Changeset 4229

Timestamp:

Feb 6, 2007, 6:26:43 PM (18 years ago)

Author:

ole

Message:

Further thoughts on alpha in write-up

File:

• anuga_core/documentation/user_manual/old_pyvolution_documentation/limiting.tex (modified) (2 diffs)

anuga_core/documentation/user_manual/old_pyvolution_documentation/limiting.tex

@@ -136 +136 @@
 is ignored we have immediately that 
 \[
-  \alpha = 1 \mbox{for} \hmin \ge \epsilon or dz=0
-\]
-where the maximal bed elevation range $dz$ is defined as
-\[
-  dz = \max_i |z_i - z|
-\]
+  \alpha = 1 \mbox{ for } \hmin \ge \epsilon %or dz=0
+\]
+%where the maximal bed elevation range $dz$ is defined as
+%\[
+%  dz = \max_i |z_i - z|
+%\]
 
 If $\hmin < \epsilon$ we want to use the 'shallow' limiter just enough that
@@ -149 +149 @@
 \]
 or 
-\[
+\begin{equation} 
   \alpha(\tilde{h_i} - \bar{h_i}) > \epsilon - \bar{h_i}, \forall i
-\]
-Rearranging and solving for $\alpha$ one obtains the bound
-\[
-  \alpha > \frac{\epsilon - \bar{h_i}}{\tilde{h_i} - \bar{h_i}}, \forall i
-\] 
-
-Ensuring this holds true for all vertices on arrives at the definition
-\begin{equation}
-  \alpha = \max_{i} \frac{\epsilon - \bar{h_i}}{\tilde{h_i} - \bar{h_i}}
+  \label{eq:limiter bound}
 \end{equation} 
-which will guarantee that no vertex 'cuts' through the bed.
+
+There are two cases:
+\begin{enumerate} 
+  \item $\bar{h_i} \le \tilde{h_i}$: The deep water (limited using stage) 
+  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
+  \[
+    \tilde{h_i} > \epsilon  
+  \]
+  \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. 
+  In this case we need to find an $alpha$ that will ensure a positive depth.   
+  Rearranging Equation \ref{eq:limiter bound} and solving for $\alpha$ one 
+  obtains the bound
+  \[
+    \alpha < \frac{\epsilon - \bar{h_i}}{\tilde{h_i} - \bar{h_i}}, \forall i
+  \] 
+\end{enumerate} 
+
+Ensuring Equation \ref{eq:limiter bound} holds true for all vertices one 
+arrives at the definition
+\[
+  \alpha = \min_{i} \frac{\bar{h_i} - \epsilon}{\bar{h_i} - \tilde{h_i}}
+\]
+which will guarantee that no vertex 'cuts' through the bed. Finally, should
+$\bar{h_i} < \epsilon$ and therefore $\alpha < 0$, we suggest setting $alpha=0$.
+
+%Furthermore,
+%dropping the $\epsilon$ ensures that alpha is always positive and also  
+%provides a numerical safety {??)
+  
+  
+
+
+