Changes between Version 35 and Version 36 of FluxLimitedDiffusion

Timestamp:

03/21/13 10:28:49 (13 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -1 +1 @@
 [[PageOutline()]]
+[[latex(\exp(x))]]
+[[latex(exp(x))]]
 
 Most of what follows is taken from [http://adsabs.harvard.edu/abs/2007ApJ...667..626K Krumholz et al. 2007]
@@ -193 +195 @@
 
 
+Now since the second equation has no spatial dependence, we can solve it for 
+   [[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 \} )]]   
+and plug the result into the first equation to get a matrix equation involving only one variable.
+
+   [[latex(\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i-\frac{\phi^n_i \psi \epsilon^n_i}{ 1 +\psi \phi^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 +\frac{\psi \phi^n_i}{ 1 +\psi \phi^n_i}\left ( \bar{\psi} \epsilon^n_i \right ) E^n_i 
+
++ \frac{ \phi^n_i}{ 1 +\psi \phi^n_i}  e^n_i
+
++ \frac{1}{ 1 +\psi \phi^n_i}\theta^i_n
+
+)]]   
+
+
+
 If we ignore the change in the Planck function due to heating during the implicit solve, it is equivalent to replacing the term with [[latex(\psi \phi^n_i e^{n+1}_i)]] with [[latex(\psi \phi^n_i e^n_i)]] in which case the equations simplify to
 
@@ -199 +218 @@
 
 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... ===