Changes between Version 31 and Version 32 of FluxLimitedDiffusion

Timestamp:

03/20/13 13:55:27 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -116 +116 @@
 === Expanding about e,,0,, ===
 
-Of course even if the opacity is independent of energy and radiation energy, the above combined system of equations is still non-linear due to the dependence on Temperature of the Planck Function [[latex(B(T))]].
+Of course even if the opacity is independent of energy and radiation energy, the above combined system of equations is still non-linear due to the dependence on Temperature of the Planck Function [[latex(B(T))]]
 
 If we ignore the changes in the Temperature due to heating during the implicit step which would feed back into the source function.  We can improve this by writing 
@@ -130 +130 @@
    [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} \left [4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]]   
    [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]]   
+
+which will be accurate as long as [[latex(4\Gamma \frac{e-e_0}{T_0} < \xi << 1)]] or [[latex(\Delta e = e-e_0 < xi \frac{T_0}{4 \Gamma})]]
+
+We can calculate the time scale for this to be true using the evolution equation for the energy density
+
+[[latex(\Delta e = -\Delta t \kappa_{0P} \left [ 4 \pi B_0 -cE \right ] < \xi \frac{T_0}{4 \Gamma})]]
+
+which gives [[latex(\Delta t < \xi \frac{T_0}{4 \Gamma \kappa_{0P} \left ( 4 \pi B_0 - cE \right ) })]]