% Use the standard \LaTeXe\ article style in 12pt Computer Modern font
% on A4 paper by
\documentclass[12pt,a4paper]{article}
% Do \emph{not} change the width nor the height of the text from the
% defaults set by this document class.
%
% The alternative which is closer to what we actually use is
% \documentclass[11pt,a5paper]{article}
% \usepackage[a5paper]{geometry}
% Because it is a great size for on screen reading
% and prints nicely on a4paper either 2up or booklet.

% The preamble is to contain your own \LaTeX\ commands and to say
% what packages to use.  Three useful packages are the following:
\usepackage{graphicx} % avoid epsfig or earlier such packages
\usepackage{url}      % for URLs and DOIs
\newcommand{\doi}[1]{\url{http://dx.doi.org/#1}}
\usepackage{amsmath}  % many want amsmath extensions
\usepackage{amsfonts}
\usepackage{underscore}
% Avoid loading unused packages (as done by some \LaTeX\ editors).

% Create title and authors using \verb|\maketitle|.  Separate authors by
% \verb|\and| and put addresses in \verb|\thanks| command with
% \verb|\url| command \verb|\protect|ed.
\title{Open Source Software for Computational Modelling of Fluid Flow}

\author{
O.~M.~Nielsen\thanks{Risk Assessment Methods Project, Geospatial and
Earth Monitoring Division, Geoscience Australia, Symonston,
\textsc{Australia}. \protect\url{mailto:Ole.Nielsen@ga.gov.au}}\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{28 December 2006}

\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}

% Use default \verb|\maketitle|.
\maketitle

% Use the \verb|abstract| environment.
\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 we describe the model, the architecture and some applications.
ANUGA has recently been released 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

% Use \verb|\section|, \verb|\subsection|, \verb|\subsubsection| and
% possibly \verb|\paragraph| to structure your document.  Make sure
% you \verb|\label| them for cross-referencing with \verb|\ref|\,.

%\clearpage
\section{Introduction}
\label{sec:intro}

The Indian Ocean tsunami on 26 December 2004 demonstrated the
potentially catastrophic consequences of natural hazards.  While the
scale of the impact from such events is not common, smaller-scale
tsunami regularly threaten coastal communities
around the world. Earthquakes which occur in the Java Trench near
Indonesia (e.g.~\cite{TsuMIS1995} or \cite{Baldwin-2006}) and along
the Puysegur Ridge to the south of New Zealand (e.g.~\cite{LebKC1998})
have potential to generate tsunami that may threaten Australia's
northwestern and southeastern coastlines. In addition, the
preferential development of Australian coastal corridors means that
inundation from hydrological disasters such as tsunami or storm-surge
of even a few hundred metres beyond the shoreline has increased
potential to cause significant disruption and loss. 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.

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.  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:fvm}.  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.
Section~\ref{sec:software} describes the software implementation and
the API while section~\ref{sec:validation} presents some
validation results.


\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.

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

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}

\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]{tsunami-fig-1}
\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]{tsunami-fig-2}
\caption{A hypothetical runup scenario.}
\label{fig:beach runup}
\end{center}
\end{figure}



\section{Validation}
\label{sec:validation} The process of validating the \AnuGA{}
application is in its early stages, however initial indications are
encouraging.

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.

\begin{figure}[htbp]
\centerline{\includegraphics[width=4in]{tsunami-fig-3.eps}}
\caption{Comparison of wave tank and \AnuGA{} water stages at gauge
5.}\label{fig:val}
\end{figure}


\begin{figure}[htbp]
\centerline{\includegraphics[width=4in]{tsunami-fig-4.eps}}
\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.

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. Subsequent validation will be conducted as
additional datasets become available.


\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.
\AnuGA{} is already being used to model the behaviour of
hydrodynamic natural hazards. 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}).
The \AnuGA{} source code is available
at \url{http://sourceforge.net/projects/anuga}.

\bibliographystyle{plain}
\bibliography{database1}




\end{document}