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1%Anuga validation publication
2%
3%Geoscience Australia and others 2007-2008
4       
5% Use the Elsevier LaTeX document class
6%\documentclass{elsart3p} % Two column
7%\documentclass{elsart1p} % One column
8\documentclass{elsart} % Basic
9
10% Useful packages
11\usepackage{graphicx} % avoid epsfig or earlier such packages
12\usepackage{url}      % for URLs and DOIs
13\usepackage{amsmath}  % many want amsmath extensions
14\usepackage{amsfonts}
15\usepackage{underscore}
16\usepackage{natbib}   % Suggested by the Elsevier style
17                      % Use \citep and \citet instead of \cite
18
19% Local LaTeX commands
20%\newcommand{\Python}{\textsc{Python}}
21%\newcommand{\VPython}{\textsc{VPython}}
22\newcommand{\pypar}{\textsc{mpi}}
23\newcommand{\Metis}{\textsc{Metis}}
24\newcommand{\mpi}{\textsc{mpi}}
25
26\newcommand{\UU}{\mathbf{U}}
27\newcommand{\VV}{\mathbf{V}}
28\newcommand{\EE}{\mathbf{E}}
29\newcommand{\GG}{\mathbf{G}}
30\newcommand{\FF}{\mathbf{F}}
31\newcommand{\HH}{\mathbf{H}}
32\newcommand{\SSS}{\mathbf{S}}
33\newcommand{\nn}{\mathbf{n}}
34
35\newcommand{\code}[1]{\texttt{#1}}
36
37
38
39
40\begin{document}
41
42
43\begin{frontmatter}
44\title{On The Validation of A Hydrodynamic Model}
45
46
47\author[GA]{D.~S.~Gray}
48\ead{Duncan.Gray@ga.gov.au}
49\author[GA]{O.~M.~Nielsen}
50\ead{Ole.Nielsen@ga.gov.au}
51\author[GA]{M.~J.~Sexton}
52\ead{Jane.Sexton@ga.gov.au}
53\author[GA]{L.~Fountain}
54\author[GA]{K.~VanPutten}
55\author[ANU]{S.~G.~Roberts}
56\ead{Stephen.Roberts@anu.edu.au}
57\author[UQ]{T.~Baldock}
58\ead{Tom.Baldock@uq.edu.au}
59\author[UQ]{M.~Barnes}
60\ead{Matthew.Barnes@uq.edu.au}
61
62\address[GA]{Natural Hazard Impacts Project,
63 Geospatial and Earh Monitoring Division,
64 Geoscience Australia, Canberra, Australia} 
65 
66\address[ANU]{Department of Mathematics,
67Australian National University, Canberra, Australia} 
68
69\address[UQ]{University of Queensland, Brisbane, Australia}
70
71
72% Use the \verb|abstract| environment.
73\begin{abstract}
74Modelling the effects on the built environment of natural hazards such
75as riverine flooding, storm surges and tsunami is critical for
76understanding their economic and social impact on our urban
77communities.  Geoscience Australia and the Australian National
78University have developed a hydrodynamic inundation modelling tool
79called ANUGA to help simulate the impact of these hazards.
80The core of ANUGA is a Python implementation of a finite-volume method
81for solving the conservative form of the Shallow Water Wave equation.
82
83In this paper, a number of tests are performed to validate ANUGA. These tests
84range from benchmark problems to wave and flume tank examples.
85ANUGA is available as Open Source to enable
86free access to the software and allow the scientific community to
87use, validate and contribute to the software in the future.
88
89%This method allows the study area to be represented by an unstructured
90%mesh with variable resolution to suit the particular problem.  The
91%conserved quantities are water level (stage) and horizontal momentum.
92%An important capability of ANUGA is that it can robustly model the
93%process of wetting and drying as water enters and leaves an area. This
94%means that it is suitable for simulating water flow onto a beach or
95%dry land and around structures such as buildings.
96
97\end{abstract}
98
99
100\begin{keyword}
101% keywords here, in the form: keyword \sep keyword
102% PACS codes here, in the form: \PACS code \sep code
103
104Hydrodynamic Modelling \sep Model validation \sep
105Finite-volumes \sep Shallow water wave equation
106
107\end{keyword}
108
109\date{\today()}
110\end{frontmatter}
111
112
113
114
115% Begin document in earnest
116\section{Introduction}
117\label{sec:intro}
118
119Hydrodynamic modelling allows impacts from flooding, storm-surge and
120tsunami to be better understood, their impacts to be anticipated and,
121with appropriate planning, their effects to be mitigated.  A significant
122proportion of the Australian population reside in the coastal
123corridors, thus the potential of significant disruption and loss
124is real.  The extent of
125inundation is critically linked to the event, tidal conditions,
126bathymetry and topography and it not feasible to make impact
127predictions using heuristics alone.
128Geoscience
129Australia in collaboration with the Mathematical Sciences Institute,
130Australian National University, is developing a software application
131called ANUGA to model the hydrodynamics of floods, storm surges and
132tsunami. These hazards are modelled using the conservative shallow
133water equations which are described in section~\ref{sec:model}. In
134ANUGA these equations are solved using a finite volume method as
135described in section~\ref{sec:model}.  A more complete discussion of the
136method can be found in \citet{Nielsen2005} where the model and solution
137technique is validated on a standard tsunami benchmark data set
138or in \citet{Roberts2007} where the numerical method and parallelisation
139of ANUGA is discussed.
140This modelling capability is part of
141Geoscience Australia's ongoing research effort to model and
142understand the potential impact from natural hazards in order to
143reduce their impact on Australian communities \citep{Nielsen2006}.
144ANUGA is currently being trialled for flood
145modelling \citep{Rigby2008}.
146
147The validity of other hydrodynamic models have been reported
148elsewhere, with \citet{Hubbard02} providing an
149excellent review of 1D and 2D models and associated validation
150tests. They described the evolution of these models from fixed, nested
151to adaptive grids and the ability of the solvers to cope with the
152moving shoreline. They highlighted the difficulty in verify the
153nonlinear shallow water equations themselves as the only standard
154analytical solution is that of \citet{Carrier58} that is strictly for
155non-breaking waves. Further,
156whilst there is a 2D analytic solution from \citet{Thacker81}, it appears
157that the circular island wave tank example of Briggs et al will become
158the standard data set to verify the equations.
159
160This paper will describe the validation outputs in a similar way to
161\citet{Hubbard02} to
162present an exhaustive validation of the numerical model.
163Further to these tests, we will
164incorporate a test to verify friction values. The tests reported in
165this paper are:
166\begin{itemize}
167  \item Verification against the 1D analytical solution of Carrier and
168  Greenspan (p~\pageref{sec:carrier})
169  \item Testing against 1D (flume) data sets to verify wave height and
170  velocity (p~\pageref{sec:stage and velocity})
171  \item Determining friction values from 1D flume data sets
172  (p~\pageref{sec:friction})
173  \item Validation against a genuinely 2D analytical
174  solution of the model equations (p~\ref{sec:XXX})
175  \item Testing against the 2D Okushiri benchmark problem
176  (p~\pageref{sec:okushiri})   
177  \item Testing against the 2D data sets modelling wave run-up around a circular island by Briggs et al.
178  (p~\pageref{sec:circular island})
179\end{itemize}   
180
181
182Throughout the paper, qualitative comparisons will be drawn against
183other models.  Moreover, all source code necessary to reproduce the
184results reported in this paper is available as part of the ANUGA
185distribution in the form of a test suite. It is thus possible for
186anyone to readily verify that the implementation meets the
187requirements set out by these benchmarks.
188 
189
190%Hubbard and Dodd's model, OTT-2D, has some similarities to ANUGA, and
191%whilst the mesh can be refined, it is based on rectangular mesh.
192
193%The ANUGA model and numerical scheme is briefly described in
194%section~\ref{sec:model}.  A more detailed description of the numerical
195%scheme and software implementation can be found in \citet{Nielsen2005} and
196%\citet{Roberts2007}.
197The six case studies to validation and verify ANUGA
198will be presented in section~\ref{sec:validation}, with the
199conclusions outlined in section~\ref{sec:conclusions}.
200
201
202\section{Mathematical model, numerical scheme and implementation}
203\label{sec:model}
204
205The ANUGA model is based on the shallow water wave equations which are
206widely regarded as suitable for modelling 2D flows subject to the
207assumptions that horizontal scales (e.g. wave lengths) greatly exceed
208the depth, vertical velocities are negligible and the fluid is treated
209as inviscid and incompressible. See e.g. the classical texts
210\citet{Stoker57} and \citet{Peregrine67} for the background or
211\citet{Roberts1999} for more details on the mathematical model
212used by ANUGA.
213
214The conservation form of the shallow water wave
215equations used in ANUGA are:
216\[
217\frac{\partial \UU}{\partial t}+\frac{\partial \EE}{\partial
218x}+\frac{\partial \GG}{\partial y}=\SSS
219\]
220where $\UU=\left[ {{\begin{array}{*{20}c}
221 h & {uh} & {vh} \\
222\end{array} }} \right]^T$ is the vector of conserved quantities; water depth
223$h$, $x$-momentum $uh$ and $y$-momentum $vh$. Other quantities
224entering the system are bed elevation $z$ and stage (absolute water
225level above a reference datum such as Mean Sea Level) $w$,
226where the relation $w = z + h$ holds true at all times.
227The fluxes in the $x$ and $y$ directions, $\EE$ and $\GG$ are given
228by
229\[
230\EE=\left[ {{\begin{array}{*{20}c}
231 {uh} \hfill \\
232 {u^2h+gh^2/2} \hfill \\
233 {uvh} \hfill \\
234\end{array} }} \right]\mbox{ and }\GG=\left[ {{\begin{array}{*{20}c}
235 {vh} \hfill \\
236 {vuh} \hfill \\
237 {v^2h+gh^2/2} \hfill \\
238\end{array} }} \right]
239\]
240and the source term (which includes gravity and friction) is given
241by
242\[
243\SSS=\left[ {{\begin{array}{*{20}c}
244 0 \hfill \\
245 -{gh(z_{x} + S_{fx} )} \hfill \\
246 -{gh(z_{y} + S_{fy} )} \hfill \\
247\end{array} }} \right]
248\]
249where $S_f$ is the bed friction. The friction term is modelled using
250Manning's resistance law
251\[
252S_{fx} =\frac{u\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}\mbox{ and }S_{fy}
253=\frac{v\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}
254\]
255in which $\eta$ is the Manning resistance coefficient.
256
257%%As demonstrated in our papers, \cite{modsim2005,Roberts1999} these
258%%equations provide an excellent model of flows associated with
259%%inundation such as dam breaks and tsunamis. Question - how do we
260%%know it is excellent?
261
262ANUGA uses a finite-volume method as
263described in \citet{Roberts2007} where the study area is represented by an
264unstructured triangular mesh in which the vector of conserved quantities
265$\UU$ is maintained and updated over time. The flexibility afforded by
266allowing unstructed meshes rather than fixed resolution grids
267is the ability for the user to refine the mesh in areas of interest
268while leaving other areas coarse and thereby conserving computational
269resources.
270
271
272The approach used in ANUGA are distinguished from many
273other implementations (e.g. \citet{Hubbard02} or \citet{Zhang07}) by the
274following features:
275\begin{itemize}
276    \item The fluxes across each edge are computed using the semi-discrete
277    central-upwind scheme for approximating the Riemann problem
278    proposed by \citet{KurNP2001}. This scheme deals with different
279    flow regimes such as shocks, rarefactions and sub to super
280    critical flow transitions using one general approach. We have
281    found this scheme to be pleasingly simple, robust and efficient.
282    \item ANUGA does not employ a shoreline detection algorithm as the
283    central-upwind scheme is capable of resolving fluxes arising between
284    wet and dry cells. ANUGA does optionally bypass unnecessary
285    computations for dry-dry cell boundaries purely to improve performance.
286    \item ANUGA employs a second order spatial reconstruction of triangles
287    to produce a piece-wise linear function construction of the conserved
288    quantities. This function is allowed to be discontinuous across the
289    edges of the cells, but the slope of this function is limited to avoid
290    artificially introduced oscillations. This approach provides good
291    approximation of steep gradients in the solution. However,
292    where the depths are very small compared to the bed-slope a linear
293    combination between second order and first order reconstructions is
294    employed to guarantee numerical stability that may arise form very
295    small depths.
296\end{itemize}     
297   
298In the computations presented in this paper we use an explicit Euler
299time stepping method with variable timestepping subject to the
300CFL condition:
301\[
302  \delta t = \min_k \frac{r_k}{v_k} 
303\]
304where $r_k$ refers to the radius of the inscribed circle of triangle
305$k$, $v_k$ refers to the maximal velocity calculated from fluxes
306passing in or out of triangle $k$ and $\delta t$ is the resulting
307'safe' timestep to be used for the next iteration.
308
309
310ANUGA utilises a general velocity limiter described in the
311manual which guarantees a gradual compression of computed velocities
312in the presence of very shallow depths:
313\begin{equation}
314  \hat{u} = \frac{\mu}{h + h_0/h}, \bigskip \hat{v} = \frac{\nu}{h + h_0/h},
315\end{equation}
316where $h_0$ is a regularisation parameter that controls the minimal
317magnitude of the denominator. The default value is $h_0 = 10^{-6}$.
318
319
320ANUGA is mostly written in the object-oriented programming
321language Python with computationally intensive parts implemented
322as highly optimised shared objects written in C.
323
324Python is known for its clarity, elegance, efficiency and
325reliability. Complex software can be built in Python without undue
326distractions arising from idiosyncrasies of the underlying software
327language syntax. In addition, Python's automatic memory management,
328dynamic typing, object model and vast number of libraries means that
329ANUGA scripts can be produced quickly and can be adapted fairly easily to
330changing requirements.
331
332
333
334\section{Validation}
335\label{sec:validation} Validation is an ongoing process and the purpose of this paper
336is to describe a range of tests that validate ANUGA as a hydrodynamic model.
337This section will describe the six tests outlined in section~\ref{sec:intro}.
338Run times where specified measure the model time only and exclude model setup,
339data conversions etc. All examples were timed on a a 2GHz 64-bit
340Dual-Core AMD Opteron(tm) series 2212 Linux server
341running at 2GHz. %This is a tornado compute node (cat /proc/cpuinfo).   
342
343
344\subsection{1D analytical validation}
345
346Tom Baldock has done something here for that NSW report
347
348\subsection{Stage and Velocity Validation in a Flume}
349\label{sec:stage and velocity}
350This section will describe tilting flume tank experiments that were
351conducted at the Gordon McKay Hydraulics Laboratory at the University of
352Queensland that confirm ANUGA's ability to estimate wave height
353and velocity. The same flume tank simulations were also used
354to explore Manning's friction and this will be described in the next section.
355
356The flume was set up for dam-break experiments, having a
357water reservior at one end.  The flume was glass-sided, 3m long, 0.4m
358in wide, and 0.4m deep, with a PVC bottom. The reservoir in the flume
359was 0.75m long.  For this experiment the reservoir water was 0.2m
360deep. At time zero the reservoir gate is manually opened and the water flows
361into the other side of the flume.  The water ran up a flume slope of
3620.03 m/m.  To accurately model the bed surface a Manning's friction
363value of 0.01, representing PVC was used.
364
365% Neale, L.C. and R.E. Price.  Flow characteristics of PVC sewer pipe.
366% Journal of the Sanitary Engineering Division, Div. Proc 90SA3, ASCE.
367% pp. 109-129.  1964.
368
369Acoustic displacement sensors that produced a voltage that changed
370with the water depth was positioned 0.4m from the reservoir gate. The
371water velocity was measured with an Acoustic Doppler Velocimeter 0.45m
372from the reservoir gate.  This sensor only produced reliable results 4
373seconds after the reservoir gate opened, due to limitations of the sensor.
374
375
376% Validation UQ flume
377% at X:\anuga_validation\uq_sloped_flume_2008
378% run run_dam.py to create sww file and .csv files
379% run plot.py to create graphs heere automatically
380% The Coasts and Ports '2007 paper is in TRIM d2007-17186
381\begin{figure}[htbp]
382\centerline{\includegraphics[width=4in]{uq-flume-depth}}
383\caption{Comparison of wave tank and ANUGA water height at .4 m
384  from the gate}\label{fig:uq-flume-depth}
385\end{figure}
386
387\begin{figure}[htbp]
388\centerline{\includegraphics[width=4in]{uq-flume-velocity}}
389\caption{Comparison of wave tank and ANUGA water velocity at .45 m
390  from the gate}\label{fig:uq-flume-velocity}
391\end{figure}
392
393Figure~\ref{fig:uq-flume-depth} shows that ANUGA predicts the actual
394water depth very well, although there is an initial drop in water depth
395within the first second that is not simulated by ANUGA.
396Water depth and velocity are coupled as described by the nonlinear
397shallow water equations, thus if one of these quantities accurately
398estimates the measured values, we would expect the same for the other
399quantity. This is demonstrated in Figure~\ref{fig:uq-flume-velocity}
400where the water velocity is also predicted accurately. Sediment
401transport studies rely on water velocity estimates in the region where
402the sensors cannot provide this data.  With water velocity being
403accurately predicted, studies such as sediment transport can now use
404reliable estimates.
405
406
407\subsection{1D flume tank to verify friction}
408\label{sec:friction}
409The same tilting flume tank was used to validate stage and velocity
410was used to validate the ANUGA friction model. A ground slope of 1:20,
411reservior lenght of 0.85m and damn depth of 0.4 m was used to verify
412the friction. The PVC bottom of the tank is equivalent to a friction
413value of 0.01.  {\bf Add ref } Depth sensors were placed 0.2, 0.3,
4140.4, 0.5 and 0.6 m from the bed gate.
415
416 
417As described in the model equations in ~\ref{sec:model}, the bed
418friction is modelled using the Manning's model. {\bf Add the formula}
419Validation of this model was carried out by comparing results
420from ANUGA against experimental results from flume wave tanks.
421 
422This experiment was simulated twice by ANUGA: without using the
423friction model {\bf Duncan: It really used the friction model, with a
424value of 0.0, representing no friction model.  Is it ok to say
425'without using the model?'} and using the friction model with a
426Manning's friction value of 0.01.  The results from both of these
427simulations were compared against the experimental flume tank results
428using the Root Mean Square Relative Error (RMSRE). The RMSRE was
429summed over all of the depth sensors, for the first 30 seconds of the
430experiment.  This resulted in one number which represents the error
431between tow data sets, with a lower number representing less
432differences.  The RMSRE for no friction model was 0.380, the RMSRE for
433the friction model with a Manning's friction value of 0.01 was
4340.358. So for this experiment using a friction value given from a
435standard fricition table improved the accuracy of the ANUGA
436simulation.  {\bf Add ref to table}
437
438% Validation UQ friction
439% at X:\anuga_validation\uq_friction_2007
440% run run_dam.py to create sww file and .csv files
441% run plot.py to create graphs, and move them here
442\begin{figure}[htbp]
443\centerline{\includegraphics[width=4in]{uq-friction-depth}}
444\caption{Comparison of wave tank and ANUGA water height at .4 m
445  from the gate, simulated using a Mannings friction of 0.0 and
446  0.1.}\label{fig:uq-friction-depth}
447\end{figure}
448
449\subsection{Okushiri Wavetank Validation}
450\label{sec:okushiri}
451As part of the Third International Workshop on Long-wave Runup
452Models in 2004 (\url{http://www.cee.cornell.edu/longwave}), four
453benchmark problems were specified to allow the comparison of
454numerical, analytical and physical models with laboratory and field
455data. One of these problems describes a wave tank simulation of the
4561993 Okushiri Island tsunami off Hokkaido, Japan \cite{MatH2001}. A
457significant feature of this tsunami was a maximum run-up of 32~m
458observed at the head of the Monai Valley. This run-up was not
459uniform along the coast and is thought to have resulted from a
460particular topographic effect. Among other features, simulations of
461the Hokkaido tsunami should capture this run-up phenomenon.
462
463This dataset has been used by to validate tsunami models by
464a number of tsunami scientists. Examples include Titov ... lit review
465here on who has used this example for verification (Leharne?)
466
467\begin{figure}[htbp]
468%\centerline{\includegraphics[width=4in]{okushiri-gauge-5.eps}}
469\centerline{\includegraphics[width=4in]{ch5.png}}
470\centerline{\includegraphics[width=4in]{ch7.png}}
471\centerline{\includegraphics[width=4in]{ch9.png}}
472\caption{Comparison of wave tank and ANUGA water stages at gauge
4735,7 and 9.}\label{fig:val}
474\end{figure}
475
476
477\begin{figure}[htbp]
478\centerline{\includegraphics[width=4in]{okushiri-model.jpg}}
479\caption{Complex reflection patterns and run-up into Monai Valley
480simulated by ANUGA and visualised using our netcdf OSG
481viewer.}\label{fig:run}
482\end{figure}
483
484The wave tank simulation of the Hokkaido tsunami was used as the
485first scenario for validating ANUGA. The dataset provided
486bathymetry and topography along with initial water depth and the
487wave specifications. The dataset also contained water depth time
488series from three wave gauges situated offshore from the simulated
489inundation area. The ANUGA model comprised $41404$ triangles
490and took about $1330$ s to run on the test platform described in
491Section~\ref{sec:validation}.
492
493The script to run this example is available in the ANUGA distribution in the subdirectory
494\code{anuga_validation/automated_validation_tests/okushiri_tank_validation}.
495
496
497Figure~\ref{fig:val} compares the observed wave tank and modelled
498ANUGA water depth (stage height) at one of the gauges. The plots
499show good agreement between the two time series, with ANUGA
500closely modelling the initial draw down, the wave shoulder and the
501subsequent reflections. The discrepancy between modelled and
502simulated data in the first 10 seconds is due to the initial
503condition in the physical tank not being uniformly zero. Similarly
504good comparisons are evident with data from the other two gauges.
505Additionally, ANUGA replicates exceptionally well the 32~m Monai
506Valley run-up, and demonstrates its occurrence to be due to the
507interaction of the tsunami wave with two juxtaposed valleys above
508the coastline. The run-up is depicted in Figure~\ref{fig:run}.
509
510This successful replication of the tsunami wave tank simulation on a
511complex 3D beach is a positive first step in validating the ANUGA
512modelling capability.
513
514\subsection{Runup of solitary wave on circular island wavetank validation}
515\label{sec:circular island}
516This section will describe the ANUGA results for the experiments
517conducted by Briggs et al (1995). Here, a 30x25m basin with a conical
518island is situated near the centre and a directional wavemaker is used
519to produce planar solitary waves of specified crest lenghts and
520heights. A series of gauges were distributed within the experimental
521setup. As described by Hubbard and Dodd \cite{Hubbard02}, a number of
522researchers have used this benchmark problem to test their numerical
523models. {\bf Jane: check whether these results are now avilable as
524they were not in 2002}. Hubbard and Dodd \cite{Hubbard02} note that a
525particular 3D model appears to obtain slightly better results than the
5262D ones reported but that 3D models are unlikely to be competitive in
527terms of computing power for applications in coastal engineering at
528least. Choi et al \cite{Choi07} use a 3D RANS model (based on the
529Navier-Stokes equations) for the same problem and find a very good
530comparison with laboratory and 2D numerical results. An obvious
531advantage of the 3D model is its ability to investigate the velocity
532field and Choi et al also report on the limitation of depth-averaged
5332D models for run-up simulations of this type.
534
535Once results are availble, need to compare to Hubbard and Dodd and draw any conclusions
536from nested rectangular grid vs unstructured gird.
537Figure \ref{fig:circular screenshots} shows a sequence of screenshots depicting the evolution of the solitary wave as it hits the circular island.
538
539\begin{figure}[htbp]
540\centerline{
541  \includegraphics[width=5cm]{circular1.png}
542  \includegraphics[width=5cm]{circular2.png}}
543\centerline{
544  \includegraphics[width=5cm]{circular3.png}
545  \includegraphics[width=5cm]{circular4.png}}
546\centerline{
547  \includegraphics[width=5cm]{circular5.png}
548  \includegraphics[width=5cm]{circular6.png}}
549\centerline{
550  \includegraphics[width=5cm]{circular7.png}
551  \includegraphics[width=5cm]{circular8.png}}
552\centerline{
553  \includegraphics[width=5cm]{circular9.png}
554  \includegraphics[width=5cm]{circular10.png}}
555\caption{Screenshots of the evolution of solitary wave around circular island.}
556\label{fig:circular screenshots}
557\end{figure}
558
559
560\clearpage
561
562\section{Conclusions}
563\label{sec:conclusions}
564ANUGA is a flexible and robust modelling system
565that simulates hydrodynamics by solving the shallow water wave
566equation in a triangular mesh. It can model the process of wetting
567and drying as water enters and leaves an area and is capable of
568capturing hydraulic shocks due to the ability of the finite-volume
569method to accommodate discontinuities in the solution.
570ANUGA can take as input bathymetric and topographic datasets and
571simulate the behaviour of riverine flooding, storm surge,
572tsunami or even dam breaks.
573Initial validation using wave tank data supports ANUGA's
574ability to model complex scenarios. Further validation will be
575pursued as additional datasets become available.
576The ANUGA source code and validation case studies reported here are available
577at \url{http://sourceforge.net/projects/anuga}.
578
579something about use on flood modelling community and their validation initiatives
580
581
582%\bibliographystyle{plainnat}
583\bibliographystyle{elsart-harv}
584\bibliography{anuga-bibliography}
585
586\end{document}
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