Changes between Version 132 and Version 133 of FluxLimitedDiffusion

Timestamp:

03/31/13 10:22:57 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -439 +439 @@
 [[latex(\Gamma = \frac{\partial T}{\partial e} = \frac{(\gamma-1)}{n k_B})]]
 
-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})]]
+and we can identify [[latex(\phi = -\Delta t \frac{\partial f}{\partial e} = 4 \pi \kappa_{0P} \Delta t \frac{\partial B}{\partial e} = 16 \pi \kappa_{0P} \Delta t B_0 \frac{\Gamma}{T_0})]]
 
 Then the equation for e becomes
@@ -455 +455 @@
 
 === Implicit Discretization 2 ===
-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 ) )]]   
-   [[latex(e^{n+1}_i-e^{n}_i = \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 ) )]]   
+Now we can discretize 
+[[latex(g(E*, \nabla E*) =  \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 )]]
+
+as
+
+[[latex(g = \epsilon \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) + \omega v^n_x \left ( \psi E^{n+1}_{i+1} - \psi E^{n+1}_{i-1} + \bar{\psi} E^n_{i+1}- \bar{\psi} E^n_{i-1} \right ) - \xi \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right )]]
+
+which along with the other terms gives
+
+   [[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha^n_{i+1/2} \left ( \psi E^{n}_{i+1} + \bar{\psi} E^{n+1}_{i+1}- \psi E^{n+1}_{i} - \psi E^n_{i} \right ) - \alpha^n_{i-1/2} \left ( \psi E^{n+1}_{i} + \bar{\psi} E^n_i - \psi E^{n+1}_{i-1} - \bar{\psi}E^{n}_{i-1} \right ) \right ] + \left [ \zeta^n_{i+1/2} \left ( \psi E^{n}_{i+1} + \bar{\psi} E^{n+1}_{i+1} + \psi E^{n+1}_{i} + \psi E^n_{i} \right ) - \zeta^n_{i-1/2} \left ( \psi E^{n+1}_{i} + \bar{\psi} E^n_i + \psi E^{n+1}_{i-1} + \bar{\psi}E^{n}_{i-1} \right ) \right ] - \frac{1}{1-\psi \phi} \left [ \epsilon \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) - \omega v^n_x \left ( \psi E^{n+1}_{i+1} - \psi E^{n+1}_{i-1} + \bar{\psi} E^n_{i+1}- \bar{\psi} E^n_{i-1} \right ) + \xi \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) \right ] )]]   
+
 
 where the diffusion coefficient is given by