Version 40 (modified by 12 years ago) ( diff ) | ,
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Most of what follows is taken from Krumholz et al. 2007

# Physics of Radiation Transfer

streaming limit | |

static diffusion limit | |

dynamic diffusion limit |

## Equations of Radiation Hydrodynamics

where the moments of the specific intensity are defined as

and the radiation 4-momentum is given by

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

but in general 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

where

comoving-frame Planck function weighted opacity | |

comoving-frame radiation energy weighted opacity | |

comoving-frame radiation flux weighted opacity |

- If the radiation has a blackbody spectrum then
- If the radiation is optically thick, then

which implies that

- In the optically thin regime, so we would have however assuming a blackbody temperature in the optically thin limit may be any more accurate than assuming that

## Flux limited diffusion

The flux limited diffusion approximation drops the radiation momentum equation in favor of

where

is the flux-limiterwhich corresponds to a pressure tensor

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

# 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:

For static diffusion, the terms in blue with 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).

## 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 e
_{0}, and E_{0}using d_{1}, e_{1}, E_{1} - Update E
_{2*mbc}using Edot_{2*mbc} - Update e
_{2*mbc}using E_{2*mbc}, Edot_{2*mbc}, and e_{2*mbc} - Update Edot
_{0}using pre IR and post IR E_{0} - 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 e
_{0}, and E_{0}using d_{1}, e_{1}, E_{1} - Update Edot
_{0}using pre IR and post IR E_{0} - Update E
_{1}using Edot_{1} - Ghost e
_{mbc}, E_{1}, Edot_{1} - Do second EH
_{0} - Do ER
_{0}—- 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.

## Implicit Equation

For now we will assume that

and 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:where

### 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

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

where

Then the system of equations becomes

which will be accurate as long as

orWe can calculate the time scale for this to be true using the evolution equation for the energy density

which gives

### Implicit Discretization

Which we can discretize for (1D) as

where the diffusion coefficient is given by

and where

and

represents the number of absorption/emissions during the time step

and

and we can think of the radiative flux as

### 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
and for Crank Nicholson we have
or we can parameterize the solution
where

Backward Euler has

and Crank Nicholson hasForward Euler has

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

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

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

If we ignore the change in the Planck function due to heating during the implicit solve, it is equivalent to replacing the term with

with in which case the equations simplify toIn 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…).

## 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

Clearly if we set

and we should get the correct flux.This corresponds to an

So we would just modify

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.

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.

### Reflecting Boundary

Reflecting boundary should be fairly straightforward. This an be achieved by setting

which zeros out any flux - and has the same effect as setting or### Constant radiative flux

To have a constant radiative flux we must have

Which we can solve for

but when we plug this into the coefficient matrix the terms with

cancel and we just get in the source vector### Summary

Boundary | ||

Open | 0 | |

User-Defined opacity and Temperature | ||

Reflecting | ||

User-Defined Flux |

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