Changes between Version 40 and Version 41 of FluxLimitedDiffusion

Timestamp:

03/21/13 11:37:39 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -194 +194 @@
 
 Now since the second equation has no spatial dependence, we can solve it for 
-   [[latex(\color{red}{e^{n+1}_i = \frac{1}{ 1 +\psi \phi^n_i}\left \{ \left ( \psi \epsilon^n_i \right )E^{n+1}_i + \left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n \right \}} )]]   
+   [[latex(\color{purple}{e^{n+1}_i = \frac{1}{ 1 +\psi \phi^n_i}\left \{ \left ( \psi \epsilon^n_i \right )E^{n+1}_i + \left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n \right \}} )]]   
 
 and plug the result into the first equation to get a matrix equation involving only one variable.
 
-   [[latex(\color{red}{\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \frac{\epsilon^n_i}{ 1 +\psi \phi^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 [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2}  +\frac{\epsilon^n_i }{ 1 +\psi \phi^n_i} \right ) \right ] E^n_i + \frac{ \phi^n_i}{ 1 +\psi \phi^n_i}  e^n_i+ \frac{1}{ 1 +\psi \phi^n_i}\theta^i_n})]]   
+   [[latex(\color{purple}{\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \frac{\epsilon^n_i}{ 1 +\psi \phi^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 [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2}  +\frac{\epsilon^n_i }{ 1 +\psi \phi^n_i} \right ) \right ] E^n_i + \frac{ \phi^n_i}{ 1 +\psi \phi^n_i}  e^n_i+ \frac{1}{ 1 +\psi \phi^n_i}\theta^i_n})]]