Changes between Version 76 and Version 77 of FluxLimitedDiffusion

Timestamp:

03/27/13 14:59:01 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -277 +277 @@
 == Physical Boundary Conditions ==
 
-=== Open boundaries ===
+=== 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 [[latex(E_g = 0)]] and [[latex(\lambda_g =\kappa_g \Delta x)]] we should get the correct flux.
 
@@ -289 +288 @@
 So we would just modify [[latex(\alpha)]] 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.
-
-[[latex(E_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.  [[latex(E^{n+1}_g=E^{n+1}_i \mbox{ and } E^n_g=E^n_i)]] 
-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 [[latex(\alpha_g = \alpha_i) \mbox { and } E_g=2*E_i-E_j)]]  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 [[latex(\alpha_g \mbox{ from } \alpha_i)]] but this will effectively zero out the matrix connections to the interior as well.  This can also be done by setting [[latex(\alpha_g = \alpha_i = 0)]]
 
-=== Reflecting Boundary ===
+=== 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 [[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 have 
-[[latex(\alpha_g \left ( E^{*}_i-E^{*}_g \right ) = F_0 \frac{\Delta t}{\Delta x})]]
-
-Which we can solve for 
-[[latex(E^{*}_g = E^{*}_i - \frac{F_0 \Delta t}{\alpha_g \Delta x})]]
-
-but when we plug this into the coefficient matrix the terms with [[latex(\alpha_g)]] cancel and we just get [[latex(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 ===