Changes between Version 92 and Version 93 of FluxLimitedDiffusion

Timestamp:

03/27/13 23:46:42 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -495 +495 @@
    [[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 \left ( \alpha^n_{i+1/2} - \omega_i v_{x,i} \right ) \right ) E^{n+1}_{i+1} - \left ( \psi \left ( \alpha^n_{i-1/2} + \omega_i v_{x,i} \right ) \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 + \left ( \bar{\psi} \left ( \alpha^n_{i+1/2} - \omega_i v_{x,i} \right ) \right ) E^{n}_{i+1} + \left ( \bar{\psi} \left ( \alpha^n_{i-1/2} + \omega_i v_{x,i} \right ) \right ) E^{n}_{i-1} +\bar{\psi}\phi^n_i e^n_i + \theta^n_i)]]   
 
-   [[latex(\left ( 1 +\psi \phi^n_i \right ) e^{n+1}_i - \left ( \psi \epsilon^n_i \right )E^{n+1}_i - \psi \omega_i v_{x,i} E^{n+1}_{i+1} + \psi \omega_i v_{x,i} E^{n+1}_{i-1} =\left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n - \bar{\psi} \omega_i v_{x,i}  \left ( E^{n}_{i+1}- E^{n}_{i-1} \right ) )]]   
+   [[latex(\left ( 1 +\psi \phi^n_i \right ) e^{n+1}_i - \left ( \psi \epsilon^n_i \right )E^{n+1}_i - \psi \omega_i v_{x,i} E^{n+1}_{i+1} + \psi \omega_i v_{x,i} E^{n+1}_{i-1} =\left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n + \bar{\psi} \omega_i v_{x,i}  \left ( E^{n}_{i+1}- E^{n}_{i-1} \right ) )]]   
 
 
 Now since the second equation has no spatial dependence, we can solve it for 
-   [[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 ( \psi \omega_i v_{x,i} \right ) E^{n+1}_{i+1} + \left ( \psi \omega_i v_{x,i} \right ) E^{n+1}_{i-1} + \left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n - \bar{\psi} \omega_i v_{x,i}  \left ( E^{n}_{i+1}- E^{n}_{i-1} \right ) \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 ( \psi \omega_i v_{x,i} \right ) E^{n+1}_{i+1} - \left ( \psi \omega_i v_{x,i} \right ) E^{n+1}_{i-1} + \left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n + \bar{\psi} \omega_i v_{x,i}  \left ( E^{n}_{i+1}- E^{n}_{i-1} \right ) \right \}} )]]   
 
 and plug the result into the first equation to get a matrix equation involving only one variable.
 
-   [[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 + \left ( \bar{\psi} \alpha^n_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha^n_{i-1/2} \right ) E^{n}_{i-1} + \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} + \epsilon^n_i \right ) \right ] E^{n+1}_i - \left ( \psi \left ( \alpha^n_{i+1/2} - \omega_i v_{x,i} \right ) \right ) E^{n+1}_{i+1} - \left ( \psi \left ( \alpha^n_{i-1/2} + \omega_i v_{x,i} \right ) \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 + \left ( \bar{\psi} \left ( \alpha^n_{i+1/2} - \omega_i v_{x,i} \right ) \right ) E^{n}_{i+1} + \left ( \bar{\psi} \left ( \alpha^n_{i-1/2} + \omega_i v_{x,i} \right ) \right ) E^{n}_{i-1} +\bar{\psi}\phi^n_i e^n_i + \theta^n_i})]]   
+
+   [[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 \left ( \alpha^n_{i+1/2}- \frac{\omega_i v_{x,i}}{1+\psi \phi^n_i} \right ) \right ) E^{n+1}_{i+1} - \left ( \psi \left ( \alpha^n_{i-1/2} + \frac{\omega_i v_{x,i}}{1+\psi \phi^n_i} \right ) \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 + \left ( \bar{\psi} \left ( \alpha^n_{i+1/2} - \frac{\omega_i v_{x,i}}{1+\psi \phi^n_i} \right ) \right ) E^{n}_{i+1} + \left ( \bar{\psi} \left ( \alpha^n_{i-1/2} + \frac{\omega_i v_{x,i}}{1+\psi \phi^n_i} \right ) \right ) E^{n}_{i-1} + \frac{ \phi^n_i}{ 1 +\psi \phi^n_i}  e^n_i+ \frac{1}{ 1 +\psi \phi^n_i}\theta^i_n})]]