Changes between Version 20 and Version 21 of FluxLimitedDiffusion

Timestamp:

03/20/13 10:30:29 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -135 +135 @@
 Which we can discretize for (1D) as
 
-||   [[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 \left ( \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( 1 - 4\Gamma \frac{e^n_i}{T^n_i} + 4\Gamma \frac{e^{*}_i}{T^n_i} \right ) -E^{*}_i \right ) )]]   ||
-||   [[latex(e^{n+1}_i-e^{n}_i = - \epsilon^n_i \left ( \frac{4 \pi}{c} B\left ( T^n_i \right ) \left ( 1 - 4\Gamma \frac{e^n_i}{T^n_i} + 4\Gamma \frac{e^{*}_i}{T^n_i} \right ) - E^{*}_i \right ) )]]   ||
-
-
-
-
-
-||   [[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{n+1}_{i+1}-E^{n+1}_{i} \right ) - \alpha^n_{i-1/2} \left ( E^{n+1}_{i}-E^{n+1}_{i-1} \right ) \right ] + \epsilon^n_i \left ( \frac{4 \pi}{c} B(T^n_i)-E^{n+1}_i \right ) )]]   ||
-||   [[latex(e^{n+1}_i-e^{n}_i = - \epsilon^n_i \left ( \frac{4 \pi}{c} B(T^n_i)-E^{n+1}_i \right ) )]]   ||
-
+||   [[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) )]]   ||
+||   [[latex(e^{n+1}_i-e^{n}_i = + \epsilon^n_i E^{*}_i  - \phi^n_i e^{*}_i  - \theta^n_i )]]   ||
 
 where
 
-
+[[latex(\epsilon^n_i=c\Delta t \kappa^n_{0P,i})]]
+
+represents the number of absorption/emissions during the time step
+
+and the diffusion coefficient is given by 
+
+[[latex(\alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2}  \frac{c \lambda_{i+1/2}}{\kappa_{i+1/2}} \mbox{ where } \kappa_{i+1/2} = \frac{\kappa_{i}+\kappa_{i+1}}{2} \mbox{ and } \lambda_{i+1/2} = f \left ( R_{i+1/2} \right ))]]
+
+and 
+
+[[latex(\theta = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( 1 - 4\Gamma \frac{e^n_i}{T^n_i} \right ) )]]
+[[latex(\phi = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) )]]
+
+
+and we have
 
 [[latex(\frac{\Delta t}{\Delta x}\mathbf{F}^n_{i+1/2} = \alpha^n_{i+1/2} \left ( E^{n+1}_{i+1} - E^{n+1}_i \right ) )]]
-
-and
-
-[[latex(\epsilon^n_i=c\Delta t \kappa^n_{0P,i})]]
-
-represents the number of absorption/emissions during the time step
-
-and the diffusion coefficient is given by 
-
-[[latex(\alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2}  \frac{c \lambda_{i+1/2}}{\kappa_{i+1/2}} \mbox{ where } \kappa_{i+1/2} = \frac{\kappa_{i}+\kappa_{i+1}}{2} \mbox{ and } \lambda_{i+1/2} = f \left ( R_{i+1/2} \right ))]]
-
-where
 
 [[latex(R_{i+1/2} = \frac{\left | E_{i+1}-E_{i} \right | }{2 \kappa_{i+1/2} \left ( E_i+E_{i+1} \right )})]]