Changes between Version 179 and Version 180 of FluxLimitedDiffusion

Timestamp:

04/03/13 17:09:18 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -521 +521 @@
 Then the equation for e becomes
 
-   [[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 ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E )]]   
+   [[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 ] + \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)]]   
 
 which will be accurate as long as \(4\Gamma \frac{e-e_0}{T_0} < \xi << 1\) or \(\Delta e = e-e_0 < \xi \frac{T_0}{4 \Gamma}\)
@@ -527 +527 @@
 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 ) })]]
+[[latex(e-e_0=\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 | \left ( \frac{\partial e}{\partial t} \right )_0 | })]]