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\title{On The Validation of A Hydrodynamic Model}

\author{
D.~S.~Gray\thanks{Natural Hazard Impacts Project, Geospatial and
Earth Monitoring Division, Geoscience Australia, Symonston,
\textsc{Australia}. \protect\url{mailto:Duncan.Gray@ga.gov.au}}\footnotemark[1]
\and
T.~Baldock\thanks{University of Queensland, Brisbane, \textsc{Australia}. 
\protect\url{mailto:tom.baldock@uq.edu.au}}\footnotemark[2]
\and
O.~M.~Nielsen\footnotemark[1]
\and 
M.~J.~Sexton\footnotemark[1]
\and
N.~Bartzis\footnotemark[1]
\and
S.~G.~Roberts\thanks{Department of Mathematics, Australian National University,
Canberra, \textsc{Australia}. \protect\url{mailto:stephen.roberts@anu.edu.au}}}

\date{30 May 2008}

\newcommand{\ANUGA}{\textsc{ANUGA}}
\newcommand{\Python}{\textsc{Python}}
\newcommand{\VPython}{\textsc{VPython}}
\newcommand{\pypar}{\textsc{mpi}}
\newcommand{\Metis}{\textsc{Metis}}
\newcommand{\mpi}{\textsc{mpi}}

\newcommand{\UU}{\mathbf{U}}
\newcommand{\VV}{\mathbf{V}}
\newcommand{\EE}{\mathbf{E}}
\newcommand{\GG}{\mathbf{G}}
\newcommand{\FF}{\mathbf{F}}
\newcommand{\HH}{\mathbf{H}}
\newcommand{\SSS}{\mathbf{S}}
\newcommand{\nn}{\mathbf{n}}

\newcommand{\code}[1]{\texttt{#1}}




\begin{document}

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\begin{abstract}
Modelling the effects on the built environment of natural hazards such
as riverine flooding, storm surges and tsunami is critical for
understanding their economic and social impact on our urban
communities.  Geoscience Australia and the Australian National
University have developed a hydrodynamic inundation modelling tool
called \ANUGA{} to help simulate the impact of these hazards.
The core of \ANUGA{} is a \Python{} implementation of a finite-volume method
for solving the conservative form of the Shallow Water Wave equation.

In this paper, a number of tests are performed to validate \ANUGA{}. These tests
range from benchmark problems to wave and flume tank examples.
\ANUGA{} is available as Open Source to enable
free access to the software and allow the scientific community to
use, validate and contribute to the software in the future.

%This method allows the study area to be represented by an unstructured
%mesh with variable resolution to suit the particular problem.  The
%conserved quantities are water level (stage) and horizontal momentum.
%An important capability of ANUGA is that it can robustly model the
%process of wetting and drying as water enters and leaves an area. This
%means that it is suitable for simulating water flow onto a beach or
%dry land and around structures such as buildings.


\end{abstract}

% By default we include a table of contents in each paper.
%\tableofcontents

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% possibly \verb|\paragraph| to structure your document.  Make sure
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%\clearpage
\section{Introduction}
\label{sec:intro}

Hydrodynamic modelling allows impacts from flooding, storm-surge and
tsunami to be better understood, their impacts to be anticipated and,
with appropriate planning, their effects to be mitigated.  A significant
proportion of the Australian population reside in the coastal
corridors, thus the potential of significant disruption and loss
is real.  The extent of
inundation is critically linked to the event, tidal conditions,
bathymetry and topography and it not feasible to make impact
predictions using heuristics alone.
Geoscience
Australia in collaboration with the Mathematical Sciences Institute,
Australian National University, is developing a software application
called \ANUGA{} to model the hydrodynamics of floods, storm surges and
tsunami. These hazards are modelled using the conservative shallow
water equations which are described in section~\ref{sec:model}. In
\ANUGA{} these equations are solved using a finite volume method as
described in section~\ref{sec:model}.  A more complete discussion of the
method can be found in \cite{modsim2005} where the model and solution
technique is validated on a standard tsunami benchmark data set 
or in \cite{Roberts2007} where parallelisation of ANUGA is discussed.
This modelling capability is part of
Geoscience Australia's ongoing research effort to model and
understand the potential impact from natural hazards in order to
reduce their impact on Australian communities (see \cite{Nielsen2006}).
\ANUGA{} is currently being trialled for flood 
modelling (see \cite{Rigby2008}).

The validity of other hydrodynamic models have been reported elsewhere, 
with Hubbard and Dodd (2002) \cite{Hubbard02} providing
an excellent review of 1D and 2D models and associated validation tests. They
described the evolution of these models from fixed, nested to adaptive grids 
and the ability of the solvers to cope with the moving shoreline. They highlighted the
difficulty in verify the nonlinear shallow water equations themselves as the only
standard analytical solution is that of Carrier and Greenspan (1958) 
\cite{Carrier58} that is strictly
for non-breaking waves. Further, whilst there is a 2D analytic solution from Thacker (1981), it
appears that the circular island wave tank example of Briggs et al will become
the standard data set to verify the equations.

This paper will describe the validation outputs in a similar way to Hubbard and Dodd 
\cite{Hubbard02} to
present an exhaustive validation of the numerical model. Further to these tests, we will
incorporate a test to verify friction values. The six tests are:
(1) verification against the 1D analytical solution of Carrier and Greenspan; 
(2) testing against 1D (flume) data sets to verify wave height and velocity
(3) determining friction values from 1D flume data sets; 
(4) validation against a genuinely 2D analytical solution of the model equations; 
(5) testing against the 2D Okushiri benchmark problem; and 
(6) testing against the 2D data sets modelling wave run-up around a circular island by Briggs et al. 
Throughout the paper, qualitative comparisons will be drawn against other models.

%Hubbard and Dodd's model, OTT-2D, has some similarities to \ANUGA{}, and 
%whilst the mesh can be refined, it is based on rectangular mesh.

The \ANUGA{} model and numerical scheme is briefly described in section~\ref{sec:model}.
A detailed description of the numerical scheme and software implementation can be found in 
the MODSIM, CTAC etc papers. The six case studies to validation and verify \ANUGA{} will be
presented in section~\ref{sec:validation}, with the conclusions 
outlined in section~\ref{sec:conclusions}. 

{\bf question - if the Okushiri result has already been presented in the
MODSIM paper, how should it be presented in this paper? - simply refer to it I think}

\section{Model}
\label{sec:model}

The shallow water wave equations are a system of differential
conservation equations which describe the flow of a thin layer of
fluid over terrain. The form of the equations are:
\[
\frac{\partial \UU}{\partial t}+\frac{\partial \EE}{\partial
x}+\frac{\partial \GG}{\partial y}=\SSS
\]
where $\UU=\left[ {{\begin{array}{*{20}c}
 h & {uh} & {vh} \\
\end{array} }} \right]^T$ is the vector of conserved quantities; water depth
$h$, $x$-momentum $uh$ and $y$-momentum $vh$. Other quantities
entering the system are bed elevation $z$ and stage (absolute water
level) $w$, where the relation $w = z + h$ holds true at all times.
The fluxes in the $x$ and $y$ directions, $\EE$ and $\GG$ are given
by
\[
\EE=\left[ {{\begin{array}{*{20}c}
 {uh} \hfill \\
 {u^2h+gh^2/2} \hfill \\
 {uvh} \hfill \\
\end{array} }} \right]\mbox{ and }\GG=\left[ {{\begin{array}{*{20}c}
 {vh} \hfill \\
 {vuh} \hfill \\
 {v^2h+gh^2/2} \hfill \\
\end{array} }} \right]
\]
and the source term (which includes gravity and friction) is given
by
\[
\SSS=\left[ {{\begin{array}{*{20}c}
 0 \hfill \\
 -{gh(z_{x} + S_{fx} )} \hfill \\
 -{gh(z_{y} + S_{fy} )} \hfill \\
\end{array} }} \right]
\]
where $S_f$ is the bed friction. The friction term is modelled using
Manning's resistance law
\[
S_{fx} =\frac{u\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}\mbox{ and }S_{fy}
=\frac{v\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}
\]
in which $\eta$ is the Manning resistance coefficient.

As demonstrated in our papers, \cite{modsim2005,Rob99l} these
equations provide an excellent model of flows associated with
inundation such as dam breaks and tsunamis. Question - how do we
know it is excellent? 

\ANUGA{} uses a finite-volume method as 
described in \cite{Rob991} where the study area is represented by an
unstructured triangular mesh. The flexibility afforded by this approach
is the ability for the user to refine the mesh in areas of interest.
\ANUGA{} is mostly written in the object-oriented programming
language \Python{} with computationally intensive parts implemented
as highly optimised shared objects written in C. The API is a 
\Python{} script where the user sets up the scenario. This script
defines the study area, mesh refinement as well as initial and boundary conditions. 
The user is free to update quantity values or boundary conditions through
the simulation. Reference to user manual

Could include here a brief overview of the numerical
technique and reference the CTAC 2006 paper and has Steve written something
that could be included?

\section{Finite Volume Method}
\label{sec:fvm}

{\bf Jane: I don't think this section is needed here, but the 
content is referred to at the end of section 1}

We use a finite-volume method for solving the shallow water wave
equations \cite{Rob99l}. The study area is represented by a mesh of
triangular cells as in Figure~\ref{fig:mesh} in which the conserved
quantities of  water depth $h$, and horizontal momentum $(uh, vh)$,
in each volume are to be determined. The size of the triangles may
be varied within the mesh to allow greater resolution in regions of
particular interest.

\begin{figure}
\begin{center}
%\includegraphics[width=5.0cm,keepaspectratio=true]{step-five}
\caption{Triangular mesh used in our finite volume method. Conserved
quantities $h$, $uh$ and $vh$ are associated with the centroid of
each triangular cell.} \label{fig:mesh}
\end{center}
\end{figure}

The equations constituting the finite-volume method are obtained by
integrating the differential conservation equations over each
triangular cell of the mesh. Introducing some notation we use $i$ to
refer to the $i$th triangular cell $T_i$, and ${\cal N}(i)$ to the
set of indices referring to the cells neighbouring the $i$th cell.
Then $A_i$ is the area of the $i$th triangular cell and $l_{ij}$ is
the length of the edge between the $i$th and $j$th cells.

By applying the divergence theorem we obtain for each volume an
equation which describes the rate of change of the average of the
conserved quantities within each cell, in terms of the fluxes across
the edges of the cells and the effect of the source terms. In
particular, rate equations associated with each cell have the form
$$
 \frac{d\UU_i }{dt}+ \frac1{A_i}\sum\limits_{j\in{\cal N}(i)} \HH_{ij} l_{ij} = \SSS_i
$$
where
\begin{itemize}
\item $\UU_i$ the vector of conserved quantities averaged over the $i$th cell,
\item $\SSS_i$ is the source term associated with the $i$th cell,
and
\item $\HH_{ij}$ is the outward normal flux of
material across the \textit{ij}th edge.
\end{itemize}


%\item $l_{ij}$ is the length of the edge between the $i$th and $j$th
%cells
%\item $m_{ij}$ is the midpoint of
%the \textit{ij}th edge,
%\item
%$\mathbf{n}_{ij} = (n_{ij,1} , n_{ij,2})$is the outward pointing
%normal along the \textit{ij}th edge, and The

The flux $\HH_{ij}$ is evaluated using a numerical flux function
$\HH(\cdot, \cdot ; \ \cdot)$ which is consistent with the shallow
water flux in the sense that for all conservation vectors $\UU$ and normal vectors $\nn$
$$
H(\UU,\UU;\ \nn) = \EE(\UU) n_1 + \GG(\UU) n_2 .
$$

Then
$$
\HH_{ij}  = \HH(\UU_i(m_{ij}),
\UU_j(m_{ij}); \mathbf{n}_{ij})
$$
where $m_{ij}$ is the midpoint of the \textit{ij}th edge and
$\mathbf{n}_{ij}$ is the outward pointing normal, with respect to the $i$th cell, on the
\textit{ij}th edge. The function $\UU_i(x)$ for $x \in
T_i$ is obtained from the vector $\UU_k$ of conserved average values for the $i$th and
neighbouring  cells.

We use a second order reconstruction to produce a piece-wise linear
function construction of the conserved quantities for  all $x \in
T_i$ for each cell (see Figure~\ref{fig:mesh:reconstruct}. This
function is allowed to be discontinuous across the edges of the
cells, but the slope of this function is limited to avoid
artificially introduced oscillations.

Godunov's method (see \cite{Toro-92}) involves calculating the
numerical flux function $\HH(\cdot, \cdot ; \ \cdot)$ by exactly
solving the corresponding one dimensional Riemann problem normal to
the edge. We use the central-upwind scheme of \cite{KurNP2001} to
calculate an approximation of the flux across each edge.

\begin{figure}
\begin{center}
%\includegraphics[width=5.0cm,keepaspectratio=true]{step-reconstruct}
\caption{From the values of the conserved quantities at the centroid
of the cell and its neighbouring cells, a discontinuous piecewise
linear reconstruction of the conserved quantities is obtained.}
\label{fig:mesh:reconstruct}
\end{center}
\end{figure}

In the computations presented in this paper we use an explicit Euler
time stepping method with variable timestepping adapted to the
observed CFL condition.


\section{Software Implementation}
\label{sec:software}

{\bf Jane: I don't think this section is needed here, but the 
content is referred to at the end of section 1}
 
\ANUGA{} is mostly written in the object-oriented programming
language \Python{} with computationally intensive parts implemented
as highly optimised shared objects written in C.

\Python{} is known for its clarity, elegance, efficiency and
reliability. Complex software can be built in \Python{} without undue
distractions arising from idiosyncrasies of the underlying software
language syntax. In addition, \Python{}'s automatic memory management,
dynamic typing, object model and vast number of libraries means that
software can be produced quickly and can be readily adapted to
changing requirements throughout its lifetime.

The fundamental object in \ANUGA{} is the \code{Domain} which
inherits functionality from a hierarchy of increasingly specialised
classes starting with a basic structural Mesh to classes implementing
the finite-volume scheme described in section \ref{sec:fvm}. Other classes
are \code{Quantity} which represents values of one variable across the mesh
along with their associated operations, \code{Geospatial_data} which
represents georeferenced elevation data and a collection of \code{Boundary}
classes which allows for a 'pluggable' way of driving the model.
The conserved quantities updated automatically by the numerical scheme
are stage (water level) $w$, $x$-momentum $uh$ and $y$-momentum
$vh$. The quanitites elevation $z$ and friction $\eta$ are
quantities that are not updated automatically but can be changed explicitly
during run-time if the user wishes to do so.

To set up a scenario the user specifies the study area along with any internal
regions where increased mesh resolution is required. External edges may
be labelled using symbolic tags which are subsequently used to bind
boundary condition objects to tagged segments of the mesh boundary.
The mesh is then generated using \ANUGA{}'s built-in mesh generator and
converted into the \code{Domain} object which provides all methods used to
setup and run the flow simulation. Figure \ref{fig:anuga mesh} shows an example of a mesh generated by \ANUGA{}.
\begin{figure}
\begin{center}
\includegraphics[width=4in,keepaspectratio=true]{triangular-mesh}
\caption{Triangular mesh used in our finite volume method. Conserved
quantities $h$, $uh$ and $vh$ are associated with each triangular cell.}
\label{fig:anuga mesh}
\end{center}
\end{figure}



Next step is to setup initial conditions for each \code{Quantity} object. For
the elevation $z$ this is typically obtained from bathymetric and
topographic data sets. Setting initial values for quantities is done
through the method \code{domain.set_quantity(name, X, location,
region)} where name is the name of the quantity (e.g.\ 'stage',
'xmomentum', 'ymomentum', 'elevation' or 'friction').  The variable X
represents the source data for populating the quantity and may take
one of the following forms:

\begin{itemize}
\item A constant value as in \code{domain.set_quantity('stage', 1)} which
will set the initial water level to 1 m everywhere.
\item Another quantity or a linear combination of quantities.  If \code{q1}
and \code{q2} are two arbitrary quantities defined within the same domain,
the expression \code{domain.set_quantity('stage', q1*(3*q2 + 5))} will set the stage
quantity accordingly.  One common application of this would be to
assign the stage as a constant depth above the bed elevation.
\item An arbitrary function (or a callable object), \code{f(x, y)}, where \code{x}
and \code{y} are assumed to be vectors. The quantity will be assigned values by
evaluating \code{f} at each location within the mesh.
\item An arbitrary set of points and associated values (wrapped into a
Geospatial_data object). The points need not coincide with triangle
vertices or centroids and a penalised least squares technique is
employed to populate the quantity in a smooth and stable way.
Since the least squares technique can be time consuming for large
problems, \code{set_quantity} employs a caching technique which automatically
decides whether to perform the computations or retrieve them from a
cache.  This will typically speed up the build by several orders of
magnitude after each computation has been performed once.
\item A filename containing points and attributes.
\item A Numerical Python array (or a list of numbers) ordered
according to the internal data structure.
\end{itemize}
The parameter \code{location} determines whether the values should be
assigned to triangle edge, midpoints or vertices and \code{region} allows the
operation to be restricted to a region specified by a symbolic tag or
a set of indices.

Boundary conditions are bound to symbolic tags through the method
\code{domain.set_boundary} which takes as input a lookup table (implemented
as a Python dictionary) of the form \code{\{tag:~boundary_object\}}.
The boundary objects are all assumed to be callable functions of vectors x
and y.  Several predefined standard boundary objects are available and
it is relatively straightforward to define problem-specific custom
boundaries if needed.  The predefined boundary conditions include
Dirichlet, Reflective, Transmissive, Temporal, and Spatio-Temporal
boundaries.

Forcing terms can be written according to a fixed protocol and added
to the model using the idiom \code{domain.forcing_terms.append(F)} where \code{F} is
assumed to be a user-defined callable object.

When the simulation is running, the length of each time step is
determined from the maximal speeds encountered and the sizes of
triangles in order not to violate the CFL condition which specifies
that no information should skip any triangles in one time step.  With
large speeds and small triangles, time steps can become very small.
In order to access the state of the simulation at regular time
intervals, \ANUGA{} uses the method evolve:
\begin{verbatim}
For t in domain.evolve(yieldstep, duration):
    <model interrogation and modification>
\end{verbatim}
The parameter \code{duration} specifies the time period over which
evolve operates, and control is passed to the body of the for-loop at
each fixed time step called \code{yieldstep}.  The internal workings
of the numerical scheme and its variable time stepping are thus
decoupled from the fixed time stepping of the evolve loop.  This means
that the user of the API may access the model at fixed timesteps to
e.g.\ store model outputs, interrogate quantities or change the model
itself at runtime. The evolve method has been implemented using a
Python generator hence the reference to 'yield' in the parameter name.

%Figure \ref{fig:beach runup} shows a simulation of water flowing onto a
%hypothetical beach with obstacles.
%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.
%See the \ANUGA{} User Manual at \url{http://sourceforge.net/projects/anuga} for more details and examples.
%\begin{figure}
%\begin{center}
%\includegraphics[width=4in,keepaspectratio=true]{runup}
%\caption{A hypothetical runup scenario.}
%\label{fig:beach runup}
%\end{center}
%\end{figure}



\section{Validation}
\label{sec:validation} Validation is an ongoing process and the purpose of this paper
is to describe a range of tests that validate \ANUGA{} as a hydrodynamic model. 
This section will describe the six tests outlined in section~\ref{sec:intro}.

\subsection{1D analytical validation}

Tom Baldock has done something here for that NSW report

\subsection{Stage and Velocity Validation in a Flume}
This section will describe flume tank experiments that were
conducted at the Gordon McKay Hydraulics Laboratory at the University of
Queensland that confirm \ANUGA{}'s ability to estimate wave height
and velocity. The same flume tank simulations were also used
to explore Manning's friction and this will be described in the next section.

The flume was set up for dam-break experiments, having a
water reservior at one end.  The flume was glass-sided, 3m long, 0.4m
in wide, and 0.4m deep, with a PVC bottom. The reservoir in the flume
was 0.75m long.  For this experiment the reservoir water was 0.2m
deep. At time zero the reservoir gate is opened and the water flows
into the other side of the flume.  The water ran up a flume slope of
0.03 m/m.  To accurately model the bed surface a Manning's friction
value of 0.01, representing PVC was used.

% Neale, L.C. and R.E. Price.  Flow characteristics of PVC sewer pipe.
% Journal of the Sanitary Engineering Division, Div. Proc 90SA3, ASCE.
% pp. 109-129.  1964.

Acoustic displacement sensors that produced a voltage that changed
with the water depth was positioned 0.4m from the reservoir gate. The
water velocity was measured with an Acoustic Doppler Velocimeter 0.45m
from the reservoir gate.  This sensor only produced reliable results 4
seconds after the reservoir gate opened, due to limitations of the sensor.


% Validation UQ flume
% at X:\anuga_validation\uq_sloped_flume_2008
% run run_dam.py to create sww file and .csv files
% run plot.py to create graphs heere automatically
% The Coasts and Ports '2007 paper is in TRIM d2007-17186
\begin{figure}[htbp]
\centerline{\includegraphics[width=4in]{uq-flume-depth}}
\caption{Comparison of wave tank and \ANUGA{} water height at .4 m
  from the gate}\label{fig:uq-flume-depth}
\end{figure}

\begin{figure}[htbp]
\centerline{\includegraphics[width=4in]{uq-flume-velocity}}
\caption{Comparison of wave tank and \ANUGA{} water velocity at .45 m
  from the gate}\label{fig:uq-flume-velocity}
\end{figure}

Figure~\ref{fig:uq-flume-depth} shows that ANUGA predicts the actual
water depth very well, with the exception of the fluid tip-region {\bf
Duncan - what does that mean? About where on the graph is that). 
Water depth and velocity are coupled as described by the nonlinear shallow water equations, thus
if one of these quantities accurately estimates the measured values, we would expect 
the same for the other quantity. This is demonstrated in figure~\ref{fig:uq-flume-velocity)
where the water velocity is also predicted accurately. Sediment transport studies
rely on water velocity estimates in the region where the sensors cannot provide this data.
With water velocity being accurately predicted, studies such as sediment transport can now use
reliable estimates.


\subsection{1D flume tank to verify friction}

The same flume tank experimental setup was used to obtain friction values for
use in hydrodynamic models. A number of bed friction scenarios were simulated in 
the flume tank. The PVC bottom of the tank is equivalent to a friction value of 0 (i.e
completely smooth) and small pebbles were used to cover the base of the tank and the
aim of the experiment was to determine what the Manning's friction value is for 
this case.
 
As described in the model equations in \section~\ref{sec:model}, the bed
friction is modelled using the Manning's model. 
Validation of this model was carried out by comparing results
from ANUGA against experimental results from flume wave tanks. 
 
% Validation UQ friction
% at X:\anuga_validation\uq_friction_2007
% run run_dam.py to create sww file and .csv files
% run plot.py to create graphs, and move them here
\begin{figure}[htbp]
\centerline{\includegraphics[width=4in]{uq-friction-depth}}
\caption{Comparison of wave tank and \ANUGA{} water height at .4 m
  from the gate, simulated using a Mannings friction of 0.0 and 0.1.}\label{fig:uq-friction-depth}
\end{figure}

\subsection{Okushiri Wavetank Validation}

As part of the Third International Workshop on Long-wave Runup
Models in 2004 (\url{http://www.cee.cornell.edu/longwave}), four
benchmark problems were specified to allow the comparison of
numerical, analytical and physical models with laboratory and field
data. One of these problems describes a wave tank simulation of the
1993 Okushiri Island tsunami off Hokkaido, Japan \cite{MatH2001}. A
significant feature of this tsunami was a maximum run-up of 32~m
observed at the head of the Monai Valley. This run-up was not
uniform along the coast and is thought to have resulted from a
particular topographic effect. Among other features, simulations of
the Hokkaido tsunami should capture this run-up phenomenon.

This dataset has been used by to validate tsunami models by
a number of tsunami scientists. Examples include Titov ... lit review
here on who has used this example for verification

\begin{figure}[htbp]
%\centerline{\includegraphics[width=4in]{okushiri-gauge-5.eps}}
\centerline{\includegraphics[width=4in]{ch5.png}}
\centerline{\includegraphics[width=4in]{ch7.png}}
\centerline{\includegraphics[width=4in]{ch9.png}}
\caption{Comparison of wave tank and \ANUGA{} water stages at gauge
5,7 and 9.}\label{fig:val}
\end{figure}


\begin{figure}[htbp]
\centerline{\includegraphics[width=4in]{okushiri-model.jpg}}
\caption{Complex reflection patterns and run-up into Monai Valley
simulated by \ANUGA{} and visualised using our netcdf OSG
viewer.}\label{fig:run}
\end{figure}

The wave tank simulation of the Hokkaido tsunami was used as the
first scenario for validating \ANUGA{}. The dataset provided
bathymetry and topography along with initial water depth and the
wave specifications. The dataset also contained water depth time
series from three wave gauges situated offshore from the simulated
inundation area. The \ANUGA{} model comprised $41404$ triangles 
and took about $2000$ s to run on a standard PC or $1500$ s 
on a 64-bit Opteron 2000 series Linux server.  

Figure~\ref{fig:val} compares the observed wave tank and modelled
\ANUGA{} water depth (stage height) at one of the gauges. The plots
show good agreement between the two time series, with \ANUGA{
closely modelling the initial draw down, the wave shoulder and the
subsequent reflections. The discrepancy between modelled and
simulated data in the first 10 seconds is due to the initial
condition in the physical tank not being uniformly zero. Similarly
good comparisons are evident with data from the other two gauges.
Additionally, \ANUGA{} replicates exceptionally well the 32~m Monai
Valley run-up, and demonstrates its occurrence to be due to the
interaction of the tsunami wave with two juxtaposed valleys above
the coastline. The run-up is depicted in Figure~\ref{fig:run}.

This successful replication of the tsunami wave tank simulation on a
complex 3D beach is a positive first step in validating the \ANUGA{}
modelling capability. 

\subsection{Runup of solitary wave on circular island wavetank validation}

This section will describe the ANUGA results for the experiments conducted
by Briggs et al (1995). Here, a 30x25m basin with a conical island is situated near
the centre and a directional wavemaker is used to produce planar solitary waves of
specified crest lenghts and heights. A series of gauges were distributed within the
experimental setup. As described by Hubbard and Dodd \cite{Hubbard02}, a number of researchers
have used this benchmark problem to test their numerical models. {\bf Jane: check
whether these results are now avilable as they were not in 2002}. Hubbard and Dodd 
\cite{Hubbard02} note that 
a particular 3D model appears to obtain slightly better results than the 2D ones reported 
but that 3D models are unlikely to be competitive in terms of computing power for
applications in coastal engineering at least. Choi et al \cite{Choi07) use a 3D RANS model
(based on the Navier-Stokes equations)
for the same problem and find a very good comparison with laboratory and 2D numerical 
results. An obvious advantage of the 3D model is its ability to investigate the
velocity field and Choi et al also report on the limitation of depth-averaged
2D models for run-up simulations of this type.

Once results are availble, need to compare to Hubbard and Dodd and draw any conclusions
from nested rectangular grid vs unstructured gird.
Figure \ref{fig:circular screenshots} shows a sequence of screenshots depicting the evolution of the solitary wave as it hits the circular island.

\begin{figure}[htbp]
\centerline{
  \includegraphics[width=5cm]{circular1.png}
  \includegraphics[width=5cm]{circular2.png}}
\centerline{
  \includegraphics[width=5cm]{circular3.png}
  \includegraphics[width=5cm]{circular4.png}}
\centerline{
  \includegraphics[width=5cm]{circular5.png}
  \includegraphics[width=5cm]{circular6.png}}
\centerline{
  \includegraphics[width=5cm]{circular7.png}
  \includegraphics[width=5cm]{circular8.png}}
\centerline{
  \includegraphics[width=5cm]{circular9.png}
  \includegraphics[width=5cm]{circular10.png}}
\caption{Screenshots of the evolution of solitary wave around circular island.}
\label{fig:circular screenshots}
\end{figure}


\clearpage

\section{Conclusions}
\label{sec:6}
\ANUGA{} is a flexible and robust modelling system
that simulates hydrodynamics by solving the shallow water wave
equation in a triangular mesh. It can model the process of wetting
and drying as water enters and leaves an area and is capable of
capturing hydraulic shocks due to the ability of the finite-volume
method to accommodate discontinuities in the solution.
\ANUGA{} can take as input bathymetric and topographic datasets and
simulate the behaviour of riverine flooding, storm surge,
tsunami or even dam breaks.
Initial validation using wave tank data supports \ANUGA{}'s
ability to model complex scenarios. Further validation will be
pursued as additional datasets become available.
The \ANUGA{} source code is available
at \url{http://sourceforge.net/projects/anuga}.

something about use on flood modelling community and their validation initiatives

\bibliographystyle{plain}
\bibliography{anuga-bibliography}

\end{document}