Changes between Version 189 and Version 190 of FluxLimitedDiffusion

Timestamp:

04/05/13 13:25:25 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -488 +488 @@
 [[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 ) )]]
 
+or
+
+[[latex(e^{n+1}_i = e^{n}_i + \frac{1}{1 + \psi \phi_i} \left ( -\theta_i + \epsilon_i E^*_i \omega_i v^n_{x,i} \left ( E^*_{i+1}-E^*_{i-1} \right ) - \xi_i E^* \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, \nabla E \right ) - \frac{1}{\Delta t} \left ( \psi \phi  e^n_i -  \psi \phi e^{n+1}_i \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 ( \psi \phi_i  e^n_i -  \psi \phi_i e^{n+1}_i \right ) )]]  
 
 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^n_i \right ) - g \left (E, \nabla E \right ) - \frac{1}{\Delta t} \left ( \psi \phi 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 ) )]]  
+  [[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 ( \psi \phi_i e^n_i - \psi \phi_i e^n_i - \frac{\psi \phi_i }{1+\psi \phi_i} \left ( f \left ( e^n_i \right ) + g \left (E,\nabla E \right ) \right ) \right ) )]]  
 
 which simplifies to 
 
-  [[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 ) - \frac{1}{1+\psi \phi} \left ( f \left ( e^n_i \right ) + g \left ( 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 ) - \frac{1}{1+\psi \phi_i} \left ( f \left ( e^n_i \right ) + g \left ( E,\nabla E \right ) \right ) )]]  
 
 Now we have 1 equation with 1 variable that we can solve implicitly using hypre, and then we can use \(E^{n+1}\) and \(E^n\) to construct \(E^*\) which we can plug into the equation for \(e^{n+1}\)
@@ -526 +531 @@
 
 which gives [[latex(\Delta t < \xi \frac{T_0}{4 \Gamma \left | \frac{\partial e}{\partial t} \right | } = \xi \frac{T_0}{4 \Gamma \left | - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E -\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E  \right | } )]]
+
+or in discretized form
+
+[[latex(\xi \frac{T^n_i}{4 \Gamma \left | \frac{1}{1 + \psi \phi_i} \left ( -\theta_i + \epsilon_i E^*_i \omega_i v^n_{x,i} \left ( E^*_{i+1}-E^*_{i-1} \right ) - \xi_i E^* \right ) \right |} )]]
 
 
@@ -556 +565 @@
 and 
 
-[[latex(\epsilon^n_i=c\Delta t \kappa^n_{0P,i})]]
-
-and 
-
-[[latex(\phi = \Delta t 4 \pi \kappa^n_{0P,i} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) )]]
-
-and
-
-[[latex(\theta = - \Delta t f(e^n_i) =  \Delta t 4 \pi \kappa^n_{0P,i} B \left ( T^n_i \right ) )]]
+[[latex(\epsilon_i=c\Delta t \kappa_{0P,i})]]
+
+and 
+
+[[latex(\phi_i- = \Delta t 4 \pi \kappa_{0P,i} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) )]]
+
+and
+
+[[latex(\theta_i = - \Delta t f(e^n_i) =  \Delta t 4 \pi \kappa_{0P,i} B \left ( T^n_i \right ) )]]
 
 and
@@ -572 +581 @@
 and
 
-[[latex(\xi = \Delta t \frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c} )]]
-and
-
-where [[latex( \kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,i+1}}{2} )]]
+[[latex(\xi_i = \Delta t \frac{3-R_{2,i}}{2}\kappa_{0P,i}\frac{v_i^2}{c} )]]
+and
+
+where [[latex( \kappa_{0R,i+1/2} = \frac{\kappa_{0R,i}+\kappa_{0R,i+1}}{2} )]]
 
 and