Changes between Version 7 and Version 8 of u/johannjc/scratchpad

Timestamp:

08/12/13 21:19:11 (11 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad

@@ -12 +12 @@
 [[latex($ \frac{c}{3\kappa} \nabla E = F$)]]
 
+so that
 
-If we discretize this equation we get (E_g is ghost zone, E_i is internal zone at left edge)
+[[latex($\frac{\partial E}{\partial t} = \nabla \cdot \left ( \frac{c}{3\kappa} \nabla E \right ) = \nabla \cdot F$)]]
+
+In any even, if we discretize this mixed equation we get (E_g is ghost zone, E_i is internal zone at left edge)
 
 [[latex($E_g + E_i - \frac{4}{3\kappa \Delta x} \left ( E_i - E_g \right ) = \frac{8}{c} F$)]]
@@ -24 +27 @@
 [[latex($E_g - E_i = \frac{\left ( -\frac{6\kappa \Delta x}{4} \right ) E_i + \frac{3\kappa \Delta x}{4} \left ( \frac{8}{c} F \right )}{1 + \frac{3 \kappa \Delta x}{4}}$)]]
 
-and
 
-[[latex($E_g + E_i = \frac{ 2 E_i + \frac{3\kappa \Delta x}{4} \left ( \frac{8}{c} F \right )}{1 + \frac{3 \kappa \Delta x}{4}}$)]]
-
-
-If we then plug this into the time evolution equation:
+If we then plug this into the time evolution equation ignoring portions that are unchanged we get:
 
 [[latex($E^{n+1}_i-E^{n}_i = \alpha \left ( E_g^*-E_i^* \right )$)]]
 
+where [[latex($\alpha = \frac{c \Delta t}{3 \kappa \Delta x^2}$)]]
 we get
 
@@ -61 +61 @@
 
 [[latex($E = \frac{4F}{c}$)]]
+
+and that if [[latex($3 \kappa \Delta x = 4$)]] we revert to what we would expect if there was no Radiation energy in the ghost zone.
+