Changes between Version 21 and Version 22 of FluxLimitedDiffusion

Timestamp:

03/20/13 11:13:01 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -138 +138 @@
 ||   [[latex(e^{n+1}_i-e^{n}_i = + \epsilon^n_i E^{*}_i  - \phi^n_i e^{*}_i  - \theta^n_i )]]   ||
 
-where
+where 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^n_{i}+\kappa^n_{i+1}}{2} \mbox{ and } \lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2} - \frac{1}{R_{i+1/2}} \right ) )]]
+and where
+[[latex(R_{i+1/2} = \frac{\left | E^n_{i+1}-E^n_{i} \right | }{2 \kappa_{i+1/2} \left ( E^n_i+E^n_{i+1} \right )})]]
+
+
+and 
 
 [[latex(\epsilon^n_i=c\Delta t \kappa^n_{0P,i})]]
@@ -144 +151 @@
 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(\phi = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) )]]
+
 [[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 ) )]]
-
-[[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 )})]]
-
-
-This gives matrix coefficients 
+
+
+and we can think of the radiative flux as
+
+[[latex(\frac{\Delta t}{\Delta x}\mathbf{F}^n_{i+1/2} = \alpha^n_{i+1/2} \left ( E^{*}_{i+1} - E^{*}_i \right ) )]]
+
+
+Now all the terms on the right hand side that are linear in E or e have been written as E^*^ because there are different ways to approximate E^*^.  For Backward Euler we have
+[[latex(E^*_i = E^{n+1}_i)]]
+and for Crank Nicholson we have
+[[latex(E^*_i = \frac{1}{2} \left ( E^{n+1}_i + E^n_i \right ) )]]
+or we can parameterize the solution
+[[latex(E^*_i = \upsilon E^{n+1}_i + \bar{\upsilon}E^n_i)]]
+where [[latex(\bar{\upsilon} = 1-\upsilon)]]
+
+Backward Euler has [[latex(\upsilon=1)]] and Crank Nicholson has [[latex(\upsilon=1/2)]]
+
+Forward Euler has [[latex(\upsilon=0)]]
+
+In any event in 1D we have the following matrix coefficients
 
 ||   [[latex(\left ( 1 + \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) E^{n+1}_i - \left ( \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1}=E^n_i+\frac{4\pi \epsilon^n_i}{c}B \left (T^n_i \right ) )]]   ||