Changes between Version 22 and Version 23 of FluxLimitedDiffusion

Timestamp:

03/20/13 11:37:49 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -136 +136 @@
 
 ||   [[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{*}_{i+1}-E^{*}_{i} \right ) - \alpha^n_{i-1/2} \left ( E^{*}_{i}-E^{*}_{i-1} \right ) \right ] - \epsilon^n_i E^{*}_i  + \phi^n_i e^{*}_i  + \theta^n_i) )]]   ||
-||   [[latex(e^{n+1}_i-e^{n}_i = + \epsilon^n_i E^{*}_i  - \phi^n_i e^{*}_i  - \theta^n_i )]]   ||
+||   [[latex(e^{n+1}_i-e^{n}_i = \epsilon^n_i E^{*}_i  - \phi^n_i e^{*}_i  - \theta^n_i )]]   ||
 
 where the diffusion coefficient is given by 
@@ -164 +164 @@
 
 
-Now all the terms on the right hand side that are linear in E or e have been written as E^*^ because there are different ways to approximate E^*^.  For Backward Euler we have
+Now all the terms on the right hand side that are linear in E or e have been written as E^*^ or e^*^ because there are different ways to approximate E^*^ (e^*^).  For Backward Euler we have
 [[latex(E^*_i = E^{n+1}_i)]]
 and for Crank Nicholson we have
 [[latex(E^*_i = \frac{1}{2} \left ( E^{n+1}_i + E^n_i \right ) )]]
 or we can parameterize the solution
-[[latex(E^*_i = \upsilon E^{n+1}_i + \bar{\upsilon}E^n_i)]]
-where [[latex(\bar{\upsilon} = 1-\upsilon)]]
-
-Backward Euler has [[latex(\upsilon=1)]] and Crank Nicholson has [[latex(\upsilon=1/2)]]
-
-Forward Euler has [[latex(\upsilon=0)]]
+[[latex(E^*_i = \psi E^{n+1}_i + \bar{\psi}E^n_i)]]
+where [[latex(\bar{\psi} = 1-\psi)]]
+
+Backward Euler has [[latex(\psi=1)]] and Crank Nicholson has [[latex(\psi=1/2)]]
+
+Forward Euler has [[latex(\psi=0)]]
 
 In any event in 1D we have the following matrix coefficients
 
-||   [[latex(\left ( 1 + \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) E^{n+1}_i - \left ( \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1}=E^n_i+\frac{4\pi \epsilon^n_i}{c}B \left (T^n_i \right ) )]]   ||
-||   [[latex(\left ( 1 \right ) e^{n+1}_i - \left ( \epsilon^n_i \right )E^{n+1}_i =e^n_i-\frac{4\pi \epsilon^n_i}{c}B \left (T^n_i \right ) )]]   ||
+||   [[latex(\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} - \left ( \psi \phi^n_i \right ) e^{n+1}_i=\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^n_i+\bar{\psi}\phi e^n_i + \theta^n_i)]]   ||
+||   [[latex(\left ( 1 +\psi \phi \right ) e^{n+1}_i - \left ( \psi \epsilon^n_i \right )E^{n+1}_i =\left ( 1 - \bar{\psi} \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-theta^i_n )]]   ||
 
 Clearly the second equation is trivial to solve after the first system of equations has been solved.  So if we treat the temperature as being constant we can calculate the local heating/cooling due to radiative emission/absorption