Changeset 4377

Timestamp:

Apr 15, 2007, 7:03:53 PM (18 years ago)

Author:

steve

Message:

 

Location:

anuga_core/documentation/user_manual

Files:

• anuga_user_manual.tex (modified) (18 diffs)
• definitions.tex (modified) (2 diffs)
• graphics/step-five.pdf (added)
• graphics/step-reconstruct.pdf (added)

anuga_core/documentation/user_manual/anuga_user_manual.tex

@@ -1731 +1731 @@
     Module: \module{shallow\_water.shallow\_water\_domain}
 
-    Set the minimum depth (in meters) that will be recognised in 
+    Set the minimum depth (in meters) that will be recognised in
     the numerical scheme (including limiters and flux computations)
-    
-    Default value is $10^{-3}$ m, but by setting this to a greater value, 
-    e.g.\ for large scale simulations, the computation time can be 
-    significantly reduced. 
+
+    Default value is $10^{-3}$ m, but by setting this to a greater value,
+    e.g.\ for large scale simulations, the computation time can be
+    significantly reduced.
 \end{methoddesc}
 
@@ -2256 +2256 @@
   Here the numbers in \code{steps=12 (12)} indicate the number of steps taken and
   the number of first-order steps, respectively.\\
-  
-  The optional keyword argument \code{track_speeds=True} will 
-  generate a histogram of speeds generated by each triangle. The 
-  speeds relate to the size of the timesteps used by ANUGA and 
-  this diagnostics may help pinpoint problem areas where excessive speeds 
-  are generated.  
-  
+
+  The optional keyword argument \code{track_speeds=True} will
+  generate a histogram of speeds generated by each triangle. The
+  speeds relate to the size of the timesteps used by ANUGA and
+  this diagnostics may help pinpoint problem areas where excessive speeds
+  are generated.
+
   \end{funcdesc}
 
@@ -2669 +2669 @@
 
 
-  
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -2679 +2679 @@
 
 This chapter outlines the mathematics underpinning \anuga.
-  
-  
+
+
 
 \section{Model}
@@ -2728 +2728 @@
 in which $\eta$ is the Manning resistance coefficient.
 
-As demonstrated in our papers, \cite{modsim2005,Rob99l} these
+As demonstrated in our papers, \cite{ZR1999,nielsen2005} these
 equations provide an excellent model of flows associated with
 inundation such as dam breaks and tsunamis.
@@ -2736 +2736 @@
 
 We use a finite-volume method for solving the shallow water wave
-equations \cite{Rob99l}. The study area is represented by a mesh of
+equations \cite{ZR1999}. The study area is represented by a mesh of
 triangular cells as in Figure~\ref{fig:mesh} in which the conserved
 quantities of  water depth $h$, and horizontal momentum $(uh, vh)$,
@@ -2745 +2745 @@
 \begin{figure}
 \begin{center}
-\includegraphics[width=5.0cm,keepaspectratio=true]{step-five}
+\includegraphics[width=8.0cm,keepaspectratio=true]{graphics/step-five}
 \caption{Triangular mesh used in our finite volume method. Conserved
 quantities $h$, $uh$ and $vh$ are associated with the centroid of
@@ -2811 +2811 @@
 artificially introduced oscillations.
 
-Godunov's method (see \cite{Toro-92}) involves calculating the
+Godunov's method (see \cite{Toro1992}) involves calculating the
 numerical flux function $\HH(\cdot, \cdot ; \ \cdot)$ by exactly
 solving the corresponding one dimensional Riemann problem normal to
@@ -2819 +2819 @@
 \begin{figure}
 \begin{center}
-\includegraphics[width=5.0cm,keepaspectratio=true]{step-reconstruct}
+\includegraphics[width=8.0cm,keepaspectratio=true]{graphics/step-reconstruct}
 \caption{From the values of the conserved quantities at the centroid
 of the cell and its neighbouring cells, a discontinuous piecewise
@@ -2834 +2834 @@
 \section{Flux limiting}
 
-The shallow water equations are solved numerically using a 
+The shallow water equations are solved numerically using a
 finite volume method on unstructured triangular grid.
-The upwind central scheme due to Kurganov and Petrova is used as an 
+The upwind central scheme due to Kurganov and Petrova is used as an
 approximate Riemann solver for the computation of inviscid flux functions.
-This makes it possible to handle discontinuous solutions. 
-
-To alleviate the problems associated with numerical instabilities due to 
+This makes it possible to handle discontinuous solutions.
+
+To alleviate the problems associated with numerical instabilities due to
 small water depths near a wet/dry boundary we employ a new flux limiter that
 ensures that unphysical fluxes are never encounted.
 
 
-Let $u$ and $v$ be the velocity components in the $x$ and $y$ direction, 
+Let $u$ and $v$ be the velocity components in the $x$ and $y$ direction,
 $w$ the absolute water level (stage) and
-$z$ the bed elevation. The latter are assumed to be relative to the 
-same height datum. 
-The conserved quantities tracked by ANUGA are momentum in the 
-$x$-direction ($\mu = uh$), momentum in the $y$-direction ($\nu = vh$) 
+$z$ the bed elevation. The latter are assumed to be relative to the
+same height datum.
+The conserved quantities tracked by ANUGA are momentum in the
+$x$-direction ($\mu = uh$), momentum in the $y$-direction ($\nu = vh$)
 and depth ($h = w-z$).
 
-The flux calculation requires access to the velocity vector $(u, v)$ 
+The flux calculation requires access to the velocity vector $(u, v)$
 where each component is obtained as $u = \mu/h$ and $v = \nu/h$ respectively.
-In the presence of very small water depths, these calculations become 
+In the presence of very small water depths, these calculations become
 numerically unreliable and will typically cause unphysical speeds.
 
-We have employed a flux limiter which replaces the calculations above with 
+We have employed a flux limiter which replaces the calculations above with
 the limited approximations.
 \begin{equation}
-  \hat{u} = \frac{\mu}{h + h_0/h}, \bigskip \hat{v} = \frac{\nu}{h + h_0/h}, 
+  \hat{u} = \frac{\mu}{h + h_0/h}, \bigskip \hat{v} = \frac{\nu}{h + h_0/h},
 \end{equation}
-where $h_0$ is a regularisation parameter that controls the minimal 
+where $h_0$ is a regularisation parameter that controls the minimal
 magnitude of the denominator. Taking the limits we have for $\hat{u}$
 \[
-  \lim_{h \rightarrow 0} \hat{u} = 
+  \lim_{h \rightarrow 0} \hat{u} =
   \lim_{h \rightarrow 0} \frac{\mu}{h + h_0/h} = 0
 \]
-and 
+and
 \[
-  \lim_{h \rightarrow \infty} \hat{u} = 
+  \lim_{h \rightarrow \infty} \hat{u} =
   \lim_{h \rightarrow \infty} \frac{\mu}{h + h_0/h} = \frac{\mu}{h} = u
 \]
@@ -2886 +2886 @@
 
 
-ANUGA has a global parameter $H_0$ that controls the minimal depth which 
+ANUGA has a global parameter $H_0$ that controls the minimal depth which
 is considered in the various equations. This parameter is typically set to
 $10^{-3}$. Setting
@@ -2897 +2897 @@
   \left[ \frac{\mu}{h + h_0/h} \right]_{h = H_0} = \frac{\mu}{2 H_0}
 \]
-In general, for multiples of the minimal depth $N H_0$ one obtains 
+In general, for multiples of the minimal depth $N H_0$ one obtains
 \[
-  \left[ \frac{\mu}{h + h_0/h} \right]_{h = N H_0} = 
+  \left[ \frac{\mu}{h + h_0/h} \right]_{h = N H_0} =
   \frac{\mu}{H_0 (1 + 1/N^2)}
 \]
-which converges quadratically to the true value with the multiple N. 
+which converges quadratically to the true value with the multiple N.
 
 
@@ -2924 +2924 @@
 \]
 
-Let $\tilde{w_i}$ be the stage obtained from a gradient limiter 
+Let $\tilde{w_i}$ be the stage obtained from a gradient limiter
 limiting on stage only. The corresponding depth is then defined as
 \[
   \tilde{h_i} = \tilde{w_i} - z_i
 \]
-We would use this limiter in deep water which we will define (somewhat boldly) 
+We would use this limiter in deep water which we will define (somewhat boldly)
 as
 \[
@@ -2936 +2936 @@
 
 
-Similarly, let $\bar{w_i}$ be the stage obtained from a gradient 
+Similarly, let $\bar{w_i}$ be the stage obtained from a gradient
 limiter limiting on depth respecting the bed slope.
 The corresponding depth is defined as
@@ -2956 +2956 @@
 where $\alpha \in [0, 1]$.
 
-Since $\tilde{w_i}$ is obtained in 'deep' water where the bedslope 
-is ignored we have immediately that 
+Since $\tilde{w_i}$ is obtained in 'deep' water where the bedslope
+is ignored we have immediately that
 \[
   \alpha = 1 \mbox{ for } \hmin \ge \epsilon %or dz=0
@@ -2971 +2971 @@
   \alpha \tilde{h_i} + (1-\alpha) \bar{h_i} > \epsilon, \forall i
 \]
-or 
-\begin{equation} 
+or
+\begin{equation}
   \alpha(\tilde{h_i} - \bar{h_i}) > \epsilon - \bar{h_i}, \forall i
   \label{eq:limiter bound}
-\end{equation} 
+\end{equation}
 
 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 
+\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 can accept any $alpha$. 
-  
+  $\bar{h_i}$ and can accept any $alpha$.
+
   E.g.\ $\alpha=1$ reduces Equation \ref{eq:limiter bound} to
   \[
-    \tilde{h_i} > \epsilon  
+    \tilde{h_i} > \epsilon
   \]
   whereas $\alpha=0$ yields
   \[
-    \bar{h_i} > \epsilon        
+    \bar{h_i} > \epsilon
   \]
   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. 
-  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 
+  \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 
+  \]
+\end{enumerate}
+
+Ensuring Equation \ref{eq:limiter bound} holds true for all vertices one
 arrives at the definition
 \[
@@ -3009 +3009 @@
 \]
 which will guarantee that no vertex 'cuts' through the bed. Finally, should
-$\bar{h_i} < \epsilon$ and therefore $\alpha < 0$, we suggest setting 
+$\bar{h_i} < \epsilon$ and therefore $\alpha < 0$, we suggest setting
 $alpha=0$ and similarly capping $\alpha$ at 1 just in case.
 
 %Furthermore,
-%dropping the $\epsilon$ ensures that alpha is always positive and also  
+%dropping the $\epsilon$ ensures that alpha is always positive and also
 %provides a numerical safety {??)
-  
-  
+
+
 
 
@@ -3951 +3951 @@
 http://www.ga.gov.au/meta/ANZCW0703008022.html
 
-
+\bibitem[ZR1999]{ZR1999}
+\newblock {Catastrophic Collapse of Water Supply Reservoirs in Urban Areas}.
+\newblock C.~Zoppou and S.~Roberts.
+\newblock {\em ASCE J. Hydraulic Engineering}, 125(7):686--695, 1999.
+
+\bibitem[Toro1999]{Toro1992}
+\newblock Riemann problems and the waf method for solving the two-dimensional
+  shallow water equations.
+\newblock E.~F. Toro.
+\newblock {\em Philosophical Transactions of the Royal Society, Series A},
+  338:43--68, 1992.
+  
+\bibitem{KurNP2001}
+\newblock Semidiscrete central-upwind schemes for hyperbolic conservation laws
+  and hamilton-jacobi equations.
+\newblock A.~Kurganov, S.~Noelle, and G.~Petrova.
+\newblock {\em SIAM Journal of Scientific Computing}, 23(3):707--740, 2001.
 \end{thebibliography}{99}
 

anuga_core/documentation/user_manual/definitions.tex

@@ -17 +17 @@
 \newcommand{\mpi}{\textsc{mpi}}
 
-\newcommand{\hmin}{h_{\mbox{{\scriptstyle min}}}}
+\newcommand{\hmin}{h_{\min}}
 
 \newcommand{\UU}{\mathbf{U}}
@@ -39 +39 @@
 \newenvironment{displayedcode}
    {\begin{tabular}{p{0.41cm}l}}
-   {\end{tabular}}  
+{\end{tabular}}