Changes between Version 25 and Version 26 of FluxLimitedDiffusion

Timestamp:

03/20/13 12:22:41 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -185 +185 @@
 
 ||   [[latex(\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+\phi^n_i e^n_i + \theta^n_i)]]   ||
-||   [[latex(\left (1 \right ) e^{n+1}_i = \left ( \psi \epsilon^n_i \right )E^{n+1}_i + \left ( 1 - \phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n )]]   ||
-
-
-
-Clearly the second equation is trivial to solve after the first system of equations has been solved.  So if we treat the temperature as being constant we can calculate the local heating/cooling due to radiative emission/absorption
-
-[[latex(e^{n+1}_i =  e^n_i + \epsilon^n_i \left (  E^{n+1}_i -\frac{4\pi}{c}B \left (T^n_i \right ) \right ))]] 
-
+||   [[latex(e^{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 the matrix coefficients would be
-
-|| [[latex(1+\alpha^n_{i+1/2,j}+\alpha^n_{i-1/2,j}+\alpha^n_{i,j+1/2}+\alpha^n_{i,j-1/2}+\epsilon^n_i )]] || [[latex(E^{n+1}_{i,j})]] ||
-|| [[latex(-\alpha^n_{i-1/2,j})]] || [[latex(E^{n+1}_{i-1,j})]] ||
-|| [[latex(-\alpha^n_{i+1/2,j})]] || [[latex(E^{n+1}_{i+1,j})]] ||
-|| [[latex(-\alpha^n_{i,j-1/2})]] || [[latex(E^{n+1}_{i,j-1})]] ||
-|| [[latex(-\alpha^n_{i,j+1/2})]] || [[latex(E^{n+1}_{i,j+1})]] ||
-
-and the source term would be unchanged from 1D
-
-||  [[latex(E^n_i+\frac{4\pi \epsilon^n_i}{c}B \left (T^n_i \right ) )]]
-
+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...).
+
+[[latex(E^{n+1}_i = E^{n}_i+\dot{E}^{n}_i \Delta t)]]
+[[latex(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 } )]]
 
 == 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.  
 
- What about radiative implicit heating in coarse ghost cells which we will need to do hydro update?  Could do this explicitly using E and Edot
+ What about radiative implicit heating in coarse ghost cells which we will need to do hydro update?  Could do this implicitly using E and Edot (don't need hypre for this implicit solve though)