# source:anuga_work/publications/anuga_2007/anuga_validation.tex@5667

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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}
51\author[GA]{O.~M.~Nielsen}
53\author[GA]{M.~J.~Sexton}
55\author[GA]{L.~Fountain}
56\author[GA]{K.~VanPutten}
57\author[ANU]{S.~G.~Roberts}
59\author[UQ]{T.~Baldock}
61\author[UQ]{M.~Barnes}
63
65 Geospatial and Earh Monitoring Division,
66 Geoscience Australia, Canberra, Australia}
67
69Australian National University, Canberra, Australia}
70
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
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{Okushiri Wavetank Validation}
442\label{sec:okushiri}
443As part of the Third International Workshop on Long-wave Runup
444Models in 2004 (\url{http://www.cee.cornell.edu/longwave}), four
445benchmark problems were specified to allow the comparison of
446numerical, analytical and physical models with laboratory and field
447data. One of these problems describes a wave tank simulation of the
4481993 Okushiri Island tsunami off Hokkaido, Japan \cite{MatH2001}. A
449significant feature of this tsunami was a maximum run-up of 32~m
450observed at the head of the Monai Valley. This run-up was not
451uniform along the coast and is thought to have resulted from a
452particular topographic effect. Among other features, simulations of
453the Hokkaido tsunami should capture this run-up phenomenon.
454
455This dataset has been used by to validate tsunami models by
456a number of tsunami scientists. Examples include Titov ... lit review
457here on who has used this example for verification (Leharne?)
458
459\begin{figure}[htbp]
460%\centerline{\includegraphics[width=4in]{okushiri-gauge-5.eps}}
461\centerline{\includegraphics[width=4in]{ch5.png}}
462\centerline{\includegraphics[width=4in]{ch7.png}}
463\centerline{\includegraphics[width=4in]{ch9.png}}
464\caption{Comparison of wave tank and ANUGA water stages at gauge
4655,7 and 9.}\label{fig:val}
466\end{figure}
467
468
469\begin{figure}[htbp]
470\centerline{\includegraphics[width=4in]{okushiri-model.jpg}}
471\caption{Complex reflection patterns and run-up into Monai Valley
472simulated by ANUGA and visualised using our netcdf OSG
473viewer.}\label{fig:run}
474\end{figure}
475
476The wave tank simulation of the Hokkaido tsunami was used as the
477first scenario for validating ANUGA. The dataset provided
478bathymetry and topography along with initial water depth and the
479wave specifications. The dataset also contained water depth time
480series from three wave gauges situated offshore from the simulated
481inundation area. The ANUGA model comprised $41404$ triangles
482and took about $1330$ s to run on the test platform described in
483Section~\ref{sec:validation}.
484
485The script to run this example is available in the ANUGA distribution in the subdirectory
486\code{anuga_validation/automated_validation_tests/okushiri_tank_validation}.
487
488
489Figure~\ref{fig:val} compares the observed wave tank and modelled
490ANUGA 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{Runup of solitary wave on circular island wavetank validation}
507\label{sec:circular island}
508This section will describe the ANUGA results for the experiments
509conducted by Briggs et al (1995). Here, a 30x25m basin with a conical
510island is situated near the centre and a directional wavemaker is used
511to produce planar solitary waves of specified crest lenghts and
512heights. A series of gauges were distributed within the experimental
513setup. As described by Hubbard and Dodd \cite{Hubbard02}, a number of
514researchers have used this benchmark problem to test their numerical
515models. {\bf Jane: check whether these results are now avilable as
516they were not in 2002}. Hubbard and Dodd \cite{Hubbard02} note that a
517particular 3D model appears to obtain slightly better results than the
5182D ones reported but that 3D models are unlikely to be competitive in
519terms of computing power for applications in coastal engineering at
520least. Choi et al \cite{Choi07} use a 3D RANS model (based on the
521Navier-Stokes equations) for the same problem and find a very good
522comparison with laboratory and 2D numerical results. An obvious
523advantage of the 3D model is its ability to investigate the velocity
524field and Choi et al also report on the limitation of depth-averaged
5252D models for run-up simulations of this type.
526
527Once results are availble, need to compare to Hubbard and Dodd and
528draw any conclusions from nested rectangular grid vs unstructured
529gird.  Figure \ref{fig:circular screenshots} shows a sequence of
530screenshots depicting the evolution of the solitary wave as it hits
531the circular island.
532
533\begin{figure}[htbp]
534\centerline{
535  \includegraphics[width=5cm]{circular1.png}
536  \includegraphics[width=5cm]{circular2.png}}
537\centerline{
538  \includegraphics[width=5cm]{circular3.png}
539  \includegraphics[width=5cm]{circular4.png}}
540\centerline{
541  \includegraphics[width=5cm]{circular5.png}
542  \includegraphics[width=5cm]{circular6.png}}
543\centerline{
544  \includegraphics[width=5cm]{circular7.png}
545  \includegraphics[width=5cm]{circular8.png}}
546\centerline{
547  \includegraphics[width=5cm]{circular9.png}
548  \includegraphics[width=5cm]{circular10.png}}
549\caption{Screenshots of the evolution of solitary wave around circular island.}
550\label{fig:circular screenshots}
551\end{figure}
552
553
554\subsection{Flume tank validation before and after breaking waves}
555
556To explicitly determine if ANUGA can model waves after breaking
557several experiments were conducted at the Monash University Institute for
558Sustainable Water Resources using a wave flume.  The experiment was
559designed to produce a variety of breaking waves.  The experiment was
560conducted on a 2.5$^\circ$ and a 1.5$^\circ$ plane beach slope set-up
561in a glass-sided wave flume of 40m in length, 1.0m wide and 1.6m deep.
562The wave generator can generate waves up to 0.6m in height, with a
563period range of 0.3 - 7.0 seconds.
564
565Four tests with different combinations of wave height and wave period
566were used, with each test being repeated once.
567
568A variety of measurements were taken during the simulation.  The wave
569height at breaking, when present, were measured using a video
570recorder. Mid-depth water velocity and wave height were measured on
571the approach section.  The water height at several points along the
572flume were measured using pressure transducers.
573
574Details of the tests performed are given in Table \ref{tab:hinwoodSummary}.
575
576\begin{table}
577\caption{Details of the Monash University experiments.} % Can't get right
578\begin{center}
579  \begin{tabular}{ c   p{3cm}  p{3cm}   p{3cm}  }
580
581    \hline
582    Test Name & Beach slope nominal, \emph{degrees}   & Water depth offshore,
583    \emph{mm } & Wave frequency nominal, \emph{Hz}  \\ \hline
584    T1R3 & 3.5 & 400 & 0.200 \\ \hline
585    T1R5 & 3.5 & 400 & 0.200 \\ \hline
586    T2R7 & 3.5 & 400 & 0.125 \\ \hline
587    T2R8 & 3.5 & 400 & 0.125 \\ \hline
588    T3R28 & 1.5 & 336 & 0.200 \\ \hline
589    T3R29 & 1.5 & 336 & 0.200 \\ \hline
590    T4R31 & 1.5 & 336 & 0.125 \\ \hline
591    T4R32 & 1.5 & 336 & 0.125 \\ \hline
592
593  \end{tabular}
594  \label{tab:hinwoodSummary}
595
596\end{center}
597\end{table}
598
599 All of these experiments were simulated using ANUGA. The Mid-depth
600water velocity and wave height measured on the approach section were
601as boundary conditions for the ANUGA simulations.  For both the
602experimental and simulation results the zero data was the still water
603line.  To quantify the difference between the simulated stage and the
604experimental stage the Root Mean Square Deviation (RMSD)
605(\cite{Kobayshi2000}) was used
606
607$608RMSD =\sqrt {\frac{1 }{n} \displaystyle\sum_{i=1}^{n}{(x_i - y_i)}^2} 609$
610
611
612\label{sec:Hinwood}
613
614
615
616
617\clearpage
618
619\section{Conclusions}
620\label{sec:conclusions}
621ANUGA is a flexible and robust modelling system
622that simulates hydrodynamics by solving the shallow water wave
623equation in a triangular mesh. It can model the process of wetting
624and drying as water enters and leaves an area and is capable of
625capturing hydraulic shocks due to the ability of the finite-volume
626method to accommodate discontinuities in the solution.
627ANUGA can take as input bathymetric and topographic datasets and
628simulate the behaviour of riverine flooding, storm surge,
629tsunami or even dam breaks.
630Initial validation using wave tank data supports ANUGA's
631ability to model complex scenarios. Further validation will be
632pursued as additional datasets become available.
633The ANUGA source code and validation case studies reported here are available
634at \url{http://sourceforge.net/projects/anuga}.
635
636something about use on flood modelling community and their validation initiatives
637
638
639%\bibliographystyle{plainnat}
640\bibliographystyle{elsart-harv}
641\bibliography{anuga-bibliography}
642
643\end{document}
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