Changes between Version 119 and Version 120 of FluxLimitedDiffusion

Timestamp:

03/30/13 22:28:21 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -389 +389 @@
 which we can also write as
 
-  [[latex(\frac{\partial e}{\partial t}  = f \left ( e,E,\nabla E \right )
-  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E  - f \left (e,E, \nabla E \right ) \right ) )]]  
+  [[latex(\frac{\partial e}{\partial t}  = f \left ( e \right ) + g \left (E,\nabla E \right )
+  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E  - f \left ( e \right ) - g \left (e,E, \nabla E \right ) \right ) )]]  
   where
-  [[latex( f \left (e,E, \nabla E \right ) = -\kappa_{0P}(4 \pi B-cE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E )]]
+[[latex( f \left ( e \right ) = - 4 \pi \kappa_{0P} e )]]
+ and
+  [[latex( g \left (e,E, \nabla E \right ) = \kappa_{0P}(cE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E )]]
 
 Now we can linearize f about e,,0,,
-[[latex( f \left (e, E, \nabla E \right ) = f (\left e_0, E, \nabla E \right ) + \phi e )]]
+[[latex( f \left ( e \right ) = f \left ( e_0 ) + \partial{f}\partial{e} \left ( e - e_0 \right ) )]]
 
 so that the first equation can be written as 
 
-  [[latex(\frac{\partial e}{\partial t}  = f \left ( e_0,E,\nabla E \right ) + \phi e)]]
+  [[latex(\frac{\partial e}{\partial t}  = f \left ( e_0 \right ) + \partial{f}\partial{e} \left ( e - e_0 \right ) )]]
 
 and then discretized as 
 
-  [[latex(e^{n+1}_i-e^{n}_i = \frac{\Delta t}{\Delta x}  f \left ( e_0,E*,\nabla E* \right ) \right ) + \bar{\psi} \phi e^n_i + \psi \phi e^{n+1}_i \right )]]
-
-which can be solved for [[latex(e^{n+1} = \frac{1}{1-\psi \phi} \left ( \bar{\psi} \phi e^n_i + f \left ( e_0,E*,\nabla E* \right ) \right ) )]]
+  [[latex(e^{n+1}_i-e^{n}_i = \frac{\Delta t}{\Delta x}  \left ( f \left ( e_n \right ) + g(e_n,E^*,\nabla E^* \right ) + \right ) \left ( \left ( \bar {\psi} - 1 \right ) \phi \right ) e_n +  \psi \phi e^{n+1}_i ]]
+
+which can be solved for [[latex(e^{n+1} = \frac{1}{1-\psi \phi} \left ( \bar{\psi} \phi e^n_i + f \left ( e^n_i \right ) + g \left ( e^n_i,E^*,\nabla E^* \right ) \right ) )]]
 
 Then if we take the semi-discretized equation for E 
 
-  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - f \left (e_0,E, \nabla E \right ) - \bar{\psi}\phi e^n_i -\psi \phi e^{n+1}_i )]]  
+  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - f \left ( e^n_i \right ) + g \left (e^n_i,E, \nabla E \right ) + \bar{\psi}\phi e^n_i - \psi \phi e^{n+1}_i )]]  
 
 and then plugin the solution for e^n+1^,,i,, we get
 
-  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - f \left (e_0,E, \nabla E \right ) - \bar{\psi}\phi e^n_i - \frac{\psi \phi }{1-\psi \phi} \left ( \bar{\psi} \phi e^n_i + f \left ( e_0,E*,\nabla E* \right ) \right ) )]]  
+  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - f \left (e_0,E, \nabla E \right ) - \bar{\psi}\phi e^n_i - \frac{\psi \phi }{1-\psi \phi} \left ( \bar{\psi} \phi e^n_i + f \left ( e_0,E,\nabla E \right ) \right ) )]]  
 
 which simplifies to 
 
-  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - \frac{1}{1-\psi \phi} \left ( \bar{\psi} \phi e^n_i + f \left ( e_0,E*,\nabla E* \right ) \right ) )]]  
-
-
-
-=== Expanding about e,,0,, ===
+  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - \frac{1}{1-\psi \phi} \left ( \bar{\psi} \phi e^n_i + f \left ( e_0,E,\nabla E \right ) \right ) )]]  
+
+
+=== Expanding f about e,,0,, ===
 
 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 [[latex(B(T))]]