Changes between Version 131 and Version 132 of FluxLimitedDiffusion

Timestamp:

03/31/13 09:59:07 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -392 +392 @@
   [[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 \right ) - g \left (e,E, \nabla E \right )  )]]  
   where
-[[latex( f \left ( e \right ) = - 4 \pi \kappa_{0P} B(T(e)) )]]
+[[latex( f \left ( e \right ) = -4 \pi \kappa_{0P} B(T(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 )]]
@@ -401 +401 @@
 so that the first equation can be written as 
 
-  [[latex(\frac{\partial e}{\partial t}  = f \left ( e_0 \right ) + g \left ( E,\nabla E \right )  + \frac{\partial f}{\partial e} \left ( e - e_0 \right ) )]]
+  [[latex(\frac{\partial e}{\partial t}  =  f \left ( e_0 \right ) + g \left ( E,\nabla E \right )  +  \frac{\partial f}{\partial e} \left ( e - e_0 \right ) )]]
 
 and then discretized as 
 
-  [[latex(e^{n+1}_i-e^{n}_i = \Delta t  \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) + \left ( \left ( \bar{\psi} - 1  \right ) \phi \right ) e^n_i +  \psi \phi e^{n+1}_i )]]
+  [[latex(e^{n+1}_i-e^{n}_i = \Delta t  \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) - \left ( \left ( \bar{\psi} - 1  \right ) \phi \right ) e^n_i -  \psi \phi e^{n+1}_i )]]
+
+where
+
+[[latex(\phi = -\Delta t \frac{\partial f}{\partial e} = 4 \pi \kappa_{0P} \Delta t \frac{\partial B(T(e))}{\partial e})]]
 
 which can be solved for 
-[[latex(e^{n+1}_i = e^{n}_i + \frac{\Delta t}{1 - \psi \phi} \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) )]]
+[[latex(e^{n+1}_i = e^{n}_i + \frac{\Delta t}{1 + \psi \phi} \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) )]]
 
 Then if we take the semi-discretized equation for E 
@@ -422 +426 @@
   [[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 ( f \left ( e^n_i \right ) + g \left ( E,\nabla E \right ) \right ) )]]  
 
+Now we have 1 equation with 1 variable that we can solve implicitly using hypre, and then we can use E^n+1^ and E^n^ to construct E^*^ which we can plug into the equation for e^n+1^
+
 
 === 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))]]
-
-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 
-
-[[latex(B(T) = B \left ( T_0+dT \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right | _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right ) )]]
+Expanding 
+
+[[latex(B(T(e)) = B \left ( T_0+dT(e) \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right | _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right ) )]]
 
 where
@@ -435 +439 @@
 [[latex(\Gamma = \frac{\partial T}{\partial e} = \frac{(\gamma-1)}{n k_B})]]
 
-Then the system of equations becomes
-
-   [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} \left [4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] - \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E )]]   
+and we can identify [[latex(\phi = -\Delta t \frac{\partial f}{\parial e} = 4 \pi \kappa_{0P} \Delta t \frac{\partial B}{\partial e} = 16 \pi \kappa \Delta t B_0 \frac{\Gamma}{T_0})]]
+
+Then the equation for e becomes
+
    [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E )]]   
 
@@ -450 +455 @@
 
 === Implicit Discretization 2 ===
-Which we can discretize for (1D) as
+Now we can discretize the radiation energy equation
 
    [[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{*}_{i+1}-E^{*}_{i} \right ) - \alpha^n_{i-1/2} \left ( E^{*}_{i}-E^{*}_{i-1} \right ) \right ] - \epsilon^n_i E^{*}_i  + \phi^n_i e^{*}_i  + \theta^n_i) - \omega_{i} v_{x,i} \left ( E^{*}_{i+1}-E^*_{i-1} \right ) )]]