Changes between Version 175 and Version 176 of FluxLimitedDiffusion

Timestamp:

04/03/13 14:27:47 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -168 +168 @@
 In AstroBEAR this would look like:
 * Initialization
- * Prolongate, \(\rho\), \(\rho\mathbf{v}\), \(e\), \(E\), \(\dot{E}^I\), and \(\dot{E}^E\)
+ * Prolongate, \(\rho\), \(\rho\mathbf{v}\), \(e\), \(E\), and \(\dot{E}^I\)
 * Step 1
  * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\) and do physical BC's
  * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}^I_1\)
  * Update \(E_{2\mbox{mbc}}\) using \(\dot{E}^I_{2\mbox{mbc}}\)
- * Update \(e_{2\mbox{mbc}} using \(E_{2\mbox{mbc}}\), \(\dot{E}^I_{2\mbox{mbc}}\), and \(e_{2\mbox{mbc}}\)
+ * Update \(e_{2\mbox{mbc}}\) using \(E_{2\mbox{mbc}}\), \(\dot{E}^I_{2\mbox{mbc}}\), and \(e_{2\mbox{mbc}}\)
  * Update \(\dot{E}^I_0\) using pre IR and post IR \(E_0\)
  * Ghost \(e_{2\mbox{mbc}}\), \(E_{\mbox{mbc}+1}\), \(\dot{E}^I_{\mbox{mbc}+1}\)
@@ -187 +187 @@
  * Do second EH,,0,,
  * Do ER,,mbc,, --- Terms with \(\nabla E\) can be done without ghosting since EH did not 
-
  * Do ER,,0,, --- Terms with grad E can be done without ghosting since EH did not change \(E\).  The \(\nabla \cdot \mathbf{v}E\) term needs time centered face centered velocities which can be stored during the hydro update.
-
 
 [[CollapsibleEnd()]]
@@ -400 +398 @@
 == Alternative Splitting Method ==
 
+While the previous method for splitting the equation technically works, it is likely to lead to large differences in the radiation field and the energy fields at coarse fine boundaries since there is no radiative flux to coarsen from the explicit updates to keep the various AMR levels consistent.  There is one term that could potentially be coarsened and applied like any flux, but it is not the dominant term.  As a result it might be better to treat the entire radiative energy equation implicitly (though keeping the velocity constant)
+
+
   [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P\color{green}{-\lambda \nabla E})]]   
-  [[latex(\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{-\kappa_{0P}(4 \pi B-cE)} \color{red}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E} \color{red}{-\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E})]]  
-  [[latex(\frac{\partial E}{\partial t}  \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} = \color{red}{\kappa_{0P} (4 \pi B-cE)} \color{red}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E} \color{red}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )} \color{red}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E}  )]]  
-
-It is possible that with included the terms in orange in a semi-implicit method, the dynamic diffusion regime may be stable...  In any event, it costs very little to add all of the terms in orange to the implicit solve (using the old velocity).  Then the momentum update can be done explicitly - though using a time centered radiation field.
+  [[latex(\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{-\kappa_{0P}(4 \pi B-cE)} \color{magenta}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E} \color{magenta}{-\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E})]]  
+  [[latex(\frac{\partial E}{\partial t}  \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} = \color{red}{\kappa_{0P} (4 \pi B-cE)} \color{magenta}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E} \color{magenta}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )} \color{magenta}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E}  )]]  
+
+It is possible that with included the terms in magenta in a semi-implicit method, the dynamic diffusion regime may be stable...  In any event, it costs very little to add all of the terms in magenta to the implicit solve (using the old velocity).  Then the momentum update can be done explicitly - though using a time centered radiation field.
 
 
 [[CollapsibleStart(Operator Splitting 2)]]
 == Operator splitting 2 ==
-Krumholz et al. perform Implicit Radiative, Explicit Hydro, Explicit Radiative
 
 In AstroBEAR this would look like:
 * Initialization
- * Prolongate, d, p, e, E, Edot
+ * Prolongate, \(\rho\), \(\rho\mathbf{v}\), \(e\), \(E\), and \(\dot{E}\)
 * Step 1
- * Overlap d, p, e, E and do physical BC's
- * Do IR which updates e,,0,,, and E,,0,, using d,,1,,, e,,1,,, E,,1,,
- * Update E,,2*mbc,, using Edot,,2*mbc,,
- * Update e,,2*mbc,, using E,,2*mbc,,, Edot,,2*mbc,,, and e,,2*mbc,,
- * Update Edot,,0,, using pre IR and post IR E,,0,,
- * Ghost e,,2*mbc,,, E,,mbc+1,,, Edot,,mbc+1,,
- * Store EDot in child arrays to be prolongated
+ * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\) and do physical BC's
+ * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}_1\)
+ * Update \(E_{2\mbox{mbc}}\) using \(\dot{E}_{2\mbox{mbc}}\)
+ * Update \(e_{2\mbox{mbc}}\) using \(E_{2\mbox{mbc}}\), \(\dot{E}_{2\mbox{mbc}}\), and \(e_{2\mbox{mbc}}\)
+ * Update \(\dot{E}_0\) using pre IR and post IR \(E_0\)
+ * Ghost \(e_{2\mbox{mbc}}\), \(E_{\mbox{mbc}+1}\), \(\dot{E}_{\mbox{mbc}+1}\)
  * Do first EH,,mbc,,
- * Do ER,,mbc,, --- Only momentum terms.
+ * Do ER,,mbc,, --- Terms with \(\nabla E\) can be done without ghosting since EH did not change \(E\).  
+ * Store \(\dot{E}\) in child arrays to be prolongated
 * Step 2
- * Overlap d, p, e, E, and do physical BCs
- * Do IR which updates e,,0,,, and E,,0,, using d,,1,,, e,,1,,, E,,1,,
- * Update E,,mbc,, using Edot,,mbc,,
- * Update e,,2*mbc,, using E,,2*mbc,,, Edot,,2*mbc,,, and e,,2*mbc,,
- * Update Edot,,0,, using pre IR and post IR E,,0,,
- * Ghost e,,mbc,,, E,,1,,, Edot,,1,,
- * Store EDot in child arrays to be prolongated
+ * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\) and do physical BC's
+ * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}_1\)
+ * Update \(\dot{E}_0\) using pre IR and post IR \(E_0\)
+ * Update \(E_{1}\) using \(\dot{E}_{1}\)
+ * Ghost \(e_{mbc}\), \(E_{1}\), \(\dot{E}_{1}\)
  * Do second EH,,0,,
- * Do ER,,0,, --- Only momentum terms.
+ * Do ER,,0,, --- Terms with \(\nabla E\) can be done without ghosting since EH did not change \(E\).  
+
 
 [[CollapsibleEnd()]]