Changes between Version 126 and Version 127 of FluxLimitedDiffusion

Timestamp:

03/30/13 23:14:34 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -405 +405 @@
 and then discretized as 
 
-  [[latex(e^{n+1}_i-e^{n}_i = \frac{\Delta t}{\Delta x}  \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) + \left ( \left ( \bar{\psi} - 1  \right ) \phi \right ) e^n_i +  \psi \phi e^{n+1}_i )]]
+  [[latex(e^{n+1}_i-e^{n}_i = \frac{1}{\Delta t}  \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) + \left ( \left ( \bar{\psi} - 1  \right ) \phi \right ) e^n_i +  \psi \phi e^{n+1}_i )]]
 
 which can be solved for 
-[[latex(e^{n+1}_i = e^{n}_i + \frac{1}{1 - \psi \phi} \left ( f \left ( e^n_i \right ) + g \left ( e^n_i,E^*,\nabla E^* \right ) \right ) )]]
+[[latex(e^{n+1}_i = e^{n}_i + \frac{\Delta t}{1 - \psi \phi} \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) )]]
 
 Then if we take the semi-discretized equation for E 
 
-  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - f \left ( e^n_i \right ) + g \left (e^n_i,E, \nabla E \right ) + \bar{\psi}\phi e^n_i - \psi \phi e^{n+1}_i )]]  
+  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - f \left ( e^n_i \right ) - g \left (E, \nabla E \right ) - \left ( \left ( \bar{\psi} - 1  \right ) \phi \right ) e^n_i -  \psi \phi e^{n+1}_i ) ]]  
 
 and then plugin the solution for e^n+1^,,i,, we get
 
-  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - f \left (e_0,E, \nabla E \right ) - \bar{\psi}\phi e^n_i - \frac{\psi \phi }{1-\psi \phi} \left ( \bar{\psi} \phi e^n_i + f \left ( e_0,E,\nabla E \right ) \right ) )]]  
+  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - f \left (e^n_i \right ) - g \left (E, \nabla E \right ) - \frac{1}{\Delta t} \left ( \left ( \left ( \bar{\psi} - 1  \right ) \phi \right ) e^n_i  - \psi \phi e^n_i - \frac{\psi \phi }{1-\psi \phi} \left ( f \left ( e^n_i \right ) + g \left (E,\nabla E \right ) \right ) \right ) )]]  
 
 which simplifies to