Changes between Version 23 and Version 24 of u/johannjc/scratchpad

Timestamp:

04/22/14 09:18:49 (12 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad

@@ -10 +10 @@
 Now if
 
-[[latex($E=a_RT^4$)]]
+[[latex($E=a_RT^4=a_R\left(\frac{e}{\rho c_v}\right)^4$)]]
 
 and
 
-[[latex($e=\rho c_v T$)]]
+[[latex($e=\rho c_v T=\rho c_v \left(\frac{E}{a_R}\right)^{1/4}$)]]
 
-we can combine the gas and radiation diffusion equations to arrive at:
+and we just consider the implicit terms, we can combine the gas and radiation diffusion equations to arrive at:
 
-  [[latex($\frac{\partial \left (e+E\right) }{\partial t}+\nabla\cdot\left[\left(e+P\right)\mathbf{v}\right]=\color{red}{ \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} \color{green}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E\right )}$)]]
+  [[latex($\frac{\partial \left (e+E\right) }{\partial t}=\color{red}{ \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E}$)]]
+
+which simplifies to 
+
+[[latex($\left(1+\frac{\rho c_v}{4 E}\left(\frac{E}{a_R}\right )^{1/4}\right) \frac{\partial E}{\partial t}= \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E$)]]
+
+
+The second term in parenthesis represents the extra 'inertia' the radiation field has due to its coupling with the gas.  It is non-linear and this limits the time step that can be taken.
+
+[[latex($\Delta t \approx \frac{E}{\frac{\partial E}{\partial t}} = \frac{\left(E+\frac{\rho c_v}{4}\left(\frac{E}{a_R}\right )^{1/4}\right)}{\nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E}$)]]
+
+== Changes to the discretization ==
+
+