wiki:FluxLimitedDiffusion

Version 139 (modified by Jonathan, 12 years ago) ( diff )

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-limiter

which 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 1

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 1

Eplicit Update 1

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 1

Implicit Update 1

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 or

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

which gives

Implicit Discretization 1

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 has

Forward 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 (Free streaming) 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

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

Periodic Boundary

This is the same as internal zones - it just maps the neighbor cell to be across the domain. Hypre has built in functionality for this under for the Struct Interface

User defined radiation field/Coarse Fine boundary

This will be the boundary at internal coarse-fine boundaries, but could also be used at the physical boundary if the radiation energy were specified.

Reflecting/ZeroSlope 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 zero out terms involving the gradient and just add in the source vector

Summary

Numerical value Boundary
0 RAD_FREE_STREAMING
1 RAD_EXTRAPOLATE_FLUX
2 INTERNAL/PERIODIC
3 RAD_REFLECTING
4 RAD_USERDEFINED_BOUNDARY/AMR_BOUNDARY
5 RAD_USERDEFINED_FLUX

Alternative Splitting Method

It is possible that with included the terms in orange 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 orange to the implicit solve (using the old velocity). Then the momentum update can be done explicitly - though using a time centered radiation field.

Operator Splitting 2

Operator splitting 2

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
    • Store EDot in child arrays to be prolongated
    • Do first EHmbc
    • Do ERmbc —- Only momentum terms.
  • Step 2
    • Overlap d, p, e, E, and do physical BCs
    • Do IR which updates e0, and E0 using d1, e1, E1
    • Update Embc using Edotmbc
    • Update e2*mbc using E2*mbc, Edot2*mbc, and e2*mbc
    • Update Edot0 using pre IR and post IR E0
    • Ghost embc, E1, Edot1
    • Store EDot in child arrays to be prolongated
    • Do second EH0
    • Do ER0 —- Only momentum terms.

Explcit Update 2

Eplicit Update 2

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

These can be discretized as follows:

and

Implicit Update 2

Implicit Update 2

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:

which we can also write as

where

and

Now we can linearize f about e0

so that the first equation can be written as

and then discretized as

where

which can be solved for

Then if we take the semi-discretized equation for E

and then plugin the solution for en+1i we get

which simplifies to

Now we have 1 equation with 1 variable that we can solve implicitly using hypre, and then we can use En+1 and En to construct E* which we can plug into the equation for en+1

Expanding f about e0

Expanding

where

and we can identify

Then the equation for e becomes

which will be accurate as long as or

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

which gives

Implicit Discretization 2

Now we can discretize

as

which along with the other terms gives

where the diffusion coefficient is given by

and where

and

and

and

and

and

and

where

and

and

and

and

and

and

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 has

Forward 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.

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 - and the velocity components (vx) will change, 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 (Free streaming) 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

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

Periodic Boundary

This is the same as internal zones - it just maps the neighbor cell to be across the domain. Hypre has built in functionality for this under for the Struct Interface

User defined radiation field/Coarse Fine boundary

This will be the boundary at internal coarse-fine boundaries, but could also be used at the physical boundary if the radiation energy were specified.

Reflecting/ZeroSlope 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 zero out terms involving the gradient and just add in the source vector

Summary

Numerical value Boundary
0 RAD_FREE_STREAMING
1 RAD_EXTRAPOLATE_FLUX
2 INTERNAL/PERIODIC
3 RAD_REFLECTING
4 RAD_USERDEFINED_BOUNDARY/AMR_BOUNDARY
5 RAD_USERDEFINED_FLUX

And for the semi implicit term

Numerical value Boundary
0 RAD_FREE_STREAMING
1 RAD_EXTRAPOLATE_FLUX*
2 INTERNAL/PERIODIC
3 RAD_REFLECTING
4 RAD_USERDEFINED_BOUNDARY/AMR_BOUNDARY
5 RAD_USERDEFINED_FLUX

!* We can make the left boundary look like the right boundary by reflecting the domain in x. Then we just swap every i+1i-1 and vx → -vx

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