Changes between Version 135 and Version 136 of FluxLimitedDiffusion

Timestamp:

03/31/13 10:43:40 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -456 +456 @@
 === Implicit Discretization 2 ===
 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 )]]
+[[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
@@ -466 +466 @@
 #!latex
 \begin{eqnarray}
-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 ] \\
+E^{n+1}_i-E^{n}_i = & \left [ \alpha^n_{i+1/2} \left ( \psi E^{n+1}_{i+1} + \bar{\psi} E^{n}_{i+1}- \psi E^{n+1}_{i} - \bar{\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+1}_{i+1} + \bar{\psi} E^{n}_{i+1} + \psi E^{n+1}_{i} + \bar{\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 [ \theta + \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 ] \\
 \end{eqnarray}
 }}}
@@ -474 +474 @@
 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_{0R,i+1/2}} \mbox{ where } \kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,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 ) )]]
+[[latex(\alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2}  \frac{c \lambda_{i+1/2}}{\kappa_{0R,i+1/2}} )]]  
+
 and where
+
+[[latex(\zeta_{i+1/2}= \frac{\Delta t}{\Delta x}\frac{3-R_{2,i+1/2}}{2} v^n_x,{i+1/2} )]]
+
+and 
+
+[[latex(\epsilon^n_i=c\Delta t \kappa^n_{0P,i})]]
+
+and 
+
+[[latex(\phi = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) )]]
+
+and
+
+[[latex(\theta = f(e^n_i) = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) )]]
+
+and
+
+[[latex(\omega_{i} = \frac{\lambda_i \Delta t}{\Delta x} \left ( \frac{\kappa_{0P,i}}{\kappa_{0R,i}}-\frac{1}{2} \right ))]]
+
+
+and
+
+where [[latex( \kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,i+1}}{2} )]]
+
+and 
+
 [[latex(R_{i+1/2} = \frac{\left | E^n_{i+1}-E^n_{i} \right | }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )})]]
 
-
-and 
-
-[[latex(\epsilon^n_i=c\Delta t \kappa^n_{0P,i})]]
-
-represents the number of absorption/emissions during the time step
-
-
-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 ) )]]
-
-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 ) )]]
-
-and
-
-where [[latex(\omega_{i} = \frac{\lambda_i \Delta t}{\Delta x} \left ( \frac{\kappa_{0P,i}}{\kappa_{0R,i}}-\frac{1}{2} \right ))]]
-
-where
-
-[[latex(\frac{\omega}{\alpha} = \frac{v}{c} \kappa_{0R,i} \Delta x)]]
+and
+
+[[latex(\lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2} - \frac{1}{R_{i+1/2}} \right ) )]]
+
+and
+
+[[latex(R_{2,i+1/2} = \lambda_{i+1/2}+\lambda_{i+1/2}^2 R_{i+1/2}^2)]]
+
+and
+
+
+[[latex(R_{i} = \frac{\left | E^n_{i+1}-E^n_{i-1} \right | }{2 \kappa_{0R,i} E^n_{i}})]]
+
+and
+
+[[latex(\lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i} - \frac{1}{R_{i}} \right ) )]]
+
+and
+
+[[latex(R_{2,i} = \lambda_{i}+\lambda_{i}^2 R_{i}^2)]]
+
+
 === Time Discretization ===