Changes between Version 17 and Version 18 of FluxLimitedDiffusion

Timestamp:

03/19/13 15:50:25 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -152 +152 @@
 [[latex(e^{n+1}_i =  e^n_i + \epsilon^n_i \left (  E^{n+1}_i -\frac{4\pi}{c}B \left (T^n_i \right ) \right ))]] 
 
+=== Expanding about e,,0,, ===
+
+Of course the above system is only linear 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 [[latex(B(T) = \frac{c}{4 \pi} a_RT^4 = \frac{c}{4 \pi} \left ( \frac{2}{n f k_B} \right ) ^4 a_R e^4 = \frac{c}{4 \pi} A_R e^4 = A_R e_0^4 \left ( 1 + 4 \delta + 6 \delta^2 + ... \right ))]]
+
+where [[latex(e = e_0 \left ( 1 + \delta \right ) )]]
+
+Then the system of equations becomes
+
+||   [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E +  c \kappa_{0P} \left [ A_R e_0^4 \left ( 1 + 4 \delta \right ) -E \right ] )]]   ||
+||   [[latex(\frac{\partial \delta}{\partial t} = - \frac{c \kappa_{0P}}{e_0} \left [ \left (1+4 \delta \right )-E \right ] )]]   ||
+
+or equivalently
+
+||   [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E +  \kappa_{0P} \left [ 4 \pi B \left ( T_0 \right ) \left ( 1 + 4 \delta \right ) -cE \right ] )]]   ||
+||   [[latex(\frac{\partial \delta}{\partial t} = - \frac{ \kappa_{0P}}{e_0} \left [ 4 \pi B \left ( T_0 \right ) \left ( 1+4 \delta \right ) -cE \right ] )]]   ||
+
+
 
 === 2D etc... ===