Changes between Version 141 and Version 142 of FluxLimitedDiffusion

Timestamp:

03/31/13 15:04:01 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -528 +528 @@
 [[latex(R_{2,i} = \lambda_{i}+\lambda_{i}^2 R_{i}^2)]]
 
-
-=== Time Discretization ===
-
-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 = \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
+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 ] \\
+\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)]]