Changeset 4236 for anuga_core/documentation

Timestamp:

Feb 7, 2007, 3:43:46 PM (18 years ago)

Author:

ole

Message:

More thoughts on limiters

File:

• anuga_core/documentation/user_manual/old_pyvolution_documentation/limiting.tex (modified) (4 diffs)

anuga_core/documentation/user_manual/old_pyvolution_documentation/limiting.tex

@@ -7 +7 @@
 \begin{document}
 
-
+Ole's thoughts on flux and slope limiting in ANUGA - 7 February 2007.
+
+This is a rough draft, please let me know if this makes sense.
+O
 
 \section{Flux limiting}
@@ -159 +162 @@
   vertex is at least as far away from the bed than the shallow water 
   (limited using depth). In this case we won't need any contribution from
-  $\bar{h_i}$ and set $alpha = 1$ reducing Equation \ref{eq:limiter bound} to
+  $\bar{h_i}$ and can accept any $alpha$. 
+  
+  E.g.\ $\alpha=1$ reduces Equation \ref{eq:limiter bound} to
   \[
     \tilde{h_i} > \epsilon  
   \]
+  whereas $\alpha=0$ yields
+  \[
+    \bar{h_i} > \epsilon        
+  \]
+  all well and good.
   \item $\bar{h_i} > \tilde{h_i}$: In this case the the deep water vertex is 
   closer to the bed than the shallow water vertex or even below the bed. 
@@ -179 +189 @@
 \]
 which will guarantee that no vertex 'cuts' through the bed. Finally, should
-$\bar{h_i} < \epsilon$ and therefore $\alpha < 0$, we suggest setting $alpha=0$.
+$\bar{h_i} < \epsilon$ and therefore $\alpha < 0$, we suggest setting 
+$alpha=0$ and similarly capping $\alpha$ at 1 just in case.
 
 %Furthermore,
@@ -195 +206 @@
 
 
-\section{Slope limiting (old)}
+\section{Slope limiting (old version)}
 
 Let $w, z, h$  be the stage, bed elevation and depth at the centroid and