf{n}_0 _2=\lambda+\lambda^2R^

If we Lorentz boost the comoving terms into the lab frame and keep terms necessary to maintain accuracy we get:

^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 ) 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

Numerics of Flux Limited Diffusion

Numerics of Flux Limited Diffusion

If we plug the expressions for the radiation 4-momentum back into the gas equations and keep terms necessary to maintain accuracy we get:

frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P\color{green}{-\lambda \nabla E 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 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 v²/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).

Operator splitting

Krumholz et al. perform Implicit Radiative, Explicit Hydro, Explicit Radiative In AstroBEAR this would look like:

• Initialization
  • Prolongate, d, p, e, E, Edot
• Step 1
  • Overlap d, p, e, E and do physical BC's
  • Do IR which updates e₀, and E₀ using d₁, e₁, E₁
  • Update E_2*mbc using Edot_2*mbc
  • Update e_2*mbc using E_2*mbc, Edot_2*mbc, and e_2*mbc
  • Update Edot₀ using pre IR and post IR E₀
  • Ghost e_2*mbc, E_mbc+1, Edot_mbc+1
  • Do first EH_mbc
  • Do ER_mbc —- Terms with grad E can be done without ghosting since EH did not change E. The del dot vE term needs time centered face centered velocities which can be stored during the hydro update.
  • Store Edot in child arrays to be prolongated
• Step 2
  • Overlap d, p, e, E, and do physical BCs
  • Do IR which updates e₀, and E₀ using d₁, e₁, E₁
  • Update Edot₀ using pre IR and post IR E₀
  • Update E₁ using Edot₁
  • Ghost e_mbc, E₁, Edot₁
  • Do second EH₀
  • Do ER₀ —- Terms with grad E can be done without ghosting since EH did not change E. The del dot vE term needs time centered face centered velocities which can be stored during the hydro update.

Explcit Update

The extra terms in the explicit update due to radiation energy are as follows:

frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) =-\lambda \nabla frac{\partial e}{\partial t} = \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E 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:

^n+1_i=p^n_i - \frac{1}{2}\frac{\Delta t}{\Delta x} \lambda_{i} (E^n_{i+1}-E^n_{i-1

^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_x^n_i \left ( E^n_{i+1}-E^n_{i-1} \right ) \right ]

^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_x^n_i \left ( E^n_{i+1}-E^n_{i-1} \right ) \right ] + \left ( F_{i+1/2}-F_{i-1/2} \right ) \right

Implicit Update

Implicit Equation

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:

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 frac{\partial e}{\partial t} = - \kappa_{0P} (4 \pi B(T)-cE

where mathbf{F} = \frac{c\lambda}{\kappa_{0R}} \nabla

Expanding about e₀

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 (T

If we ignore the changes in the Temperature due to heating during the implicit step which would feed back into the source function. We can improve this by writing

(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

Gamma = \frac{\partial T}{\partial e} = \frac{(\gamma-1)}{n k_B

Then the system of equations becomes

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 ] 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 \Gamma \frac{e-e_0}{T_0} < \xi << or Delta e = e-e_0 < \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

Delta e = -\Delta t \kappa_{0P} \left [ 4 \pi B_0 -cE \right ] < \xi \frac{T_0}{4 \Gamma

which gives Delta t < \xi \frac{T_0}{4 \Gamma \kappa_{0P} \left ( 4 \pi B_0 - cE \right )

Implicit Discretization

Which we can discretize for (1D) as

^{n+1}_i-E^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{*}_{i+1}-E^{*}_{i} \right ) - \alpha^n_{i-1/2} \left ( E^{*}_{i}-E^{*}_{i-1} \right ) \right ] - \epsilon^n_i E^{*}_i + \phi^n_i e^{*}_i + \theta^n_i) ^{n+1}_i-e^{n}_i = \epsilon^n_i E^{*}_i - \phi^n_i e^{*}_i - \theta^n_i

where the diffusion coefficient is given by

alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2} \frac{c \lambda_{i+1/2}}{\kappa_{i+1/2}} \mbox{ where } \kappa_{i+1/2} = \frac{\kappa^n_{i}+\kappa^n_{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 _{i+1/2} = \frac{\left | E^n_{i+1}-E^n_{i} \right | }{2 \kappa_{i+1/2} \left ( E^n_i+E^n_{i+1} \right )

and

epsilon^n_i=c\Delta t \kappa^n_{0P,i

represents the number of absorption/emissions during the time step

and

phi = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right )

theta = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( 1 - 4\Gamma \frac{e^n_i}{T^n_i} \right )

and we can think of the radiative flux as

frac{\Delta t}{\Delta x}\mathbf{F}^n_{i+1/2} = \alpha^n_{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 ^*_i = E^{n+1}_ and for Crank Nicholson we have ^*_i = \frac{1}{2} \left ( E^{n+1}_i + E^n_i \right ) or we can parameterize the solution ^*_i = \psi E^{n+1}_i + \bar{\psi}E^n_ where bar{\psi} = 1-\ps

Backward Euler has psi= and Crank Nicholson has psi=1/

Forward Euler has psi=

In any event in 1D we have the following matrix coefficients

left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} - \left ( \psi \phi^n_i \right ) e^{n+1}_i=\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha^n_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha^n_{i-1/2} \right ) E^{n}_{i-1} +\bar{\psi}\phi^n_i e^n_i + \theta^n_ left ( 1 +\psi \phi^n_i \right ) e^{n+1}_i - \left ( \psi \epsilon^n_i \right )E^{n+1}_i =\left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n

Now since the second equation has no spatial dependence, we can solve it for

color{purple}{e^{n+1}_i = \frac{1}{ 1 +\psi \phi^n_i}\left \{ \left ( \psi \epsilon^n_i \right )E^{n+1}_i + \left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n \right \}}

and plug the result into the first equation to get a matrix equation involving only one variable.

color{purple}{\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \frac{\epsilon^n_i}{ 1 +\psi \phi^n_i}\right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} =\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} +\frac{\epsilon^n_i }{ 1 +\psi \phi^n_i} \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha^n_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha^n_{i-1/2} \right ) E^{n}_{i-1} + \frac{ \phi^n_i}{ 1 +\psi \phi^n_i} e^n_i+ \frac{1}{ 1 +\psi \phi^n_i}\theta^i_n

If we ignore the change in the Planck function due to heating during the implicit solve, it is equivalent to setting psi \phi = This gives the following equations:

left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} =\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^n_i+ \left ( \bar{\psi} \alpha^n_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha^n_{i-1/2} \right ) E^{n}_{i-1} +\phi^n_i e^n_i + \theta^n_ ^{n+1}_i = e^n_i + \epsilon^n_i \left [ \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) - \frac{4 \pi}{c} B \left ( T^n_i \right ) \right ]

In this case the first equation decouples and can be solved on it's own, and then the solution plugged back into the second equation to solve for the new energy.

2D etc…

For 2D or 3D we have more connections to add to the matrix elements but it is very straight forward… There will be additional alpha terms for each dimension, but everything else stays the same.

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, 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…).

^{n+1}_i = E^{n}_i+\dot{E}^{n}_i \Delta ^{n+1}_i = \frac{1}{1+\psi \phi^n_i} \left \{ \left ( \psi \epsilon^n_i \right )E^{n+1}_i + \left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n \right \}

Coarse-Fine Boundaries

Since we are doing our implicit solves first, we can use time interpolated solutions for the implicit solve for non-refined ghost zones. To do this we just need Edot. The opacities etc… in the ghost zones can be obtained from the hydro terms.

And the radiative implicit heating in coarse ghost cells can be done along with the initial solution vector so they are available for the hydro update.

Physical Boundary Conditions

Open boundaries

We would like the radiation to leave at the free streaming limit. So 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 _g = and lambda_g =\kappa_g \Delta we should get the correct flux.

This corresponds to an alpha = c \frac{\Delta t}{\Delta x

So we would just modify alph and zero out the matrix coefficient to the ghost zone

Thermally emitting boundary

Another possible boundary condition would be to have the edge of the grid be adjacent to some thermal emitter. This corresponds to setting the radiation energy density in the ghost zones to the Planck function.

_g = \frac{4 \pi}{c} B(T_g however, we still need an opacity which could be defined from the fluid properties of the ghost region… This would essentially be a thermally emitting boundary with the temperature, density, opacity, etc… derived from the hydro boundary type. If the boundary type was extrapolating, and the density and temperature uniform, one could have a constant thermal energy spectrum… However we would need to either specify the opacity and temperature of the ghost zone - or the various fluid properties needed to reconstruct the opacity and temperature of the ghost zone - or make an extra call to set physicalBC - since this may happen in between a hydro step and a physical boundary update.

ZeroSlope Boundary

Here we want the radiation energy in the ghost zone to match the radiation energy in the internal cell. ^{n+1}_g=E^{n+1}_i \mbox{ and } E^n_g=E^n_ This is identical to reflecting below

Constant Slope Boundary

Here we want the flux to be constant so energy does not pile up near the boundary. So we want alpha_g = \alpha_i) \mbox { and } E_g=2*E_i-E_ This will effectively cancel all terms related to alpha. However, we want to maintain this boundary during the implicit solve, so we also need to modify the matrix connections and subtract alpha_g \mbox{ from } \alpha_ but this will effectively zero out the matrix connections to the interior as well. This can also be done by setting alpha_g = \alpha_i =

Reflecting Boundary

Reflecting boundary should be fairly straightforward. This an be achieved by setting alpha_g = which zeros out any flux - and has the same effect as setting ^{*}_g=E^{*}_ or ^{n+1}_g=E^{n+1}_i \mbox{ and } E^{n}_g=E^{n}_

Constant radiative flux

To have a constant radiative flux we must have alpha_g \left ( E^{*}_i-E^{*}_g \right ) = F_0 \frac{\Delta t}{\Delta x

Which we can solve for ^{*}_g = E^{*}_i - \frac{F_0 \Delta t}{\alpha_g \Delta x

but when we plug this into the coefficient matrix the terms with alpha_ cancel and we just get _0 \frac{\Delta t}{\Delta x in the source vector

Summary

Always kill the matrix coefficient to a physical boundary and add an additional psi \alpha_g E_ to the source vector

Boundary	alpha_	alpha_	_
Open	\frac{\Delta t}{\Delta x
User-Defined opacity and Temperature			\frac{4 \pi}{c} B(T_g
Extrapolate Flux			X
Reflecting (Extrapolate E)			X
User-Defined Flux			X	_0 \frac{\Delta t}{\Delta x