Version 72 (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
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.
Simplifiying assumptions
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
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 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
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 E2*mbc using Edot2*mbc
- Update e2*mbc using E2*mbc, Edot2*mbc, and e2*mbc
- Update Edot0 using pre IR and post IR E0
- Ghost e2*mbc, Embc+1, Edotmbc+1
- Do first EHmbc
- Do ERmbc —- 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 embc, 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:
These can be discretized as follows:
where
where
and
and
and
and
and
Implicit 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 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
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 whereBackward 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 setting
This gives the following equations: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…).
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 zoneThermally 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.
ZeroSlope Boundary
Here we want the radiation energy in the ghost zone to match the radiation energy in the internal cell.
This is identical to reflecting belowConstant Slope Boundary
Here we want the flux to be constant so energy does not pile up near the boundary. So we want
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 but this will effectively zero out the matrix connections to the interior as well. This can also be done by settingReflecting 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 orConstant 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 vectorSummary
Numerical value | Boundary | Requires Variables | |||
0 | RAD_FREE_STREAMING | ||||
1 | RAD_EXTRAPOLATE_FLUX* | ||||
2 | INTERNAL/PERIODIC | ||||
3 | RAD_REFLECTING | ||||
4 | RAD_USERDEFINED_BOUNDARY/AMR_BOUNDARY | ||||
5 | RAD_USERDEFINED_FLUX | RadFlux |
!* Also requires zeroing out stencil and source contributions from
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