source: anuga_work/publications/anuga_2007/anuga_validation.tex @ 5599

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(1) update to validation paper - references and loose text and (2) update to urs2sts testing

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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[draft]{elsart} % Basic
9\documentclass{elsart} % Basic
10
11% Useful packages
12\usepackage{graphicx} % avoid epsfig or earlier such packages
13\usepackage{url}      % for URLs and DOIs
14\usepackage{amsmath}  % many want amsmath extensions
15\usepackage{amsfonts}
16\usepackage{underscore}
17\usepackage{natbib}   % Suggested by the Elsevier style
18                      % Use \citep and \citet instead of \cite
19                     
20
21% Local LaTeX commands
22%\newcommand{\Python}{\textsc{Python}}
23%\newcommand{\VPython}{\textsc{VPython}}
24\newcommand{\pypar}{\textsc{mpi}}
25\newcommand{\Metis}{\textsc{Metis}}
26\newcommand{\mpi}{\textsc{mpi}}
27
28\newcommand{\UU}{\mathbf{U}}
29\newcommand{\VV}{\mathbf{V}}
30\newcommand{\EE}{\mathbf{E}}
31\newcommand{\GG}{\mathbf{G}}
32\newcommand{\FF}{\mathbf{F}}
33\newcommand{\HH}{\mathbf{H}}
34\newcommand{\SSS}{\mathbf{S}}
35\newcommand{\nn}{\mathbf{n}}
36
37\newcommand{\code}[1]{\texttt{#1}}
38
39
40
41
42\begin{document}
43
44
45\begin{frontmatter}
46\title{On The Validation of A Hydrodynamic Model}
47
48
49\author[GA]{D.~S.~Gray}
50\ead{Duncan.Gray@ga.gov.au}
51\author[GA]{O.~M.~Nielsen}
52\ead{Ole.Nielsen@ga.gov.au}
53\author[GA]{M.~J.~Sexton}
54\ead{Jane.Sexton@ga.gov.au}
55\author[GA]{L.~Fountain}
56\author[GA]{K.~VanPutten}
57\author[ANU]{S.~G.~Roberts}
58\ead{Stephen.Roberts@anu.edu.au}
59\author[UQ]{T.~Baldock}
60\ead{Tom.Baldock@uq.edu.au}
61\author[UQ]{M.~Barnes}
62\ead{Matthew.Barnes@uq.edu.au}
63
64\address[GA]{Georisk Project,
65 Geospatial and Earh Monitoring Division,
66 Geoscience Australia, Canberra, Australia} 
67 
68\address[ANU]{Department of Mathematics,
69Australian National University, Canberra, Australia} 
70
71\address[UQ]{University of Queensland, Brisbane, Australia}
72
73
74% Use the \verb|abstract| environment.
75\begin{abstract}
76Modelling the effects on the built environment of natural hazards such
77as riverine flooding, storm surges and tsunami is critical for
78understanding their economic and social impact on our urban
79communities.  Geoscience Australia and the Australian National
80University have developed a hydrodynamic inundation modelling tool
81called ANUGA to help simulate the impact of these hazards.
82The core of ANUGA is a Python implementation of a finite-volume method
83for solving the conservative form of the Shallow Water Wave equation.
84
85In this paper, a number of tests are performed to validate ANUGA. These tests
86range from benchmark problems to wave and flume tank examples.
87ANUGA is available as Open Source to enable
88free access to the software and allow the scientific community to
89use, validate and contribute to the software in the future.
90
91%This method allows the study area to be represented by an unstructured
92%mesh with variable resolution to suit the particular problem.  The
93%conserved quantities are water level (stage) and horizontal momentum.
94%An important capability of ANUGA is that it can robustly model the
95%process of wetting and drying as water enters and leaves an area. This
96%means that it is suitable for simulating water flow onto a beach or
97%dry land and around structures such as buildings.
98
99\end{abstract}
100
101
102\begin{keyword}
103% keywords here, in the form: keyword \sep keyword
104% PACS codes here, in the form: \PACS code \sep code
105
106Hydrodynamic Modelling \sep Model validation \sep
107Finite-volumes \sep Shallow water wave equation
108
109\end{keyword}
110
111\date{\today()}
112\end{frontmatter}
113
114
115
116
117% Begin document in earnest
118\section{Introduction}
119\label{sec:intro}
120
121Hydrodynamic modelling allows impacts from flooding, storm-surge and
122tsunami to be better understood, their impacts to be anticipated and,
123with appropriate planning, their effects to be mitigated.  A significant
124proportion of the Australian population reside in the coastal
125corridors, thus the potential of significant disruption and loss
126is real.  The extent of
127inundation is critically linked to the event, tidal conditions,
128bathymetry and topography and it not feasible to make impact
129predictions using heuristics alone.
130Geoscience
131Australia in collaboration with the Mathematical Sciences Institute,
132Australian National University, is developing a software application
133called ANUGA to model the hydrodynamics of floods, storm surges and
134tsunami. These hazards are modelled using the conservative shallow
135water equations which are described in section~\ref{sec:model}. In
136ANUGA these equations are solved using a finite volume method as
137described in section~\ref{sec:model}.  A more complete discussion of the
138method can be found in \citet{Nielsen2005} where the model and solution
139technique is validated on a standard tsunami benchmark data set
140or in \citet{Roberts2007} where the numerical method and parallelisation
141of ANUGA is discussed.
142This modelling capability is part of
143Geoscience Australia's ongoing research effort to model and
144understand the potential impact from natural hazards in order to
145reduce their impact on Australian communities \citep{Nielsen2006}.
146ANUGA is currently being trialled for flood
147modelling \citep{Rigby2008}.
148
149The validity of other hydrodynamic models have been reported
150elsewhere, with \citet{Hubbard02} providing an
151excellent review of 1D and 2D models and associated validation
152tests. They described the evolution of these models from fixed, nested
153to adaptive grids and the ability of the solvers to cope with the
154moving shoreline. They highlighted the difficulty in verifying the
155nonlinear shallow water equations themselves as the only standard
156analytical solution is that of \citet{Carrier58} that is strictly for
157non-breaking waves. Further,
158whilst there is a 2D analytic solution from \citet{Thacker81}, it appears
159that the circular island wave tank example of Briggs et al will become
160the standard data set to verify the equations.
161
162This paper will describe the validation outputs in a similar way to
163\citet{Hubbard02} to
164present an exhaustive validation of the numerical model.
165Further to these tests, we will
166incorporate a test to verify friction values. The tests reported in
167this paper are:
168\begin{itemize}
169  \item Verification against the 1D analytical solution of Carrier and
170  Greenspan (p~\pageref{sec:carrier})
171  \item Testing against 1D (flume) data sets to verify wave height and
172  velocity (p~\pageref{sec:stage and velocity})
173  \item Determining friction values from 1D flume data sets
174  (p~\pageref{sec:friction})
175  \item Validation against a genuinely 2D analytical
176  solution of the model equations (p~\ref{sec:XXX})
177  \item Testing against the 2D Okushiri benchmark problem
178  (p~\pageref{sec:okushiri})   
179  \item Testing against the 2D data sets modelling wave run-up around a circular island by Briggs et al.
180  (p~\pageref{sec:circular island})
181\end{itemize}   
182
183
184Throughout the paper, qualitative comparisons will be drawn against
185other models.  Moreover, all source code necessary to reproduce the
186results reported in this paper is available as part of the ANUGA
187distribution in the form of a test suite. It is thus possible for
188anyone to readily verify that the implementation meets the
189requirements set out by these benchmarks.
190 
191
192%Hubbard and Dodd's model, OTT-2D, has some similarities to ANUGA, and
193%whilst the mesh can be refined, it is based on rectangular mesh.
194
195%The ANUGA model and numerical scheme is briefly described in
196%section~\ref{sec:model}.  A more detailed description of the numerical
197%scheme and software implementation can be found in \citet{Nielsen2005} and
198%\citet{Roberts2007}.
199The six case studies to validation and verify ANUGA
200will be presented in section~\ref{sec:validation}, with the
201conclusions outlined in section~\ref{sec:conclusions}.
202
203NOTE: This is just a brain dump at the moment and needs to be incorporated properly
204in the text somewhere.
205
206Need some discussion on Bousssinesq type models - Boussinesq equations get the
207nonlinearity and dispersive effects to a high degree of accuracy
208
209moving wet-dry boundary algorithms - applicability to coastal engineering
210
211Fuhrman and Madesn 2008 \cite{Fuhrman2008}do validation - they have a Boussinesq type
212model, finite
213difference (therefore needing a supercomputer), 4th order, four stage RK time stepping
214scheme.
215 
216their tests are (1) nonlinear run-up on periodic and transient waves on a sloping
217beach with excellent comparison to analytic solutions (2) 2d parabolic basin
218(3) solitary wave evolution through 2d triangular channel (4) solitary wave evolution on
219conical island (we need to compare to their computation time and note they use a
220vertical exaggeration for their images)
221
222excellent accuracy mentioned - but what is it - what does it mean?
223
224of interest is that they mention mass conservation and calculate it throughout the simulations
225
226Kim et al \cite{DaiHong2007} use Riemann solver - talk about improved accuracy by using 2nd order upwind
227scheme. Use finite volume on a structured mesh. Do parabolic basic and circular island. Needed?
228
229Delis et all 2008 \cite{Delis2008}- finite volume, Godunov-type explicit scheme coupled with Roe's
230approximate Riemann solver. It accurately describes breaking waves as bores or hydraulic jumps
231and conserves volume across flow discontinuties - is this just a result of finite volume?
232
233They also show mass conservation for most of the simulations
234
235similar range of validation tests that compare well - our job to compare to these as well
236
237\section{Mathematical model, numerical scheme and implementation}
238\label{sec:model}
239
240The ANUGA model is based on the shallow water wave equations which are
241widely regarded as suitable for modelling 2D flows subject to the
242assumptions that horizontal scales (e.g. wave lengths) greatly exceed
243the depth, vertical velocities are negligible and the fluid is treated
244as inviscid and incompressible. See e.g. the classical texts
245\citet{Stoker57} and \citet{Peregrine67} for the background or
246\citet{Roberts1999} for more details on the mathematical model
247used by ANUGA.
248
249The conservation form of the shallow water wave
250equations used in ANUGA are:
251\[
252\frac{\partial \UU}{\partial t}+\frac{\partial \EE}{\partial
253x}+\frac{\partial \GG}{\partial y}=\SSS
254\]
255where $\UU=\left[ {{\begin{array}{*{20}c}
256 h & {uh} & {vh} \\
257\end{array} }} \right]^T$ is the vector of conserved quantities; water depth
258$h$, $x$-momentum $uh$ and $y$-momentum $vh$. Other quantities
259entering the system are bed elevation $z$ and stage (absolute water
260level above a reference datum such as Mean Sea Level) $w$,
261where the relation $w = z + h$ holds true at all times.
262The fluxes in the $x$ and $y$ directions, $\EE$ and $\GG$ are given
263by
264\[
265\EE=\left[ {{\begin{array}{*{20}c}
266 {uh} \hfill \\
267 {u^2h+gh^2/2} \hfill \\
268 {uvh} \hfill \\
269\end{array} }} \right]\mbox{ and }\GG=\left[ {{\begin{array}{*{20}c}
270 {vh} \hfill \\
271 {vuh} \hfill \\
272 {v^2h+gh^2/2} \hfill \\
273\end{array} }} \right]
274\]
275and the source term (which includes gravity and friction) is given
276by
277\[
278\SSS=\left[ {{\begin{array}{*{20}c}
279 0 \hfill \\
280 -{gh(z_{x} + S_{fx} )} \hfill \\
281 -{gh(z_{y} + S_{fy} )} \hfill \\
282\end{array} }} \right]
283\]
284where $S_f$ is the bed friction. The friction term is modelled using
285Manning's resistance law
286\[
287S_{fx} =\frac{u\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}\mbox{ and }S_{fy}
288=\frac{v\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}
289\]
290in which $\eta$ is the Manning resistance coefficient.
291
292%%As demonstrated in our papers, \cite{modsim2005,Roberts1999} these
293%%equations provide an excellent model of flows associated with
294%%inundation such as dam breaks and tsunamis. Question - how do we
295%%know it is excellent?
296
297ANUGA uses a finite-volume method as
298described in \citet{Roberts2007} where the study area is represented by an
299unstructured triangular mesh in which the vector of conserved quantities
300$\UU$ is maintained and updated over time. The flexibility afforded by
301allowing unstructed meshes rather than fixed resolution grids
302is the ability for the user to refine the mesh in areas of interest
303while leaving other areas coarse and thereby conserving computational
304resources.
305
306
307The approach used in ANUGA are distinguished from many
308other implementations (e.g. \citet{Hubbard02} or \citet{Zhang07}) by the
309following features:
310\begin{itemize}
311    \item The fluxes across each edge are computed using the semi-discrete
312    central-upwind scheme for approximating the Riemann problem
313    proposed by \citet{KurNP2001}. This scheme deals with different
314    flow regimes such as shocks, rarefactions and sub to super
315    critical flow transitions using one general approach. We have
316    found this scheme to be pleasingly simple, robust and efficient.
317    \item ANUGA does not employ a shoreline detection algorithm as the
318    central-upwind scheme is capable of resolving fluxes arising between
319    wet and dry cells. ANUGA does optionally bypass unnecessary
320    computations for dry-dry cell boundaries purely to improve performance.
321    \item ANUGA employs a second order spatial reconstruction of triangles
322    to produce a piece-wise linear function construction of the conserved
323    quantities. This function is allowed to be discontinuous across the
324    edges of the cells, but the slope of this function is limited to avoid
325    artificially introduced oscillations. This approach provides good
326    approximation of steep gradients in the solution. However,
327    where the depths are very small compared to the bed-slope a linear
328    combination between second order and first order reconstructions is
329    employed to guarantee numerical stability that may arise form very
330    small depths.
331\end{itemize}     
332   
333In the computations presented in this paper we use an explicit Euler
334time stepping method with variable timestepping subject to the
335CFL condition:
336\[
337  \delta t = \min_k \frac{r_k}{v_k} 
338\]
339where $r_k$ refers to the radius of the inscribed circle of triangle
340$k$, $v_k$ refers to the maximal velocity calculated from fluxes
341passing in or out of triangle $k$ and $\delta t$ is the resulting
342'safe' timestep to be used for the next iteration.
343
344
345ANUGA utilises a general velocity limiter described in the
346manual which guarantees a gradual compression of computed velocities
347in the presence of very shallow depths:
348\begin{equation}
349  \hat{u} = \frac{\mu}{h + h_0/h}, \bigskip \hat{v} = \frac{\nu}{h + h_0/h},
350\end{equation}
351where $h_0$ is a regularisation parameter that controls the minimal
352magnitude of the denominator. The default value is $h_0 = 10^{-6}$.
353
354
355ANUGA is mostly written in the object-oriented programming
356language Python with computationally intensive parts implemented
357as highly optimised shared objects written in C.
358
359Python is known for its clarity, elegance, efficiency and
360reliability. Complex software can be built in Python without undue
361distractions arising from idiosyncrasies of the underlying software
362language syntax. In addition, Python's automatic memory management,
363dynamic typing, object model and vast number of libraries means that
364ANUGA scripts can be produced quickly and can be adapted fairly easily to
365changing requirements.
366
367
368
369\section{Validation}
370\label{sec:validation} Validation is an ongoing process and the purpose of this paper
371is to describe a range of tests that validate ANUGA as a hydrodynamic model.
372This section will describe the six tests outlined in section~\ref{sec:intro}.
373Run times where specified measure the model time only and exclude model setup,
374data conversions etc. All examples were timed on a a 2GHz 64-bit
375Dual-Core AMD Opteron(tm) series 2212 Linux server. %This is a tornado compute node (cat /proc/cpuinfo).   
376
377
378\subsection{1D analytical validation}
379
380Tom Baldock has done something here for that NSW report
381
382\subsection{Stage and Velocity Validation in a Flume}
383\label{sec:stage and velocity}
384This section will describe tilting flume tank experiments that were
385conducted at the Gordon McKay Hydraulics Laboratory at the University of
386Queensland that confirm ANUGA's ability to estimate wave height
387and velocity. The same flume tank simulations were also used
388to explore Manning's friction and this will be described in the next section.
389
390The flume was set up for dam-break experiments, having a
391water reservior at one end.  The flume was glass-sided, 3m long, 0.4m
392in wide, and 0.4m deep, with a PVC bottom. The reservoir in the flume
393was 0.75m long.  For this experiment the reservoir water was 0.2m
394deep. At time zero the reservoir gate is manually opened and the water flows
395into the other side of the flume.  The water ran up a flume slope of
3960.03 m/m.  To accurately model the bed surface a Manning's friction
397value of 0.01, representing PVC was used.
398
399% Neale, L.C. and R.E. Price.  Flow characteristics of PVC sewer pipe.
400% Journal of the Sanitary Engineering Division, Div. Proc 90SA3, ASCE.
401% pp. 109-129.  1964.
402
403Acoustic displacement sensors that produced a voltage that changed
404with the water depth was positioned 0.4m from the reservoir gate. The
405water velocity was measured with an Acoustic Doppler Velocimeter 0.45m
406from the reservoir gate.  This sensor only produced reliable results 4
407seconds after the reservoir gate opened, due to limitations of the sensor.
408
409
410% Validation UQ flume
411% at X:\anuga_validation\uq_sloped_flume_2008
412% run run_dam.py to create sww file and .csv files
413% run plot.py to create graphs heere automatically
414% The Coasts and Ports '2007 paper is in TRIM d2007-17186
415\begin{figure}[htbp]
416\centerline{\includegraphics[width=4in]{uq-flume-depth}}
417\caption{Comparison of wave tank and ANUGA water height at .4 m
418  from the gate}\label{fig:uq-flume-depth}
419\end{figure}
420
421\begin{figure}[htbp]
422\centerline{\includegraphics[width=4in]{uq-flume-velocity}}
423\caption{Comparison of wave tank and ANUGA water velocity at .45 m
424  from the gate}\label{fig:uq-flume-velocity}
425\end{figure}
426
427Figure~\ref{fig:uq-flume-depth} shows that ANUGA predicts the actual
428water depth very well, although there is an initial drop in water depth
429within the first second that is not simulated by ANUGA.
430Water depth and velocity are coupled as described by the nonlinear
431shallow water equations, thus if one of these quantities accurately
432estimates the measured values, we would expect the same for the other
433quantity. This is demonstrated in Figure~\ref{fig:uq-flume-velocity}
434where the water velocity is also predicted accurately. Sediment
435transport studies rely on water velocity estimates in the region where
436the sensors cannot provide this data.  With water velocity being
437accurately predicted, studies such as sediment transport can now use
438reliable estimates.
439
440
441\subsection{1D flume tank to verify friction}
442\label{sec:friction}
443The same tilting flume tank was used to validate stage and velocity
444was used to validate the ANUGA friction model. A ground slope of 1:20,
445reservior lenght of 0.85m and damn depth of 0.4 m was used to verify
446the friction. The PVC bottom of the tank is equivalent to a friction
447value of 0.01.  {\bf Add ref } Depth sensors were placed 0.2, 0.3,
4480.4, 0.5 and 0.6 m from the bed gate.
449
450 
451As described in the model equations in ~\ref{sec:model}, the bed
452friction is modelled using the Manning's model. {\bf Add the formula}
453Validation of this model was carried out by comparing results
454from ANUGA against experimental results from flume wave tanks.
455 
456This experiment was simulated twice by ANUGA: without using the
457friction model {\bf Duncan says: It really used the friction model, with a
458value of 0.0, representing no friction model.  Is it ok to say
459'without using the model?'} and using the friction model with a
460Manning's friction value of 0.01.  The results from both of these
461simulations were compared against the experimental flume tank results
462using the Root Mean Square Relative Error (RMSRE). The RMSRE was
463summed over all of the depth sensors, for the first 30 seconds of the
464experiment.  This resulted in one number which represents the error
465between tow data sets, with a lower number representing less
466differences.  The RMSRE for no friction model was 0.380, the RMSRE for
467the friction model with a Manning's friction value of 0.01 was
4680.358. So for this experiment using a friction value given from a
469standard fricition table improved the accuracy of the ANUGA
470simulation.  {\bf Add ref to table}
471
472% Validation UQ friction
473% at X:\anuga_validation\uq_friction_2007
474% run run_dam.py to create sww file and .csv files
475% run plot.py to create graphs, and move them here
476\begin{figure}[htbp]
477\centerline{\includegraphics[width=4in]{uq-friction-depth}}
478\caption{Comparison of wave tank and ANUGA water height at .4 m
479  from the gate, simulated using a Mannings friction of 0.0 and
480  0.1.}\label{fig:uq-friction-depth}
481\end{figure}
482
483\subsection{Okushiri Wavetank Validation}
484\label{sec:okushiri}
485As part of the Third International Workshop on Long-wave Runup
486Models in 2004 (\url{http://www.cee.cornell.edu/longwave}), four
487benchmark problems were specified to allow the comparison of
488numerical, analytical and physical models with laboratory and field
489data. One of these problems describes a wave tank simulation of the
4901993 Okushiri Island tsunami off Hokkaido, Japan \cite{MatH2001}. A
491significant feature of this tsunami was a maximum run-up of 32~m
492observed at the head of the Monai Valley. This run-up was not
493uniform along the coast and is thought to have resulted from a
494particular topographic effect. Among other features, simulations of
495the Hokkaido tsunami should capture this run-up phenomenon.
496
497This dataset has been used by to validate tsunami models by
498a number of tsunami scientists. Examples include Titov ... lit review
499here on who has used this example for verification (Leharne?)
500
501\begin{figure}[htbp]
502%\centerline{\includegraphics[width=4in]{okushiri-gauge-5.eps}}
503\centerline{\includegraphics[width=4in]{ch5.png}}
504\centerline{\includegraphics[width=4in]{ch7.png}}
505\centerline{\includegraphics[width=4in]{ch9.png}}
506\caption{Comparison of wave tank and ANUGA water stages at gauge
5075,7 and 9.}\label{fig:val}
508\end{figure}
509
510
511\begin{figure}[htbp]
512\centerline{\includegraphics[width=4in]{okushiri-model.jpg}}
513\caption{Complex reflection patterns and run-up into Monai Valley
514simulated by ANUGA and visualised using our netcdf OSG
515viewer.}\label{fig:run}
516\end{figure}
517
518The wave tank simulation of the Hokkaido tsunami was used as the
519first scenario for validating ANUGA. The dataset provided
520bathymetry and topography along with initial water depth and the
521wave specifications. The dataset also contained water depth time
522series from three wave gauges situated offshore from the simulated
523inundation area. The ANUGA model comprised $41404$ triangles
524and took about $1330$ s to run on the test platform described in
525Section~\ref{sec:validation}.
526
527The script to run this example is available in the ANUGA distribution in the subdirectory
528\code{anuga_validation/automated_validation_tests/okushiri_tank_validation}.
529
530
531Figure~\ref{fig:val} compares the observed wave tank and modelled
532ANUGA water depth (stage height) at one of the gauges. The plots
533show good agreement between the two time series, with ANUGA
534closely modelling the initial draw down, the wave shoulder and the
535subsequent reflections. The discrepancy between modelled and
536simulated data in the first 10 seconds is due to the initial
537condition in the physical tank not being uniformly zero. Similarly
538good comparisons are evident with data from the other two gauges.
539Additionally, ANUGA replicates exceptionally well the 32~m Monai
540Valley run-up, and demonstrates its occurrence to be due to the
541interaction of the tsunami wave with two juxtaposed valleys above
542the coastline. The run-up is depicted in Figure~\ref{fig:run}.
543
544This successful replication of the tsunami wave tank simulation on a
545complex 3D beach is a positive first step in validating the ANUGA
546modelling capability.
547
548\subsection{Runup of solitary wave on circular island wavetank validation}
549\label{sec:circular island}
550This section will describe the ANUGA results for the experiments
551conducted by Briggs et al (1995). Here, a 30x25m basin with a conical
552island is situated near the centre and a directional wavemaker is used
553to produce planar solitary waves of specified crest lenghts and
554heights. A series of gauges were distributed within the experimental
555setup. As described by Hubbard and Dodd \cite{Hubbard02}, a number of
556researchers have used this benchmark problem to test their numerical
557models. {\bf Jane: check whether these results are now avilable as
558they were not in 2002}. Hubbard and Dodd \cite{Hubbard02} note that a
559particular 3D model appears to obtain slightly better results than the
5602D ones reported but that 3D models are unlikely to be competitive in
561terms of computing power for applications in coastal engineering at
562least. Choi et al \cite{Choi07} use a 3D RANS model (based on the
563Navier-Stokes equations) for the same problem and find a very good
564comparison with laboratory and 2D numerical results. An obvious
565advantage of the 3D model is its ability to investigate the velocity
566field and Choi et al also report on the limitation of depth-averaged
5672D models for run-up simulations of this type.
568
569Once results are availble, need to compare to Hubbard and Dodd and draw any conclusions
570from nested rectangular grid vs unstructured gird.
571Figure \ref{fig:circular screenshots} shows a sequence of screenshots depicting the evolution of the solitary wave as it hits the circular island.
572
573\begin{figure}[htbp]
574\centerline{
575  \includegraphics[width=5cm]{circular1.png}
576  \includegraphics[width=5cm]{circular2.png}}
577\centerline{
578  \includegraphics[width=5cm]{circular3.png}
579  \includegraphics[width=5cm]{circular4.png}}
580\centerline{
581  \includegraphics[width=5cm]{circular5.png}
582  \includegraphics[width=5cm]{circular6.png}}
583\centerline{
584  \includegraphics[width=5cm]{circular7.png}
585  \includegraphics[width=5cm]{circular8.png}}
586\centerline{
587  \includegraphics[width=5cm]{circular9.png}
588  \includegraphics[width=5cm]{circular10.png}}
589\caption{Screenshots of the evolution of solitary wave around circular island.}
590\label{fig:circular screenshots}
591\end{figure}
592
593
594\clearpage
595
596\section{Conclusions}
597\label{sec:conclusions}
598ANUGA is a flexible and robust modelling system
599that simulates hydrodynamics by solving the shallow water wave
600equation in a triangular mesh. It can model the process of wetting
601and drying as water enters and leaves an area and is capable of
602capturing hydraulic shocks due to the ability of the finite-volume
603method to accommodate discontinuities in the solution.
604ANUGA can take as input bathymetric and topographic datasets and
605simulate the behaviour of riverine flooding, storm surge,
606tsunami or even dam breaks.
607Initial validation using wave tank data supports ANUGA's
608ability to model complex scenarios. Further validation will be
609pursued as additional datasets become available.
610The ANUGA source code and validation case studies reported here are available
611at \url{http://sourceforge.net/projects/anuga}.
612
613something about use on flood modelling community and their validation initiatives
614
615
616%\bibliographystyle{plainnat}
617\bibliographystyle{elsart-harv}
618\bibliography{anuga-bibliography}
619
620\end{document}
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