Changes between Version 142 and Version 143 of FluxLimitedDiffusion

Timestamp:

03/31/13 15:34:18 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -530 +530 @@
 Which we can arrange into the following form
 
-
-\left( 1 + \psi \left ( \alpha^n_{i+1/2} +  \alpha^n_{i-1/2} - \zeta^n_{i+1/2} v^n_{x,i+1/2} + zeta^n_{i-1/2} v^n_{x,i-1/2} \right ) E^{n+1}_i  - E^{n}_i & = & \left [ \alpha^n_{i+1/2} \left ( \psi E^{n+1}_{i+1} + \bar{\psi} E^{n}_{i+1} - \bar{\psi} E^n_{i} \right ) - \alpha^n_{i-1/2} \left ( \bar{\psi} E^{n}_i - \psi E^{n+1}_{i-1} - \bar{\psi}E^{n}_{i-1} \right ) \right ] \\
- & + & \left [ \zeta^n_{i+1/2} v^n_{x,i+1/2} \left ( \psi E^{n+1}_{i+1} + \bar{\psi} E^{n}_{i+1}  + \bar{\psi} E^n_{i} \right ) - \zeta^n_{i-1/2} v^n_{x,i-1/2}\left ( \bar{\psi} E^n_i + \psi E^{n+1}_{i-1} + \bar{\psi}E^{n}_{i-1} \right ) \right ] \\
- & - & \frac{1}{1-\psi \phi}  \left [ \theta + \epsilon \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) + \omega v^n_x \left ( \psi E^{n+1}_{i+1} + \bar{\psi} E^n_{i+1} - \psi E^{n+1}_{i-1} - \bar{\psi} E^n_{i-1} \right ) - \xi \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) \right ] \\
+{{{
+#!latex
+\begin{eqnarray}
+ & \left( 1 + \psi \left ( \alpha^n_{i+1/2} +  \alpha^n_{i-1/2} - \zeta^n_{i+1/2} v^n_{x,i+1/2} + zeta^n_{i-1/2} v^n_{x,i-1/2} + \frac{\epsilon + \xi}{1-\psi \phi} \right ) E^{n+1}_i   \\
++ & \left ( \psi \left (  -alpha^n_{i+1/2} - \zeta^n_{i+1/2} v^n_{x,i+1/2} - \frac{\omega v^n_x}{1-\psi \phi} \right ) \right ) E^{n+1}_{i+1}  \\
++ & \left ( \psi \left (  -alpha^n_{i-1/2} + \zeta^n_{i-1/2} v^n_{x,i-1/2} + \frac{\omega v^n_x}{1-\psi \phi} \right ) \right ) E^{n+1}_{i-1}  \\
+= & \left ( 1 + \bar{\psi} \left ( -\alpha^n_{i+1/2} - \alpha^n_{i-1/2} + \zeta^n_{i+1/2} v^n_{x,i+1/2} -  \zeta^n_{i-1/2} v^n_{x,i-1/2} - \frac{\epsilon + \xi}{1-\psi \phi} \right ) \right ) E^{n}_i  \\
++ & \left ( \bar{\psi} \left (  alpha^n_{i+1/2} + \zeta^n_{i+1/2} v^n_{x,i+1/2} + \frac{\omega v^n_x}{1-\psi \phi} \right ) \right ) E^{n}_{i+1}  \\
++ & \left ( \bar{\psi} \left (  alpha^n_{i-1/2} - \zeta^n_{i-1/2} v^n_{x,i-1/2} - \frac{\omega v^n_x}{1-\psi \phi} \right ) \right ) E^{n}_{i-1}  \\
+- & \frac{\theta}{1-\psi \phi}   \\
 \end{eqnarray}
 
-
-
-   [[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 ) )]]   
+}}}