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16% what packages to use.  Three useful packages are the following:
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18
19
20\usepackage{url}      % for URLs and DOIs
21\usepackage{amsmath}  % many want amsmath extensions
22\usepackage{amsfonts}
23\usepackage{underscore}
24\usepackage{epstopdf}
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27% Create title and authors using \verb|\maketitle|.  Separate authors by
28% \verb|\and| and put addresses in \verb|\thanks| command with
29% \verb|\url| command \verb|\protect|ed.
30\title{On The Validation of A Hydrodynamic Model}
31
32\author{
33D.~S.~Gray\thanks{Natural Hazard Impacts Project, Geospatial and
34Earth Monitoring Division, Geoscience Australia, Symonston,
35\textsc{Australia}. \protect\url{mailto:Duncan.Gray@ga.gov.au}}\footnotemark[1]
36\and
37T.~Baldock\thanks{University of Queensland, Brisbane, \textsc{Australia}.
38\protect\url{mailto:tom.baldock@uq.edu.au}}\footnotemark[2]
39\and
40O.~M.~Nielsen\footnotemark[1]
41\and
42M.~J.~Sexton\footnotemark[1]
43\and
44N.~Bartzis\footnotemark[1]
45\and
46S.~G.~Roberts\thanks{Department of Mathematics, Australian National University,
47Canberra, \textsc{Australia}. \protect\url{mailto:stephen.roberts@anu.edu.au}}}
48
49\date{30 May 2008}
50
51\newcommand{\ANUGA}{\textsc{ANUGA}}
52\newcommand{\Python}{\textsc{Python}}
53\newcommand{\VPython}{\textsc{VPython}}
54\newcommand{\pypar}{\textsc{mpi}}
55\newcommand{\Metis}{\textsc{Metis}}
56\newcommand{\mpi}{\textsc{mpi}}
57
58\newcommand{\UU}{\mathbf{U}}
59\newcommand{\VV}{\mathbf{V}}
60\newcommand{\EE}{\mathbf{E}}
61\newcommand{\GG}{\mathbf{G}}
62\newcommand{\FF}{\mathbf{F}}
63\newcommand{\HH}{\mathbf{H}}
64\newcommand{\SSS}{\mathbf{S}}
65\newcommand{\nn}{\mathbf{n}}
66
67\newcommand{\code}[1]{\texttt{#1}}
68
69
70
71
72\begin{document}
73
74% Use default \verb|\maketitle|.
75\maketitle
76
77% Use the \verb|abstract| environment.
78\begin{abstract}
79Modelling the effects on the built environment of natural hazards such
80as riverine flooding, storm surges and tsunami is critical for
81understanding their economic and social impact on our urban
82communities.  Geoscience Australia and the Australian National
83University have developed a hydrodynamic inundation modelling tool
84called \ANUGA{} to help simulate the impact of these hazards.
85The core of \ANUGA{} is a \Python{} implementation of a finite-volume method
86for solving the conservative form of the Shallow Water Wave equation.
87In this paper we describe the model, the architecture and a range of
88validations that have been carried out to establish confidence in the model.
89
90
91\ANUGA{} is available as Open Source to enable
92free access to the software and allow the scientific community to
93use, validate and contribute to the software in the future.
94
95%This method allows the study area to be represented by an unstructured
96%mesh with variable resolution to suit the particular problem.  The
97%conserved quantities are water level (stage) and horizontal momentum.
98%An important capability of ANUGA is that it can robustly model the
99%process of wetting and drying as water enters and leaves an area. This
100%means that it is suitable for simulating water flow onto a beach or
101%dry land and around structures such as buildings.
102
103
104\end{abstract}
105
106% By default we include a table of contents in each paper.
107%\tableofcontents
108
109% Use \verb|\section|, \verb|\subsection|, \verb|\subsubsection| and
110% possibly \verb|\paragraph| to structure your document.  Make sure
111% you \verb|\label| them for cross-referencing with \verb|\ref|\,.
112
113%\clearpage
114\section{Introduction}
115\label{sec:intro}
116
117The Indian Ocean tsunami on 26 December 2004 demonstrated the
118potentially catastrophic consequences of natural hazards.  While the
119scale of the impact from such events is not common, smaller-scale
120tsunami regularly threaten coastal communities
121around the world. Earthquakes which occur in the Java Trench near
122Indonesia (e.g.~\cite{TsuMIS1995} or \cite{Baldwin-2006}) and along
123the Puysegur Ridge to the south of New Zealand (e.g.~\cite{LebKC1998})
124have potential to generate tsunami that may threaten Australia's
125northwestern and southeastern coastlines. In addition, the
126preferential development of Australian coastal corridors means that
127inundation from hydrological disasters such as tsunami or storm-surge
128of even a few hundred metres beyond the shoreline has increased
129potential to cause significant disruption and loss. The extent of
130inundation is critically linked to the event, tidal conditions,
131bathymetry and topography and it not feasible to make impact
132predictions using heuristics alone.
133
134Hydrodynamic modelling allows impacts from flooding, storm-surge and
135tsunami to be better understood, their impacts to be anticipated and,
136with appropriate planning, their effects to be mitigated.  Geoscience
137Australia in collaboration with the Mathematical Sciences Institute,
138Australian National University, is developing a software application
139called \ANUGA{} to model the hydrodynamics of floods, storm surges and
140tsunami. These hazards are modelled using the conservative shallow
141water equations which are described in section~\ref{sec:model}. In
142\ANUGA{} these equations are solved using a finite volume method as
143described in section~\ref{sec:fvm}.  A more complete discussion of the
144method can be found in \cite{modsim2005} where the model and solution
145technique is validated on a standard tsunami benchmark data set
146or in \cite{Roberts2007} where parallelisation of ANUGA is discussed.
147This modelling capability is part of
148Geoscience Australia's ongoing research effort to model and
149understand the potential impact from natural hazards in order to
150reduce their impact on Australian communities (see \cite{Nielsen2006}).
151\ANUGA{} is currently being trialled for flood
152modelling (see \cite{Rigby2008}).
153
154Section~\ref{sec:software} describes the software implementation and
155the API while section~\ref{sec:validation} presents some
156validation results.
157
158
159\section{Model}
160\label{sec:model}
161
162The shallow water wave equations are a system of differential
163conservation equations which describe the flow of a thin layer of
164fluid over terrain. The form of the equations are:
165\[
166\frac{\partial \UU}{\partial t}+\frac{\partial \EE}{\partial
167x}+\frac{\partial \GG}{\partial y}=\SSS
168\]
169where $\UU=\left[ {{\begin{array}{*{20}c}
170 h & {uh} & {vh} \\
171\end{array} }} \right]^T$ is the vector of conserved quantities; water depth
172$h$, $x$-momentum $uh$ and $y$-momentum $vh$. Other quantities
173entering the system are bed elevation $z$ and stage (absolute water
174level) $w$, where the relation $w = z + h$ holds true at all times.
175The fluxes in the $x$ and $y$ directions, $\EE$ and $\GG$ are given
176by
177\[
178\EE=\left[ {{\begin{array}{*{20}c}
179 {uh} \hfill \\
180 {u^2h+gh^2/2} \hfill \\
181 {uvh} \hfill \\
182\end{array} }} \right]\mbox{ and }\GG=\left[ {{\begin{array}{*{20}c}
183 {vh} \hfill \\
184 {vuh} \hfill \\
185 {v^2h+gh^2/2} \hfill \\
186\end{array} }} \right]
187\]
188and the source term (which includes gravity and friction) is given
189by
190\[
191\SSS=\left[ {{\begin{array}{*{20}c}
192 0 \hfill \\
193 -{gh(z_{x} + S_{fx} )} \hfill \\
194 -{gh(z_{y} + S_{fy} )} \hfill \\
195\end{array} }} \right]
196\]
197where $S_f$ is the bed friction. The friction term is modelled using
198Manning's resistance law
199\[
200S_{fx} =\frac{u\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}\mbox{ and }S_{fy}
201=\frac{v\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}
202\]
203in which $\eta$ is the Manning resistance coefficient.
204
205As demonstrated in our papers, \cite{modsim2005,Rob99l} these
206equations provide an excellent model of flows associated with
207inundation such as dam breaks and tsunamis.
208
209\section{Finite Volume Method}
210\label{sec:fvm}
211
212We use a finite-volume method for solving the shallow water wave
213equations \cite{Rob99l}. The study area is represented by a mesh of
214triangular cells as in Figure~\ref{fig:mesh} in which the conserved
215quantities of  water depth $h$, and horizontal momentum $(uh, vh)$,
216in each volume are to be determined. The size of the triangles may
217be varied within the mesh to allow greater resolution in regions of
218particular interest.
219
220\begin{figure}
221\begin{center}
222%\includegraphics[width=5.0cm,keepaspectratio=true]{step-five}
223\caption{Triangular mesh used in our finite volume method. Conserved
224quantities $h$, $uh$ and $vh$ are associated with the centroid of
225each triangular cell.} \label{fig:mesh}
226\end{center}
227\end{figure}
228
229The equations constituting the finite-volume method are obtained by
230integrating the differential conservation equations over each
231triangular cell of the mesh. Introducing some notation we use $i$ to
232refer to the $i$th triangular cell $T_i$, and ${\cal N}(i)$ to the
233set of indices referring to the cells neighbouring the $i$th cell.
234Then $A_i$ is the area of the $i$th triangular cell and $l_{ij}$ is
235the length of the edge between the $i$th and $j$th cells.
236
237By applying the divergence theorem we obtain for each volume an
238equation which describes the rate of change of the average of the
239conserved quantities within each cell, in terms of the fluxes across
240the edges of the cells and the effect of the source terms. In
241particular, rate equations associated with each cell have the form
242$$
243 \frac{d\UU_i }{dt}+ \frac1{A_i}\sum\limits_{j\in{\cal N}(i)} \HH_{ij} l_{ij} = \SSS_i
244$$
245where
246\begin{itemize}
247\item $\UU_i$ the vector of conserved quantities averaged over the $i$th cell,
248\item $\SSS_i$ is the source term associated with the $i$th cell,
249and
250\item $\HH_{ij}$ is the outward normal flux of
251material across the \textit{ij}th edge.
252\end{itemize}
253
254
255%\item $l_{ij}$ is the length of the edge between the $i$th and $j$th
256%cells
257%\item $m_{ij}$ is the midpoint of
258%the \textit{ij}th edge,
259%\item
260%$\mathbf{n}_{ij} = (n_{ij,1} , n_{ij,2})$is the outward pointing
261%normal along the \textit{ij}th edge, and The
262
263The flux $\HH_{ij}$ is evaluated using a numerical flux function
264$\HH(\cdot, \cdot ; \ \cdot)$ which is consistent with the shallow
265water flux in the sense that for all conservation vectors $\UU$ and normal vectors $\nn$
266$$
267H(\UU,\UU;\ \nn) = \EE(\UU) n_1 + \GG(\UU) n_2 .
268$$
269
270Then
271$$
272\HH_{ij}  = \HH(\UU_i(m_{ij}),
273\UU_j(m_{ij}); \mathbf{n}_{ij})
274$$
275where $m_{ij}$ is the midpoint of the \textit{ij}th edge and
276$\mathbf{n}_{ij}$ is the outward pointing normal, with respect to the $i$th cell, on the
277\textit{ij}th edge. The function $\UU_i(x)$ for $x \in
278T_i$ is obtained from the vector $\UU_k$ of conserved average values for the $i$th and
279neighbouring  cells.
280
281We use a second order reconstruction to produce a piece-wise linear
282function construction of the conserved quantities for  all $x \in
283T_i$ for each cell (see Figure~\ref{fig:mesh:reconstruct}. This
284function is allowed to be discontinuous across the edges of the
285cells, but the slope of this function is limited to avoid
286artificially introduced oscillations.
287
288Godunov's method (see \cite{Toro-92}) involves calculating the
289numerical flux function $\HH(\cdot, \cdot ; \ \cdot)$ by exactly
290solving the corresponding one dimensional Riemann problem normal to
291the edge. We use the central-upwind scheme of \cite{KurNP2001} to
292calculate an approximation of the flux across each edge.
293
294\begin{figure}
295\begin{center}
296%\includegraphics[width=5.0cm,keepaspectratio=true]{step-reconstruct}
297\caption{From the values of the conserved quantities at the centroid
298of the cell and its neighbouring cells, a discontinuous piecewise
299linear reconstruction of the conserved quantities is obtained.}
300\label{fig:mesh:reconstruct}
301\end{center}
302\end{figure}
303
304In the computations presented in this paper we use an explicit Euler
305time stepping method with variable timestepping adapted to the
306observed CFL condition.
307
308
309\section{Software Implementation}
310\label{sec:software}
311
312\ANUGA{} is mostly written in the object-oriented programming
313language \Python{} with computationally intensive parts implemented
314as highly optimised shared objects written in C.
315
316\Python{} is known for its clarity, elegance, efficiency and
317reliability. Complex software can be built in \Python{} without undue
318distractions arising from idiosyncrasies of the underlying software
319language syntax. In addition, \Python{}'s automatic memory management,
320dynamic typing, object model and vast number of libraries means that
321software can be produced quickly and can be readily adapted to
322changing requirements throughout its lifetime.
323
324The fundamental object in \ANUGA{} is the \code{Domain} which
325inherits functionality from a hierarchy of increasingly specialised
326classes starting with a basic structural Mesh to classes implementing
327the finite-volume scheme described in section \ref{sec:fvm}. Other classes
328are \code{Quantity} which represents values of one variable across the mesh
329along with their associated operations, \code{Geospatial_data} which
330represents georeferenced elevation data and a collection of \code{Boundary}
331classes which allows for a 'pluggable' way of driving the model.
332The conserved quantities updated automatically by the numerical scheme
333are stage (water level) $w$, $x$-momentum $uh$ and $y$-momentum
334$vh$. The quanitites elevation $z$ and friction $\eta$ are
335quantities that are not updated automatically but can be changed explicitly
336during run-time if the user wishes to do so.
337
338To set up a scenario the user specifies the study area along with any internal
339regions where increased mesh resolution is required. External edges may
340be labelled using symbolic tags which are subsequently used to bind
341boundary condition objects to tagged segments of the mesh boundary.
342The mesh is then generated using \ANUGA{}'s built-in mesh generator and
343converted into the \code{Domain} object which provides all methods used to
344setup and run the flow simulation. Figure \ref{fig:anuga mesh} shows an example of a mesh generated by \ANUGA{}.
345\begin{figure}
346\begin{center}
347\includegraphics[width=4in,keepaspectratio=true]{triangular-mesh}
348\caption{Triangular mesh used in our finite volume method. Conserved
349quantities $h$, $uh$ and $vh$ are associated with each triangular cell.}
350\label{fig:anuga mesh}
351\end{center}
352\end{figure}
353
354
355
356Next step is to setup initial conditions for each \code{Quantity} object. For
357the elevation $z$ this is typically obtained from bathymetric and
358topographic data sets. Setting initial values for quantities is done
359through the method \code{domain.set_quantity(name, X, location,
360region)} where name is the name of the quantity (e.g.\ 'stage',
361'xmomentum', 'ymomentum', 'elevation' or 'friction').  The variable X
362represents the source data for populating the quantity and may take
363one of the following forms:
364
365\begin{itemize}
366\item A constant value as in \code{domain.set_quantity('stage', 1)} which
367will set the initial water level to 1 m everywhere.
368\item Another quantity or a linear combination of quantities.  If \code{q1}
369and \code{q2} are two arbitrary quantities defined within the same domain,
370the expression \code{domain.set_quantity('stage', q1*(3*q2 + 5))} will set the stage
371quantity accordingly.  One common application of this would be to
372assign the stage as a constant depth above the bed elevation.
373\item An arbitrary function (or a callable object), \code{f(x, y)}, where \code{x}
374and \code{y} are assumed to be vectors. The quantity will be assigned values by
375evaluating \code{f} at each location within the mesh.
376\item An arbitrary set of points and associated values (wrapped into a
377Geospatial_data object). The points need not coincide with triangle
378vertices or centroids and a penalised least squares technique is
379employed to populate the quantity in a smooth and stable way.
380Since the least squares technique can be time consuming for large
381problems, \code{set_quantity} employs a caching technique which automatically
382decides whether to perform the computations or retrieve them from a
383cache.  This will typically speed up the build by several orders of
384magnitude after each computation has been performed once.
385\item A filename containing points and attributes.
386\item A Numerical Python array (or a list of numbers) ordered
387according to the internal data structure.
388\end{itemize}
389The parameter \code{location} determines whether the values should be
390assigned to triangle edge, midpoints or vertices and \code{region} allows the
391operation to be restricted to a region specified by a symbolic tag or
392a set of indices.
393
394Boundary conditions are bound to symbolic tags through the method
395\code{domain.set_boundary} which takes as input a lookup table (implemented
396as a Python dictionary) of the form \code{\{tag:~boundary_object\}}.
397The boundary objects are all assumed to be callable functions of vectors x
398and y.  Several predefined standard boundary objects are available and
399it is relatively straightforward to define problem-specific custom
400boundaries if needed.  The predefined boundary conditions include
401Dirichlet, Reflective, Transmissive, Temporal, and Spatio-Temporal
402boundaries.
403
404Forcing terms can be written according to a fixed protocol and added
405to the model using the idiom \code{domain.forcing_terms.append(F)} where \code{F} is
406assumed to be a user-defined callable object.
407
408When the simulation is running, the length of each time step is
409determined from the maximal speeds encountered and the sizes of
410triangles in order not to violate the CFL condition which specifies
411that no information should skip any triangles in one time step.  With
412large speeds and small triangles, time steps can become very small.
413In order to access the state of the simulation at regular time
414intervals, \ANUGA{} uses the method evolve:
415\begin{verbatim}
416For t in domain.evolve(yieldstep, duration):
417    <model interrogation and modification>
418\end{verbatim}
419The parameter \code{duration} specifies the time period over which
420evolve operates, and control is passed to the body of the for-loop at
421each fixed time step called \code{yieldstep}.  The internal workings
422of the numerical scheme and its variable time stepping are thus
423decoupled from the fixed time stepping of the evolve loop.  This means
424that the user of the API may access the model at fixed timesteps to
425e.g.\ store model outputs, interrogate quantities or change the model
426itself at runtime. The evolve method has been implemented using a
427Python generator hence the reference to 'yield' in the parameter name.
428
429%Figure \ref{fig:beach runup} shows a simulation of water flowing onto a
430%hypothetical beach with obstacles.
431%A number of complex patterns are captured in this example including a shock where water reflected off the wall far (at the right hand side) meets the main flow. Other physical features are the standing waves and interference patterns.
432%See the \ANUGA{} User Manual at \url{http://sourceforge.net/projects/anuga} for more details and examples.
433%\begin{figure}
434%\begin{center}
435%\includegraphics[width=4in,keepaspectratio=true]{runup}
436%\caption{A hypothetical runup scenario.}
437%\label{fig:beach runup}
438%\end{center}
439%\end{figure}
440
441
442
443\section{Validation}
444\label{sec:validation} The process of validating the \ANUGA{}
445application is in its early stages, however initial indications are
446encouraging.
447
448\subsection{Okushiri Wavetank Validation}
449
450As part of the Third International Workshop on Long-wave Runup
451Models in 2004 (\url{http://www.cee.cornell.edu/longwave}), four
452benchmark problems were specified to allow the comparison of
453numerical, analytical and physical models with laboratory and field
454data. One of these problems describes a wave tank simulation of the
4551993 Okushiri Island tsunami off Hokkaido, Japan \cite{MatH2001}. A
456significant feature of this tsunami was a maximum run-up of 32~m
457observed at the head of the Monai Valley. This run-up was not
458uniform along the coast and is thought to have resulted from a
459particular topographic effect. Among other features, simulations of
460the Hokkaido tsunami should capture this run-up phenomenon.
461
462\begin{figure}[htbp]
463%\centerline{\includegraphics[width=4in]{okushiri-gauge-5.eps}}
464\centerline{\includegraphics[width=4in]{ch5.png}}
465\centerline{\includegraphics[width=4in]{ch7.png}}
466\centerline{\includegraphics[width=4in]{ch9.png}}
467\caption{Comparison of wave tank and \ANUGA{} water stages at gauge
4685,7 and 9.}\label{fig:val}
469\end{figure}
470
471
472\begin{figure}[htbp]
473\centerline{\includegraphics[width=4in]{okushiri-model.jpg}}
474\caption{Complex reflection patterns and run-up into Monai Valley
475simulated by \ANUGA{} and visualised using our netcdf OSG
476viewer.}\label{fig:run}
477\end{figure}
478
479
480The wave tank simulation of the Hokkaido tsunami was used as the
481first scenario for validating \ANUGA{}. The dataset provided
482bathymetry and topography along with initial water depth and the
483wave specifications. The dataset also contained water depth time
484series from three wave gauges situated offshore from the simulated
485inundation area. The \ANUGA{} model comprised $41404$ triangles
486and took about $2000$ s to run on a standard PC or $1500$ s
487on a 64-bit Opteron 2000 series Linux server.
488
489Figure~\ref{fig:val} compares the observed wave tank and modelled
490\ANUGA{} water depth (stage height) at one of the gauges. The plots
491show good agreement between the two time series, with \ANUGA{}
492closely modelling the initial draw down, the wave shoulder and the
493subsequent reflections. The discrepancy between modelled and
494simulated data in the first 10 seconds is due to the initial
495condition in the physical tank not being uniformly zero. Similarly
496good comparisons are evident with data from the other two gauges.
497Additionally, \ANUGA{} replicates exceptionally well the 32~m Monai
498Valley run-up, and demonstrates its occurrence to be due to the
499interaction of the tsunami wave with two juxtaposed valleys above
500the coastline. The run-up is depicted in Figure~\ref{fig:run}.
501
502This successful replication of the tsunami wave tank simulation on a
503complex 3D beach is a positive first step in validating the \ANUGA{}
504modelling capability.
505
506\subsection{Manning's Friction Model Validation}
507
508% Validation UQ friction
509% at X:\anuga_validation\uq_friction_2007
510% run run_dam.py to create sww file and .csv files
511% run plot.py to create graphs, and move them here
512\begin{figure}[htbp]
513\centerline{\includegraphics[width=4in]{uq-friction-depth}}
514\caption{Comparison of wave tank and \ANUGA{} water height at .4 m
515  from the gate, simulated using a Mannings friction of 0.0 and 0.1.}\label{fig:uq-friction-depth}
516\end{figure}
517
518 The bed friction is modelled in ANUGA using the Manning's
519 model. Validation of this model was carried out by comparing results
520 from ANUGA against experimental results from flume wave tanks. The
521experiments were carried out at the Gordon McKay Hydraulics Laboratory
522at St Lucia, University of Queensland.
523
524%The Manning's friction model is
525
526%To validate the friction model
527
528\subsection{Stage and Velocity Validation in a Flume}
529% Validation UQ flume
530% at X:\anuga_validation\uq_sloped_flume_2008
531% run run_dam.py to create sww file and .csv files
532% run plot.py to create graphs heere automatically
533% The Coasts and Ports '2007 paper is in TRIM d2007-17186
534\begin{figure}[htbp]
535\centerline{\includegraphics[width=4in]{uq-flume-depth}}
536\caption{Comparison of wave tank and \ANUGA{} water height at .4 m
537  from the gate}\label{fig:uq-flume-depth}
538\end{figure}
539
540\begin{figure}[htbp]
541\centerline{\includegraphics[width=4in]{uq-flume-velocity}}
542\caption{Comparison of wave tank and \ANUGA{} water velocity at .45 m
543  from the gate}\label{fig:uq-flume-velocity}
544\end{figure}
545
546Flume experiments caried out at the University of Queensland has also
547been used for validating the water height and velocity predicted by
548\ANUGA{}.  The Flume was set up for Dam-break experiments, having a
549water reservior at one end.  The flume was glass-sided, 3m long, 0.4m
550in wide, and 0.4m deep, with a PVC bottom. The reservoir in the flume
551was 0.75m long.  For this experiment the reservoir water was 0.2m
552deep. At time zero the reservoir gate is opened and the water flows
553into the other side of the flume.  The water ran up a flume slope of
5540.03 m/m.  To accurately model the bed surface a Manning's friction
555value of 0.01, representing PVC was used.
556
557% Neale, L.C. and R.E. Price.  Flow characteristics of PVC sewer pipe.
558% Journal of the Sanitary Engineering Division, Div. Proc 90SA3, ASCE.
559% pp. 109-129.  1964.
560
561Acoustic displacement sensors that produced a voltage that changed
562with the water depth was positioned 0.4m from the reservoir gate. The
563water velocity was measured with an Acoustic Doppler Velocimeter 0.45m
564from the reservoir gate.  This sensor only produced reliable results 4
565seconds after the reservoir gate opened, due to limitations of the sensor.
566
567Figure~\ref{fig:uq-flume-depth} show that ANUGA predicts the actual
568water depth very well, with the exception of the fluid tip-region. The
569water velocity is also predicted accurately.
570
571\subsection{Runup of Solitary wave on circular island wavetank validation}
572
573Figure \ref{fig:circular screenshots} shows a sequence of screenshots depicting the evolution of the solitary wave as it hits the circular island.
574
575\begin{figure}[htbp]
576\centerline{
577  \includegraphics[width=5cm]{circular1.png}
578  \includegraphics[width=5cm]{circular2.png}}
579\centerline{
580  \includegraphics[width=5cm]{circular3.png}
581  \includegraphics[width=5cm]{circular4.png}}
582\centerline{
583  \includegraphics[width=5cm]{circular5.png}
584  \includegraphics[width=5cm]{circular6.png}}
585\centerline{
586  \includegraphics[width=5cm]{circular7.png}
587  \includegraphics[width=5cm]{circular8.png}}
588\centerline{
589  \includegraphics[width=5cm]{circular9.png}
590  \includegraphics[width=5cm]{circular10.png}}
591\caption{Screenshots of the evolution of solitary wave around circular island.}
592\label{fig:circular screenshots}
593\end{figure}
594
595
596\clearpage
597
598\subsection{MAYBE, 1D analytical validation}
599
600
601
602
603\section{Conclusions}
604\label{sec:6}
605\ANUGA{} is a flexible and robust modelling system
606that simulates hydrodynamics by solving the shallow water wave
607equation in a triangular mesh. It can model the process of wetting
608and drying as water enters and leaves an area and is capable of
609capturing hydraulic shocks due to the ability of the finite-volume
610method to accommodate discontinuities in the solution.
611\ANUGA{} can take as input bathymetric and topographic datasets and
612simulate the behaviour of riverine flooding, storm surge,
613tsunami or even dam breaks.
614Initial validation using wave tank data supports \ANUGA{}'s
615ability to model complex scenarios. Further validation will be
616pursued as additional datasets become available.
617The \ANUGA{} source code is available
618at \url{http://sourceforge.net/projects/anuga}.
619
620\bibliographystyle{plain}
621\bibliography{anuga-bibliography}
622
623
624
625
626\end{document}
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