source: anuga_core/documentation/user_manual/anuga_user_manual.tex @ 5091

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\title{\anuga User Manual}
\author{Geoscience Australia and the Australian National University}

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\authoraddress{Geoscience Australia \\
  Email: \email{ole.nielsen@ga.gov.au}
}

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\begin{document}
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\begin{abstract}
\label{def:anuga}

\noindent \anuga\index{\anuga} is a hydrodynamic modelling tool that
allows users to model realistic flow problems in complex geometries.
Examples include dam breaks or the effects of natural hazards such
as riverine flooding, storm surges and tsunami.

The user must specify a study area represented by a mesh of triangular
cells, the topography and bathymetry, frictional resistance, initial
values for water level (called \emph{stage}\index{stage} within \anuga),
boundary
conditions and forces such as windstress or pressure gradients if
applicable.

\anuga tracks the evolution of water depth and horizontal momentum
within each cell over time by solving the shallow water wave equation
governing equation using a finite-volume method.

\anuga also incorporates a mesh generator %, called \code{graphical
                                %mesh generator},
that
allows the user to set up the geometry of the problem interactively as
well as tools for interpolation and surface fitting, and a number of
auxiliary tools for visualising and interrogating the model output.

Most \anuga components are written in the object-oriented programming
language Python and most users will interact with \anuga by writing
small Python programs based on the \anuga library
functions. Computationally intensive components are written for
efficiency in C routines working directly with the Numerical Python
structures.


\end{abstract}

\tableofcontents


\chapter{Introduction}


\section{Purpose}

The purpose of this user manual is to introduce the new user to the
inundation software, describe what it can do and give step-by-step
instructions for setting up and running hydrodynamic simulations.

\section{Scope}

This manual covers only what is needed to operate the software after
installation and configuration. It does not includes instructions
for installing the software or detailed API documentation, both of
which will be covered in separate publications and by documentation
in the source code.

\section{Audience}

Readers are assumed to be familiar with the operating environment
and have a general understanding of the subject matter, as well as
enough programming experience to adapt the code to different
requirements and to understand the basic terminology of
object-oriented programming.

\pagebreak
\chapter{Background}


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 are developing a hydrodynamic inundation modelling tool
called \anuga to help simulate the impact of these hazards.

The core of \anuga is the fluid dynamics module, called \code{shallow\_water},
which is based on a finite-volume method for solving the Shallow Water
Wave Equation.  The study area is represented by a mesh of triangular
cells.  By solving the governing equation within each cell, water
depth and horizontal momentum are tracked over time.

A major capability of \anuga is that it can 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.  \anuga is also capable
of modelling hydraulic jumps due to the ability of the finite-volume
method to accommodate discontinuities in the solution.

To set up a particular scenario the user specifies the geometry
(bathymetry and topography), the initial water level (stage),
boundary conditions such as tide, and any forcing terms that may
drive the system such as wind stress or atmospheric pressure
gradients. Gravity and frictional resistance from the different
terrains in the model are represented by predefined forcing terms.

The built-in mesh generator, called \code{graphical\_mesh\_generator},
allows the user to set up the geometry
of the problem interactively and to identify boundary segments and
regions using symbolic tags.  These tags may then be used to set the
actual boundary conditions and attributes for different regions
(e.g.\ the Manning friction coefficient) for each simulation.

Most \anuga components are written in the object-oriented programming
language Python.  Software written in Python can be produced quickly
and can be readily adapted to changing requirements throughout its
lifetime.  Computationally intensive components are written for
efficiency in C routines working directly with the Numerical Python
structures.  The animation tool developed for \anuga is based on
OpenSceneGraph, an Open Source Software (OSS) component allowing high
level interaction with sophisticated graphics primitives.
See \cite{nielsen2005} for more background on \anuga.

\chapter{Restrictions and limitations on \anuga}
\label{ch:limitations}

Although a powerful and flexible tool for hydrodynamic modelling, \anuga has a
number of limitations that any potential user need to be aware of. They are

\begin{itemize}
  \item The mathematical model is the 2D shallow water wave equation.
  As such it cannot resolve vertical convection and consequently not breaking
  waves or 3D turbulence (e.g.\ vorticity).
  \item The surface is assumed to be open, e.g. \anuga cannot model
  flow under ceilings or in pipes
  \item All spatial coordinates are assumed to be UTM (meters). As such,
  ANUGA is unsuitable for modelling flows in areas larger than one UTM zone
  (6 degrees wide).
  \item Fluid is assumed to be inviscid
  \item The finite volume is a very robust and flexible numerical technique,
  but it is not the fastest method around. If the geometry is sufficiently
  simple and if there is no need for wetting or drying, a finite-difference
  method may be able to solve the problem faster than \anuga.
  %\item Mesh resolutions near coastlines with steep gradients need to be...
  \item Frictional resistance is implemented using Manning's formula, but
  \anuga has not yet been fully validated in regard to bottom roughness
  \item ANUGA contains no tsunami-genic functionality relating to
  earthquakes.
\end{itemize}



\chapter{Getting Started}
\label{ch:getstarted}

This section is designed to assist the reader to get started with
\anuga by working through some examples. Two examples are discussed;
the first is a simple example to illustrate many of the ideas, and
the second is a more realistic example.

\section{A Simple Example}
\label{sec:simpleexample}

\subsection{Overview}

What follows is a discussion of the structure and operation of a
script called \file{runup.py}.

This example carries out the solution of the shallow-water wave
equation in the simple case of a configuration comprising a flat
bed, sloping at a fixed angle in one direction and having a
constant depth across each line in the perpendicular direction.

The example demonstrates the basic ideas involved in setting up a
complex scenario. In general the user specifies the geometry
(bathymetry and topography), the initial water level, boundary
conditions such as tide, and any forcing terms that may drive the
system such as wind stress or atmospheric pressure gradients.
Frictional resistance from the different terrains in the model is
represented by predefined forcing terms. In this example, the
boundary is reflective on three sides and a time dependent wave on
one side.

The present example represents a simple scenario and does not
include any forcing terms, nor is the data taken from a file as it
would typically be.

The conserved quantities involved in the
problem are stage (absolute height of water surface),
$x$-momentum and $y$-momentum. Other quantities
involved in the computation are the friction and elevation.

Water depth can be obtained through the equation

\begin{tabular}{rcrcl}
  \code{depth} &=& \code{stage} &$-$& \code{elevation}
\end{tabular}


\subsection{Outline of the Program}

In outline, \file{runup.py} performs the following steps:

\begin{enumerate}

   \item Sets up a triangular mesh.

   \item Sets certain parameters governing the mode of
operation of the model-specifying, for instance, where to store the model output.

   \item Inputs various quantities describing physical measurements, such
as the elevation, to be specified at each mesh point (vertex).

   \item Sets up the boundary conditions.

   \item Carries out the evolution of the model through a series of time
steps and outputs the results, providing a results file that can
be visualised.

\end{enumerate}

\subsection{The Code}

%FIXME: we are using the \code function here.
%This should be used wherever possible
For reference we include below the complete code listing for
\file{runup.py}. Subsequent paragraphs provide a
`commentary' that describes each step of the program and explains it
significance.

\verbatiminput{demos/runup.py}

\subsection{Establishing the Mesh}\index{mesh, establishing}

The first task is to set up the triangular mesh to be used for the
scenario. This is carried out through the statement:

{\small \begin{verbatim}
    points, vertices, boundary = rectangular_cross(10, 10)
\end{verbatim}}

The function \function{rectangular_cross} is imported from a module
\module{mesh\_factory} defined elsewhere. (\anuga also contains
several other schemes that can be used for setting up meshes, but we
shall not discuss these.) The above assignment sets up a $10 \times
10$ rectangular mesh, triangulated in a regular way. The assignment

{\small \begin{verbatim}
    points, vertices, boundary = rectangular_cross(m, n)
\end{verbatim}}

returns:

\begin{itemize}

   \item a list \code{points} giving the coordinates of each mesh point,

   \item a list \code{vertices} specifying the three vertices of each triangle, and

   \item a dictionary \code{boundary} that stores the edges on
   the boundary and associates each with one of the symbolic tags \code{`left'}, \code{`right'},
   \code{`top'} or \code{`bottom'}.

\end{itemize}

(For more details on symbolic tags, see page
\pageref{ref:tagdescription}.)

An example of a general unstructured mesh and the associated data
structures \code{points}, \code{vertices} and \code{boundary} is
given in Section \ref{sec:meshexample}.




\subsection{Initialising the Domain}

These variables are then used to set up a data structure
\code{domain}, through the assignment:

{\small \begin{verbatim}
    domain = Domain(points, vertices, boundary)
\end{verbatim}}

This creates an instance of the \class{Domain} class, which
represents the domain of the simulation. Specific options are set at
this point, including the basename for the output file and the
directory to be used for data:

{\small \begin{verbatim}
    domain.set_name('runup')
\end{verbatim}}

{\small \begin{verbatim}
    domain.set_datadir('.')
\end{verbatim}}

In addition, the following statement now specifies that the
quantities \code{stage}, \code{xmomentum} and \code{ymomentum} are
to be stored:

{\small \begin{verbatim}
    domain.set_quantities_to_be_stored(['stage', 'xmomentum',
    'ymomentum'])
\end{verbatim}}


\subsection{Initial Conditions}

The next task is to specify a number of quantities that we wish to
set for each mesh point. The class \class{Domain} has a method
\method{set\_quantity}, used to specify these quantities. It is a
flexible method that allows the user to set quantities in a variety
of ways---using constants, functions, numeric arrays, expressions
involving other quantities, or arbitrary data points with associated
values, all of which can be passed as arguments. All quantities can
be initialised using \method{set\_quantity}. For a conserved
quantity (such as \code{stage, xmomentum, ymomentum}) this is called
an \emph{initial condition}. However, other quantities that aren't
updated by the equation are also assigned values using the same
interface. The code in the present example demonstrates a number of
forms in which we can invoke \method{set\_quantity}.


\subsubsection{Elevation}

The elevation, or height of the bed, is set using a function,
defined through the statements below, which is specific to this
example and specifies a particularly simple initial configuration
for demonstration purposes:

{\small \begin{verbatim}
    def f(x,y):
        return -x/2
\end{verbatim}}

This simply associates an elevation with each point \code{(x, y)} of
the plane.  It specifies that the bed slopes linearly in the
\code{x} direction, with slope $-\frac{1}{2}$,  and is constant in
the \code{y} direction.

Once the function \function{f} is specified, the quantity
\code{elevation} is assigned through the simple statement:

{\small \begin{verbatim}
    domain.set_quantity('elevation', f)
\end{verbatim}}

NOTE: If using function to set \code{elevation} it must be vector
compatible. For example square root will not work.

\subsubsection{Friction}

The assignment of the friction quantity (a forcing term)
demonstrates another way we can use \method{set\_quantity} to set
quantities---namely, assign them to a constant numerical value:

{\small \begin{verbatim}
    domain.set_quantity('friction', 0.1)
\end{verbatim}}

This specifies that the Manning friction coefficient is set to 0.1
at every mesh point.

\subsubsection{Stage}

The stage (the height of the water surface) is related to the
elevation and the depth at any time by the equation

{\small \begin{verbatim}
    stage = elevation + depth
\end{verbatim}}


For this example, we simply assign a constant value to \code{stage},
using the statement

{\small \begin{verbatim}
    domain.set_quantity('stage', -.4)
\end{verbatim}}

which specifies that the surface level is set to a height of $-0.4$,
i.e. 0.4 units (m) below the zero level.

Although it is not necessary for this example, it may be useful to
digress here and mention a variant to this requirement, which allows
us to illustrate another way to use \method{set\_quantity}---namely,
incorporating an expression involving other quantities. Suppose,
instead of setting a constant value for the stage, we wished to
specify a constant value for the \emph{depth}. For such a case we
need to specify that \code{stage} is everywhere obtained by adding
that value to the value already specified for \code{elevation}. We
would do this by means of the statements:

{\small \begin{verbatim}
    h = 0.05 # Constant depth
    domain.set_quantity('stage', expression = 'elevation + %f' %h)
\end{verbatim}}

That is, the value of \code{stage} is set to $\code{h} = 0.05$ plus
the value of \code{elevation} already defined.

The reader will probably appreciate that this capability to
incorporate expressions into statements using \method{set\_quantity}
greatly expands its power.) See Section \ref{sec:Initial Conditions} for more
details.

\subsection{Boundary Conditions}\index{boundary conditions}

The boundary conditions are specified as follows:

{\small \begin{verbatim}
    Br = Reflective_boundary(domain)

    Bt = Transmissive_boundary(domain)

    Bd = Dirichlet_boundary([0.2,0.,0.])

    Bw = Time_boundary(domain=domain,
                f=lambda t: [(0.1*sin(t*2*pi)-0.3), 0.0, 0.0])
\end{verbatim}}

The effect of these statements is to set up a selection of different
alternative boundary conditions and store them in variables that can be
assigned as needed. Each boundary condition specifies the
behaviour at a boundary in terms of the behaviour in neighbouring
elements. The boundary conditions introduced here may be briefly described as
follows:

\begin{itemize}
    \item \textbf{Reflective boundary}\label{def:reflective boundary} Returns same \code{stage} as
      as present in its neighbour volume but momentum vector
      reversed 180 degrees (reflected).
      Specific to the shallow water equation as it works with the
      momentum quantities assumed to be the second and third conserved
      quantities. A reflective boundary condition models a solid wall.
    \item \textbf{Transmissive boundary}\label{def:transmissive boundary} Returns same conserved quantities as
      those present in its neighbour volume. This is one way of modelling
      outflow from a domain, but it should be used with caution if flow is
      not steady state as replication of momentum at the boundary
      may cause occasional spurious effects. If this occurs,
      consider using e.g. a Dirichlet boundary condition.
    \item \textbf{Dirichlet boundary}\label{def:dirichlet boundary} Specifies
      constant values for stage, $x$-momentum and $y$-momentum at the boundary.
    \item \textbf{Time boundary}\label{def:time boundary} Like a Dirichlet
      boundary but with behaviour varying with time.
\end{itemize}

\label{ref:tagdescription}Before describing how these boundary
conditions are assigned, we recall that a mesh is specified using
three variables \code{points}, \code{vertices} and \code{boundary}.
In the code we are discussing, these three variables are returned by
the function \code{rectangular}; however, the example given in
Section \ref{sec:realdataexample} illustrates another way of
assigning the values, by means of the function
\code{create\_mesh\_from\_regions}.

These variables store the data determining the mesh as follows. (You
may find that the example given in Section \ref{sec:meshexample}
helps to clarify the following discussion, even though that example
is a \emph{non-rectangular} mesh.)

\begin{itemize}
\item The variable \code{points} stores a list of 2-tuples giving the
coordinates of the mesh points.

\item The variable \code{vertices} stores a list of 3-tuples of
numbers, representing vertices of triangles in the mesh. In this
list, the triangle whose vertices are \code{points[i]},
\code{points[j]}, \code{points[k]} is represented by the 3-tuple
\code{(i, j, k)}.

\item The variable \code{boundary} is a Python dictionary that
not only stores the edges that make up the boundary but also assigns
symbolic tags to these edges to distinguish different parts of the
boundary. An edge with endpoints \code{points[i]} and
\code{points[j]} is represented by the 2-tuple \code{(i, j)}. The
keys for the dictionary are the 2-tuples \code{(i, j)} corresponding
to boundary edges in the mesh, and the values are the tags are used
to label them. In the present example, the value \code{boundary[(i,
j)]} assigned to \code{(i, j)]} is one of the four tags
\code{`left'}, \code{`right'}, \code{`top'} or \code{`bottom'},
depending on whether the boundary edge represented by \code{(i, j)}
occurs at the left, right, top or bottom of the rectangle bounding
the mesh. The function \code{rectangular} automatically assigns
these tags to the boundary edges when it generates the mesh.
\end{itemize}

The tags provide the means to assign different boundary conditions
to an edge depending on which part of the boundary it belongs to.
(In Section \ref{sec:realdataexample} we describe an example that
uses different boundary tags --- in general, the possible tags are entirely selectable by the user when generating the mesh and not
limited to `left', `right', `top' and `bottom' as in this example.)
All segments in bounding polygon must be tagged. If a tag is not supplied, the default tag name 'exterior' will be assigned by ANUGA.


Using the boundary objects described above, we assign a boundary
condition to each part of the boundary by means of a statement like

{\small \begin{verbatim}
    domain.set_boundary({'left': Br, 'right': Bw, 'top': Br, 'bottom': Br})
\end{verbatim}}

It is critical that all tags are assoctiated with a boundary conditing in this statement. If not the program will halt with a statement like

\begin{verbatim}

Traceback (most recent call last):
  File "mesh_test.py", line 114, in ?
    domain.set_boundary({'west': Bi, 'east': Bo, 'north': Br, 'south': Br})
  File "X:\inundation\sandpits\onielsen\anuga_core\source\anuga\abstract_2d_finite_volumes\domain.py", line 505, in set_boundary
    raise msg
ERROR (domain.py): Tag "exterior" has not been bound to a boundary object.
All boundary tags defined in domain must appear in the supplied dictionary.
The tags are: ['ocean', 'east', 'north', 'exterior', 'south']
\end{verbatim}


The command \code{set\_boundary} stipulates that, in the current example, the right
boundary varies with time, as defined by the lambda function, while the other
boundaries are all reflective.

The reader may wish to experiment by varying the choice of boundary
types for one or more of the boundaries. (In the case of \code{Bd}
and \code{Bw}, the three arguments in each case represent the
\code{stage}, $x$-momentum and $y$-momentum, respectively.)

{\small \begin{verbatim}
    Bw = Time_boundary(domain=domain,
                       f=lambda t: [(0.1*sin(t*2*pi)-0.3), 0.0, 0.0])
\end{verbatim}}



\subsection{Evolution}\index{evolution}

The final statement \nopagebreak[3]
{\small \begin{verbatim}
    for t in domain.evolve(yieldstep = 0.1, duration = 4.0):
        print domain.timestepping_statistics()
\end{verbatim}}

causes the configuration of the domain to `evolve', over a series of
steps indicated by the values of \code{yieldstep} and
\code{duration}, which can be altered as required.  The value of
\code{yieldstep} controls the time interval between successive model
outputs.  Behind the scenes more time steps are generally taken.


\subsection{Output}

The output is a NetCDF file with the extension \code{.sww}. It
contains stage and momentum information and can be used with the
ANUGA viewer \code{animate} (see Section \ref{sec:animate})
visualisation package
to generate a visual display. See Section \ref{sec:file formats}
(page \pageref{sec:file formats}) for more on NetCDF and other file
formats.

The following is a listing of the screen output seen by the user
when this example is run:

\verbatiminput{examples/runupoutput.txt}


\section{How to Run the Code}

The code can be run in various ways:

\begin{itemize}
  \item{from a Windows or Unix command line} as in\ \code{python runup.py}
  \item{within the Python IDLE environment}
  \item{within emacs}
  \item{within Windows, by double-clicking the \code{runup.py}
  file.}
\end{itemize}


\section{Exploring the Model Output}

The following figures are screenshots from the \anuga visualisation
tool \code{animate}. Figure \ref{fig:runupstart} shows the domain
with water surface as specified by the initial condition, $t=0$.
Figure \ref{fig:runup2} shows later snapshots for $t=2.3$ and
$t=4$ where the system has been evolved and the wave is encroaching
on the previously dry bed.  All figures are screenshots from an
interactive animation tool called animate which is part of \anuga and
distributed as in the package anuga\_viewer.
Animate is described in more detail is Section \ref{sec:animate}.

\begin{figure}[hbt]

  \centerline{\includegraphics[width=75mm, height=75mm]
    {graphics/bedslopestart.jpg}}

  \caption{Runup example viewed with the ANUGA viewer}
  \label{fig:runupstart}
\end{figure}


\begin{figure}[hbt]

  \centerline{
   \includegraphics[width=75mm, height=75mm]{graphics/bedslopeduring.jpg}
    \includegraphics[width=75mm, height=75mm]{graphics/bedslopeend.jpg}
   }

  \caption{Runup example viewed with ANGUA viewer}
  \label{fig:runup2}
\end{figure}



\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{A slightly more complex example}
\label{sec:channelexample}

\subsection{Overview}

The next example is about waterflow in a channel with varying boundary conditions and
more complex topograhies. These examples build on the
concepts introduced through the \file{runup.py} in Section \ref{sec:simpleexample}.
The example will be built up through three progressively more complex scripts.

\subsection{Overview}
As in the case of \file{runup.py}, the actions carried
out by the program can be organised according to this outline:

\begin{enumerate}

   \item Set up a triangular mesh.

   \item Set certain parameters governing the mode of
operation of the model---specifying, for instance, where to store the
model output.

   \item Set up initial conditions for various quantities such as the elevation, to be specified at each mesh point (vertex).

   \item Set up the boundary conditions.

   \item Carry out the evolution of the model through a series of time
steps and output the results, providing a results file that can be
visualised.

\end{enumerate}


\subsection{The Code}

Here is the code for the first version of the channel flow \file{channel1.py}:

\verbatiminput{demos/channel1.py}

In discussing the details of this example, we follow the outline
given above, discussing each major step of the code in turn.

\subsection{Establishing the Mesh}\index{mesh, establishing}

In this example we use a similar simple structured triangular mesh as in \code{runup.py}
for simplicity, but this time we will use a symmetric one and also
change the physical extent of the domain. The assignment

{\small \begin{verbatim}
    points, vertices, boundary = rectangular_cross(m, n,
                                                   len1=length, len2=width)
\end{verbatim}}
returns a m x n mesh similar to the one used in the previous example, except that now the
extent in the x and y directions are given by the value of \code{length} and \code{width}
respectively.

Defining m and n in terms of the extent as in this example provides a convenient way of
controlling the resolution: By defining dx and dy to be the desired size of each hypothenuse in the mesh we can write the mesh generation as follows:

{\small \begin{verbatim}
length = 10.
width = 5.
dx = dy = 1           # Resolution: Length of subdivisions on both axes

points, vertices, boundary = rectangular_cross(int(length/dx), int(width/dy),
                                               len1=length, len2=width)
\end{verbatim}}
which yields a mesh of length=10m, width=5m with 1m spacings. To increase the resolution, as we will later in this example, one merely decrease the values of dx and dy.

The rest of this script is as in the previous example.
% except for an application of the 'expression' form of \code{set\_quantity} where we use the value of \code{elevation} to define the (dry) initial condition for \code{stage}:
%{\small \begin{verbatim}
%  domain.set_quantity('stage', expression='elevation')
%\end{verbatim}}

\section{Model Output}

The following figure is a screenshot from the \anuga visualisation
tool \code{animate} of output from this example.
\begin{figure}[hbt]
  \centerline{\includegraphics[height=75mm]
    {graphics/channel1.png}}%

  \caption{Simple channel example viewed with the ANUGA viewer.}
  \label{fig:channel1}
\end{figure}


\subsection{Changing boundary conditions on the fly}
\label{sec:change boundary}

Here is the code for the second version of the channel flow \file{channel2.py}:
\verbatiminput{demos/channel2.py}
This example differs from the first version in that a constant outflow boundary condition has
been defined
{\small \begin{verbatim}
    Bo = Dirichlet_boundary([-5, 0, 0]) # Outflow
\end{verbatim}}
and that it is applied to the right hand side boundary when the water level there exceeds 0m.
{\small \begin{verbatim}
for t in domain.evolve(yieldstep = 0.2, finaltime = 40.0):
    domain.write_time()

    if domain.get_quantity('stage').get_values(interpolation_points=[[10, 2.5]]) > 0:
        print 'Stage > 0: Changing to outflow boundary'
        domain.set_boundary({'right': Bo})
\end{verbatim}}
\label{sec:change boundary code}

The if statement in the timestepping loop (evolve) gets the quantity
\code{stage} and obtain the interpolated value at the point (10m,
2.5m) which is on the right boundary. If the stage exceeds 0m a
message is printed and the old boundary condition at tag 'right' is
replaced by the outflow boundary using the method
{\small \begin{verbatim}
    domain.set_boundary({'right': Bo})
\end{verbatim}}
This type of dynamically varying boundary could for example be
used to model the
breakdown of a sluice door when water exceeds a certain level.

\subsection{Output}

The text output from this example looks like this
{\small \begin{verbatim}
...
Time = 15.4000, delta t in [0.03789902, 0.03789916], steps=6 (6)
Time = 15.6000, delta t in [0.03789896, 0.03789908], steps=6 (6)
Time = 15.8000, delta t in [0.03789891, 0.03789903], steps=6 (6)
Stage > 0: Changing to outflow boundary
Time = 16.0000, delta t in [0.02709050, 0.03789898], steps=6 (6)
Time = 16.2000, delta t in [0.03789892, 0.03789904], steps=6 (6)
...
\end{verbatim}}


\subsection{Flow through more complex topograhies}

Here is the code for the third version of the channel flow \file{channel3.py}:
\verbatiminput{demos/channel3.py}

This example differs from the first two versions in that the topography
contains obstacles.

This is accomplished here by defining the function \code{topography} as follows
{\small \begin{verbatim}
def topography(x,y):
    """Complex topography defined by a function of vectors x and y
    """

    z = -x/10

    N = len(x)
    for i in range(N):

        # Step
        if 10 < x[i] < 12:
            z[i] += 0.4 - 0.05*y[i]

        # Constriction
        if 27 < x[i] < 29 and y[i] > 3:
            z[i] += 2

        # Pole
        if (x[i] - 34)**2 + (y[i] - 2)**2 < 0.4**2:
            z[i] += 2

    return z
\end{verbatim}}

In addition, changing the resolution to dx=dy=0.1 creates a finer mesh resolving the new featurse better.

A screenshot of this model at time == 15s is
\begin{figure}[hbt]

  \centerline{\includegraphics[height=75mm]
    {graphics/channel3.png}}

  \caption{More complex flow in a channel}
  \label{fig:channel3}
\end{figure}




\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{An Example with Real Data}
\label{sec:realdataexample} The following discussion builds on the
concepts introduced through the \file{runup.py} example and
introduces a second example, \file{runcairns.py}.  This refers to
a {\bf hypothetical} scenario using real-life data,
in which the domain of interest surrounds the
Cairns region. Two scenarios are given; firstly, a
hypothetical tsunami wave is generated by a submarine mass failure
situated on the edge of the continental shelf, and secondly, a fixed wave
of given amplitude and period is introduced through the boundary.

{\bf
Each scenario has been designed to generate a tsunami which will
inundate the Cairns region. To achieve this, suitably large
parameters were chosen and were not based on any known tsunami sources
or realistic amplitudes.
}

\subsection{Overview}
As in the case of \file{runup.py}, the actions carried
out by the program can be organised according to this outline:

\begin{enumerate}

   \item Set up a triangular mesh.

   \item Set certain parameters governing the mode of
operation of the model---specifying, for instance, where to store the
model output.

   \item Input various quantities describing physical measurements, such
as the elevation, to be specified at each mesh point (vertex).

   \item Set up the boundary conditions.

   \item Carry out the evolution of the model through a series of time
steps and output the results, providing a results file that can be
visualised.

\end{enumerate}



\subsection{The Code}

Here is the code for \file{runcairns.py}:

\verbatiminput{demos/cairns/runcairns.py}

In discussing the details of this example, we follow the outline
given above, discussing each major step of the code in turn.

\subsection{Establishing the Mesh}\index{mesh, establishing}

One obvious way that the present example differs from
\file{runup.py} is in the use of a more complex method to
create the mesh. Instead of imposing a mesh structure on a
rectangular grid, the technique used for this example involves
building mesh structures inside polygons specified by the user,
using a mesh-generator.

In its simplest form, the mesh-generator creates the mesh within a single
polygon whose vertices are at geographical locations specified by
the user. The user specifies the \emph{resolution}---that is, the
maximal area of a triangle used for triangulation---and a triangular
mesh is created inside the polygon using a mesh generation engine.
On any given platform, the same mesh will be returned.
%Figure
%\ref{fig:pentagon} shows a simple example of this, in which the
%triangulation is carried out within a pentagon.


%\begin{figure}[hbt]

%  \caption{Mesh points are created inside the polygon}
  %\label{fig:pentagon}
%\end{figure}

Boundary tags are not restricted to \code{`left'}, \code{`bottom'},
\code{`right'} and \code{`top'}, as in the case of
\file{runup.py}. Instead the user specifies a list of
tags appropriate to the configuration being modelled.

In addition, the mesh-generator provides a way to adapt to geographic or
other features in the landscape, whose presence may require an
increase in resolution. This is done by allowing the user to specify
a number of \emph{interior polygons}, each with a specified
resolution. It is also
possible to specify one or more `holes'---that is, areas bounded by
polygons in which no triangulation is required.

%\begin{figure}[hbt]
%  \caption{Interior meshes with individual resolution}
%  \label{fig:interior meshes}
%\end{figure}

In its general form, the mesh-generator takes for its input a bounding
polygon and (optionally) a list of interior polygons. The user
specifies resolutions, both for the bounding polygon and for each of
the interior polygons. Given this data, the mesh-generator first creates a
triangular mesh with varying resolution.

The function used to implement this process is
\function{create\_mesh\_from\_regions}. Its arguments include the
bounding polygon and its resolution, a list of boundary tags, and a
list of pairs \code{[polygon, resolution]}, specifying the interior
polygons and their resolutions.

The resulting mesh is output to a \emph{mesh file}\index{mesh
file}\label{def:mesh file}. This term is used to describe a file of
a specific format used to store the data specifying a mesh. (There
are in fact two possible formats for such a file: it can either be a
binary file, with extension \code{.msh}, or an ASCII file, with
extension \code{.tsh}. In the present case, the binary file format
\code{.msh} is used. See Section \ref{sec:file formats} (page
\pageref{sec:file formats}) for more on file formats.)

In practice, the details of the polygons used are read from a
separate file \file{project.py}. Here is a complete listing of
\file{project.py}:

\verbatiminput{demos/cairns/project.py}

Figure \ref{fig:cairns3d} illustrates the landscape of the region
for the Cairns example. Understanding the landscape is important in
determining the location and resolution of interior polygons. The
supporting data is found in the ASCII grid, \code{cairns.asc}, which
has been sourced from the publically available Australian Bathymetry
and Topography Grid 2005, \cite{grid250}. The required resolution
for inundation modelling will depend on the underlying topography and
bathymetry; as the terrain becomes more complex, the desired resolution
would decrease to the order of tens of metres.

\begin{figure}[hbt]
\centerline{\includegraphics[scale=0.5]{graphics/cairns3.jpg}}
\caption{Landscape of the Cairns scenario.}
\label{fig:cairns3d}

\end{figure}
The following statements are used to read in the specific polygons
from \code{project.cairns} and assign a defined resolution to
each polygon.

{\small \begin{verbatim}
    islands_res = 100000
    cairns_res = 100000
    shallow_res = 500000
    interior_regions = [[project.poly_cairns, cairns_res],
                        [project.poly_island0, islands_res],
                        [project.poly_island1, islands_res],
                        [project.poly_island2, islands_res],
                        [project.poly_island3, islands_res],
                        [project.poly_shallow, shallow_res]]
\end{verbatim}}

Figure \ref{fig:cairnspolys}
illustrates the polygons used for the Cairns scenario.

\begin{figure}[hbt]

  \centerline{\includegraphics[scale=0.5]
      {graphics/cairnsmodel.jpg}}
  \caption{Interior and bounding polygons for the Cairns example.}
  \label{fig:cairnspolys}
\end{figure}

The statement


{\small \begin{verbatim}
remainder_res = 10000000
create_mesh_from_regions(project.bounding_polygon,
                         boundary_tags={'top': [0],
                                        'ocean_east': [1],
                                        'bottom': [2],
                                        'onshore': [3]},
                         maximum_triangle_area=remainder_res,
                         filename=meshname,
                         interior_regions=interior_regions,
                         use_cache=True,
                         verbose=True)
\end{verbatim}}
is then used to create the mesh, taking the bounding polygon to be
the polygon \code{bounding\_polygon} specified in \file{project.py}.
The argument \code{boundary\_tags} assigns a dictionary, whose keys
are the names of the boundary tags used for the bounding
polygon---\code{`top'}, \code{`ocean\_east'}, \code{`bottom'}, and
\code{`onshore'}--- and whose values identify the indices of the
segments associated with each of these tags.
The polygon may be arranged either clock-wise or counter clock-wise and the
indices refer to edges in the order they appear: Edge 0 connects vertex 0 and vertex 1, edge 1 connects vertex 1 and 2; and so forth.
(Here, the values associated with each boundary tag are one-element lists, but they can have as many indices as there are edges)
If polygons intersect, or edges coincide the resolution may be undefined in some regions.
Use the underlying mesh interface for such cases. See Section
\ref{sec:mesh interface}.

Note that every point on each polygon defining the mesh will be used as vertices in triangles.
Consequently, polygons with points very close together will cause triangles with very small
areas to be generated irrespective of the requested resolution.
Make sure points on polygons are spaced to be no closer than the smallest resolution requested.


\subsection{Initialising the Domain}

As with \file{runup.py}, once we have created the mesh, the next
step is to create the data structure \code{domain}. We did this for
\file{runup.py} by inputting lists of points and triangles and
specifying the boundary tags directly. However, in the present case,
we use a method that works directly with the mesh file
\code{meshname}, as follows:


{\small \begin{verbatim}
    domain = Domain(meshname, use_cache=True, verbose=True)
\end{verbatim}}

Providing a filename instead of the lists used in \file{runup.py}
above causes \code{Domain} to convert a mesh file \code{meshname}
into an instance of \code{Domain}, allowing us to use methods like
\method{set\_quantity} to set quantities and to apply other
operations.

%(In principle, the
%second argument of \function{pmesh\_to\_domain\_instance} can be any
%subclass of \class{Domain}, but for applications involving the
%shallow-water wave equation, the second argument of
%\function{pmesh\_to\_domain\_instance} can always be set simply to
%\class{Domain}.)

The following statements specify a basename and data directory, and
identify quantities to be stored. For the first two, values are
taken from \file{project.py}.

{\small \begin{verbatim}
    domain.set_name(project.basename)
    domain.set_datadir(project.outputdir)
    domain.set_quantities_to_be_stored(['stage', 'xmomentum',
        'ymomentum'])
\end{verbatim}}


\subsection{Initial Conditions}
Quantities for \file{runcairns.py} are set
using similar methods to those in \file{runup.py}. However,
in this case, many of the values are read from the auxiliary file
\file{project.py} or, in the case of \code{elevation}, from an
ancillary points file.



\subsubsection{Stage}

For the scenario we are modelling in this case, we use a callable
object \code{tsunami\_source}, assigned by means of a function
\function{slide\_tsunami}. This is similar to how we set elevation in
\file{runup.py} using a function---however, in this case the
function is both more complex and more interesting.

The function returns the water displacement for all \code{x} and
\code{y} in the domain. The water displacement is a double Gaussian
function that depends on the characteristics of the slide (length,
width, thickness, slope, etc), its location (origin) and the depth at that
location. For this example, we choose to apply the slide function
at a specified time into the simulation. {\bf Note, the parameters used
in this example have been deliberately chosen to generate a suitably
large amplitude tsunami which would inundate the Cairns region.}

\subsubsection{Friction}

We assign the friction exactly as we did for \file{runup.py}:

{\small \begin{verbatim}
    domain.set_quantity('friction', 0.0)
\end{verbatim}}


\subsubsection{Elevation}

The elevation is specified by reading data from a file:

{\small \begin{verbatim}
    domain.set_quantity('elevation',
                        filename = project.dem_name + '.pts',
                        use_cache = True,
                        verbose = True)
\end{verbatim}}

%However, before this step can be executed, some preliminary steps
%are needed to prepare the file from which the data is taken. Two
%source files are used for this data---their names are specified in
%the file \file{project.py}, in the variables \code{coarsedemname}
%and \code{finedemname}. They contain `coarse' and `fine' data,
%respectively---that is, data sampled at widely spaced points over a
%large region and data sampled at closely spaced points over a
%smaller subregion. The data in these files is combined through the
%statement

%{\small \begin{verbatim}
%combine_rectangular_points_files(project.finedemname + '.pts',
%                                 project.coarsedemname + '.pts',
%                                 project.combineddemname + '.pts')
%\end{verbatim}}
%The effect of this is simply to combine the datasets by eliminating
%any coarse data associated with points inside the smaller region
%common to both datasets. The name to be assigned to the resulting
%dataset is also derived from the name stored in the variable
%\code{combinedname} in the file \file{project.py}.

\subsection{Boundary Conditions}\index{boundary conditions}

Setting boundaries follows a similar pattern to the one used for
\file{runup.py}, except that in this case we need to associate a
boundary type with each of the
boundary tag names introduced when we established the mesh. In place of the four
boundary types introduced for \file{runup.py}, we use the reflective
boundary for each of the
eight tagged segments defined by \code{create_mesh_from_regions}:

{\small \begin{verbatim}
Bd = Dirichlet_boundary([0.0,0.0,0.0])
domain.set_boundary( {'ocean_east': Bd, 'bottom': Bd, 'onshore': Bd,
                          'top': Bd} )
\end{verbatim}}

\subsection{Evolution}

With the basics established, the running of the `evolve' step is
very similar to the corresponding step in \file{runup.py}. For the slide
scenario,
the simulation is run for 5000 seconds with the output stored every ten seconds.
For this example, we choose to apply the slide at 60 seconds into the simulation.

{\small \begin{verbatim}
    import time t0 = time.time()


    for t in domain.evolve(yieldstep = 10, finaltime = 60):
            domain.write_time()
            domain.write_boundary_statistics(tags = 'ocean_east')

        # add slide
        thisstagestep = domain.get_quantity('stage')
        if allclose(t, 60):
            slide = Quantity(domain)
            slide.set_values(tsunami_source)
            domain.set_quantity('stage', slide + thisstagestep)

        for t in domain.evolve(yieldstep = 10, finaltime = 5000,
                               skip_initial_step = True):
            domain.write_time()
        domain.write_boundary_statistics(tags = 'ocean_east')
\end{verbatim}}

For the fixed wave scenario, the simulation is run to 10000 seconds,
with the first half of the simulation stored at two minute intervals,
and the second half of the simulation stored at ten second intervals.
This functionality is especially convenient as it allows the detailed
parts of the simulation to be viewed at higher time resolution.


{\small \begin{verbatim}

# save every two mins leading up to wave approaching land
    for t in domain.evolve(yieldstep = 120, finaltime = 5000):
        domain.write_time()
        domain.write_boundary_statistics(tags = 'ocean_east')

    # save every 30 secs as wave starts inundating ashore
    for t in domain.evolve(yieldstep = 10, finaltime = 10000,
                           skip_initial_step = True):
        domain.write_time()
        domain.write_boundary_statistics(tags = 'ocean_east')

\end{verbatim}}

\section{Exploring the Model Output}

Now that the scenario has been run, the user can view the output in a number of ways.
As described earlier, the user may run animate to view a three-dimensional representation
of the simulation.

The user may also be interested in a maximum inundation map. This simply shows the
maximum water depth over the domain and is achieved with the function sww2dem (described in
Section \ref{sec:basicfileconversions}).
\file{ExportResults.py} demonstrates how this function can be used:

\verbatiminput{demos/cairns/ExportResults.py}

The script generates an maximum water depth ASCII grid at a defined
resolution (here 100 m$^2$) which can then be viewed in a GIS environment, for
example. The parameters used in the function are defined in \file{project.py}.
Figures \ref{fig:maxdepthcairnsslide} and \ref{fig:maxdepthcairnsfixedwave} show
the maximum water depth within the defined region for the slide and fixed wave scenario
respectively. {\bf Note, these inundation maps have been based on purely hypothetical
scenarios and were designed explicitly for demonstration purposes only.}
The user could develop a maximum absolute momentum or other expressions which can be
derived from the quantities.
It must be noted here that depth is more meaningful when the elevation is positive
(\code{depth} = \code{stage} $-$ \code{elevation}) as it describes the water height
above the available elevation. When the elevation is negative, depth is meauring the
water height from the sea floor. With this in mind, maximum inundation maps are
typically "clipped" to the coastline. However, the data input here did not contain a
coastline.

\begin{figure}[hbt]
\centerline{\includegraphics[scale=0.5]{graphics/slidedepth.jpg}}
\caption{Maximum inundation map for the Cairns slide scenario. \bf Note, this
inundaiton map has been based on a purely hypothetical scenario which was
designed explictiy for demonstration purposes only.}
\label{fig:maxdepthcairnsslide}
\end{figure}

\begin{figure}[hbt]
\centerline{\includegraphics[scale=0.5]{graphics/fixedwavedepth.jpg}}
\caption{Maximum inundation map for the Cairns fixed wave scenario.
\bf Note, this
inundaiton map has been based on a purely hypothetical scenario which was
designed explictiy for demonstration purposes only.}
\label{fig:maxdepthcairnsfixedwave}
\end{figure}

The user may also be interested in interrogating the solution at a particular spatial
location to understand the behaviour of the system through time. To do this, the user
must first define the locations of interest. A number of locations have been
identified for the Cairns scenario, as shown in Figure \ref{fig:cairnsgauges}.

\begin{figure}[hbt]
\centerline{\includegraphics[scale=0.5]{graphics/cairnsgauges.jpg}}
\caption{Point locations to show time series information for the Cairns scenario.}
\label{fig:cairnsgauges}
\end{figure}

These locations
must be stored in either a .csv or .txt file. The corresponding .csv file for
the gauges shown in Figure \ref{fig:cairnsgauges} is \file{gauges.csv}

\verbatiminput{demos/cairns/gauges.csv}

Header information has been included to identify the location in terms of eastings and
northings, and each gauge is given a name. The elevation column can be zero here.
This information is then passed to the function \code{sww2csv_gauges} (shown in
\file{GetTimeseries.py} which generates the csv files for each point location. The csv files
can then be used in \code{csv2timeseries_graphs} to create the timeseries plot for each desired
quantity. \code{csv2timeseries_graphs} relies on \code{pylab} to be installed which is not part
of the standard \code{anuga} release, however it can be downloaded and installed from \code{http://matplotlib.sourceforge.net/}

\verbatiminput{demos/cairns/GetTimeseries.py}

Here, the time series for the quantities stage, depth and speed will be generated for
each gauge defined in the gauge file. As described earlier, depth is more meaningful
for onshore gauges, and stage is more appropriate for offshore gauges.

As an example output,
Figure \ref{fig:reef} shows the time series for the quantity stage for the
Elford Reef location for each scenario (the elevation at this location is negative,
therefore stage is the more appropriate quantity to plot). Note the large negative stage value when the slide was
introduced. This is due to the double gaussian form of the initial surface
displacement of the slide. By contrast, the time series for depth is shown for the onshore location of the Cairns
Airport in Figure \ref{fig:airportboth}.

\begin{figure}[hbt]
\centerline{\includegraphics[scale=0.5]{graphics/gaugeElfordReefstage.png}}
\caption{Time series information of the quantity stage for the Elford Reef location for the
fixed wave and slide scenario.}
\label{fig:reef}
\end{figure}

\begin{figure}[hbt]
\centerline{\includegraphics[scale=0.5]{graphics/gaugeCairnsAirportdepth.png}}
\caption{Time series information of the quantity depth for the Cairns Airport
location for the slide and fixed wave scenario.}
\label{fig:airportboth}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{\anuga Public Interface}
\label{ch:interface}

This chapter gives an overview of the features of \anuga available
to the user at the public interface. These are grouped under the
following headings, which correspond to the outline of the examples
described in Chapter \ref{ch:getstarted}:

\begin{itemize}
    \item Establishing the Mesh
    \item Initialising the Domain
    \item Specifying the Quantities
    \item Initial Conditions
    \item Boundary Conditions
    \item Forcing Functions
    \item Evolution
\end{itemize}

The listings are intended merely to give the reader an idea of what
each feature is, where to find it and how it can be used---they do
not give full specifications; for these the reader
may consult the code. The code for every function or class contains
a documentation string, or `docstring', that specifies the precise
syntax for its use. This appears immediately after the line
introducing the code, between two sets of triple quotes.

Each listing also describes the location of the module in which
the code for the feature being described can be found. All modules
are in the folder \file{inundation} or one of its subfolders, and the
location of each module is described relative to \file{inundation}. Rather
than using pathnames, whose syntax depends on the operating system,
we use the format adopted for importing the function or class for
use in Python code. For example, suppose we wish to specify that the
function \function{create\_mesh\_from\_regions} is in a module called
\module{mesh\_interface} in a subfolder of \module{inundation} called
\code{pmesh}. In Linux or Unix syntax, the pathname of the file
containing the function, relative to \file{inundation}, would be

\begin{center}
%    \code{pmesh/mesh\_interface.py}
    \code{pmesh}$\slash$\code{mesh\_interface.py}
\end{center}
\label{sec:mesh interface}

while in Windows syntax it would be

\begin{center}
    \code{pmesh}$\backslash$\code{mesh\_interface.py}
\end{center}

Rather than using either of these forms, in this chapter we specify
the location simply as \code{pmesh.mesh\_interface}, in keeping with
the usage in the Python statement for importing the function,
namely:
\begin{center}
    \code{from pmesh.mesh\_interface import create\_mesh\_from\_regions}
\end{center}

Each listing details the full set of parameters for the class or
function; however, the description is generally limited to the most
important parameters and the reader is again referred to the code
for more details.

The following parameters are common to many functions and classes
and are omitted from the descriptions given below:

%\begin{center}
\begin{tabular}{ll}  %\hline
%\textbf{Name } & \textbf{Description}\\
%\hline
\emph{use\_cache} & Specifies whether caching is to be used for improved performance. See Section \ref{sec:caching} for details on the underlying caching functionality\\
\emph{verbose} & If \code{True}, provides detailed terminal output
to the user\\  % \hline
\end{tabular}
%\end{center}

\section{Mesh Generation}

Before discussing the part of the interface relating to mesh
generation, we begin with a description of a simple example of a
mesh and use it to describe how mesh data is stored.

\label{sec:meshexample} Figure \ref{fig:simplemesh} represents a
very simple mesh comprising just 11 points and 10 triangles.


\begin{figure}[h]
  \begin{center}
    \includegraphics[width=90mm, height=90mm]{triangularmesh.jpg}
  \end{center}

  \caption{A simple mesh}
  \label{fig:simplemesh}
\end{figure}


The variables \code{points}, \code{vertices} and \code{boundary}
represent the data displayed in Figure \ref{fig:simplemesh} as
follows. The list \code{points} stores the coordinates of the
points, and may be displayed schematically as in Table
\ref{tab:points}.


\begin{table}
  \begin{center}
    \begin{tabular}[t]{|c|cc|} \hline
      index & \code{x} & \code{y}\\  \hline
      0 & 1 & 1\\
      1 & 4 & 2\\
      2 & 8 & 1\\
      3 & 1 & 3\\
      4 & 5 & 5\\
      5 & 8 & 6\\
      6 & 11 & 5\\
      7 & 3 & 6\\
      8 & 1 & 8\\
      9 & 4 & 9\\
      10 & 10 & 7\\  \hline
    \end{tabular}
  \end{center}

  \caption{Point coordinates for mesh in
    Figure \protect \ref{fig:simplemesh}}
  \label{tab:points}
\end{table}

The list \code{vertices} specifies the triangles that make up the
mesh. It does this by specifying, for each triangle, the indices
(the numbers shown in the first column above) that correspond to the
three points at its vertices, taken in an anti-clockwise order
around the triangle. Thus, in the example shown in Figure
\ref{fig:simplemesh}, the variable \code{vertices} contains the
entries shown in Table \ref{tab:vertices}. The starting point is
arbitrary so triangle $(0,1,3)$ is considered the same as $(1,3,0)$
and $(3,0,1)$.


\begin{table}
  \begin{center}
    \begin{tabular}{|c|ccc|} \hline
      index & \multicolumn{3}{c|}{\code{vertices}}\\ \hline
      0 & 0 & 1 & 3\\
      1 & 1 & 2 & 4\\
      2 & 2 & 5 & 4\\
      3 & 2 & 6 & 5\\
      4 & 4 & 5 & 9\\
      5 & 4 & 9 & 7\\
      6 & 3 & 4 & 7\\
      7 & 7 & 9 & 8\\
      8 & 1 & 4 & 3\\
      9 & 5 & 10 & 9\\  \hline
    \end{tabular}
  \end{center}

  \caption{Vertices for mesh in Figure \protect \ref{fig:simplemesh}}
  \label{tab:vertices}
\end{table}

Finally, the variable \code{boundary} identifies the boundary
triangles and associates a tag with each.

\refmodindex[pmesh.meshinterface]{pmesh.mesh\_interface}\label{sec:meshgeneration}

\begin{funcdesc}  {create\_mesh\_from\_regions}{bounding_polygon,
                             boundary_tags,
                             maximum_triangle_area,
                             filename=None,
                             interior_regions=None,
                             poly_geo_reference=None,
                             mesh_geo_reference=None,
                             minimum_triangle_angle=28.0}
Module: \module{pmesh.mesh\_interface}

This function allows a user to initiate the automatic creation of a
mesh inside a specified polygon (input \code{bounding_polygon}).
Among the parameters that can be set are the \emph{resolution}
(maximal area for any triangle in the mesh) and the minimal angle
allowable in any triangle. The user can specify a number of internal
polygons within each of which the resolution of the mesh can be
specified. \code{interior_regions} is a paired list containing the
interior polygon and its resolution.  Additionally, the user specifies
a list of boundary tags, one for each edge of the bounding polygon.

\textbf{WARNING}. Note that the dictionary structure used for the
parameter \code{boundary\_tags} is different from that used for the
variable \code{boundary} that occurs in the specification of a mesh.
In the case of \code{boundary}, the tags are the \emph{values} of
the dictionary, whereas in the case of \code{boundary_tags}, the
tags are the \emph{keys} and the \emph{value} corresponding to a
particular tag is a list of numbers identifying boundary edges
labelled with that tag. Because of this, it is theoretically
possible to assign the same edge to more than one tag. However, an
attempt to do this will cause an error.

\textbf{WARNING}. Do not have polygon lines cross or be on-top of each
    other. This can result in regions of unspecified resolutions. Do
    not have polygon close to each other. This can result in the area
    between the polygons having small triangles.  For more control
    over the mesh outline use the methods described below.

\end{funcdesc}



\subsection{Advanced mesh generation}

For more control over the creation of the mesh outline, use the
methods of the class \class{Mesh}.


\begin{classdesc}  {Mesh}{userSegments=None,
                 userVertices=None,
                 holes=None,
                 regions=None}
Module: \module{pmesh.mesh}

A class used to build a mesh outline and generate a two-dimensional
triangular mesh. The mesh outline is used to describe features on the
mesh, such as the mesh boundary. Many of this classes methods are used
to build a mesh outline, such as \code{add\_vertices} and
\code{add\_region\_from\_polygon}.

\end{classdesc}


\subsubsection{Key Methods of Class Mesh}


\begin{methoddesc} {add\_hole}{x,y}
Module: \module{pmesh.mesh},  Class: \class{Mesh}

This method is used to build the mesh outline.  It defines a hole,
when the boundary of the hole has already been defined, by selecting a
point within the boundary.

\end{methoddesc}


\begin{methoddesc}  {add\_hole\_from\_polygon}{self, polygon, tags=None}
Module: \module{pmesh.mesh},  Class: \class{Mesh}

This method is used to add a `hole' within a region ---that is, to
define a interior region where the triangular mesh will not be
generated---to a \class{Mesh} instance. The region boundary is described by
the polygon passed in.  Additionally, the user specifies a list of
boundary tags, one for each edge of the bounding polygon.
\end{methoddesc}


\begin{methoddesc}  {add\_points_and_segments}{self, points, segments,
    segment\_tags=None}
Module: \module{pmesh.mesh},  Class: \class{Mesh}

This method is used to build the mesh outline. It adds points and
segments connecting the points.  A tag for each segment can optionally
be added.

\end{methoddesc}

\begin{methoddesc} {add\_region}{x,y}
Module: \module{pmesh.mesh},  Class: \class{Mesh}

This method is used to build the mesh outline.  It defines a region,
when the boundary of the region has already been defined, by selecting
a point within the boundary.  A region instance is returned.  This can
be used to set the resolution.

\end{methoddesc}

\begin{methoddesc}  {add\_region\_from\_polygon}{self, polygon,
segment_tags=None, region_tag=None
                                max_triangle_area=None}
Module: \module{pmesh.mesh},  Class: \class{Mesh}

This method is used to build the mesh outline.  It adds a region to a
\class{Mesh} instance.  Regions are commonly used to describe an area
with an increased density of triangles, by setting
\code{max_triangle_area}.  The
region boundary is described by the input \code{polygon}.  Additionally, the
user specifies a list of segment tags, one for each edge of the
bounding polygon.  The regional tag is set using  \code{region}.

\end{methoddesc}





\begin{methoddesc} {add\_vertices}{point_data}
Module: \module{pmesh.mesh},  Class: \class{Mesh}

Add user vertices. The point_data can be a list of (x,y) values, a numeric
array or a geospatial_data instance.
\end{methoddesc}

\begin{methoddesc} {auto\_segment}{raw_boundary=raw_boundary,
                    remove_holes=remove_holes,
                    smooth_indents=smooth_indents,
                    expand_pinch=expand_pinch}
Module: \module{pmesh.mesh},  Class: \class{Mesh}

Add segments between some of the user vertices to give the vertices an
outline.  The outline is an alpha shape. This method is
useful since a set of user vertices need to be outlined by segments
before generate_mesh is called.

\end{methoddesc}

\begin{methoddesc}  {export\_mesh_file}{self,ofile}
Module: \module{pmesh.mesh},  Class: \class{Mesh}

This method is used to save the mesh to a file. \code{ofile} is the
name of the mesh file to be written, including the extension.  Use
the extension \code{.msh} for the file to be in NetCDF format and
\code{.tsh} for the file to be ASCII format.
\end{methoddesc}

\begin{methoddesc}  {generate\_mesh}{self,
                      maximum_triangle_area=None,
                      minimum_triangle_angle=28.0,
                      verbose=False}
Module: \module{pmesh.mesh},  Class: \class{Mesh}

This method is used to generate the triangular mesh.  The  maximal
area of any triangle in the mesh can be specified, which is used to
control the triangle density, along with the
minimum angle in any triangle.
\end{methoddesc}



\begin{methoddesc}  {import_ungenerate_file}{self,ofile, tag=None}
Module: \module{pmesh.mesh},  Class: \class{Mesh}

This method is used to import a polygon file in the ungenerate
format, which is used by arcGIS. The polygons from the file are converted to
vertices and segments. \code{ofile} is the name of the polygon file.
\code{tag} is the tag given to all the polygon's segments.

This function can be used to import building footprints.
\end{methoddesc}

%%%%%%
\section{Initialising the Domain}

%Include description of the class Domain and the module domain.

%FIXME (Ole): This is also defined in a later chapter
%\declaremodule{standard}{...domain}

\begin{classdesc} {Domain} {source=None,
                 triangles=None,
                 boundary=None,
                 conserved_quantities=None,
                 other_quantities=None,
                 tagged_elements=None,
                 use_inscribed_circle=False,
                 mesh_filename=None,
                 use_cache=False,
                 verbose=False,
                 full_send_dict=None,
                 ghost_recv_dict=None,
                 processor=0,
                 numproc=1}
Module: \refmodule{abstract_2d_finite_volumes.domain}

This class is used to create an instance of a data structure used to
store and manipulate data associated with a mesh. The mesh is
specified either by assigning the name of a mesh file to
\code{source} or by specifying the points, triangle and boundary of the
mesh.
\end{classdesc}

\subsection{Key Methods of Domain}

\begin{methoddesc} {set\_name}{name}
    Module: \refmodule{abstract\_2d\_finite\_volumes.domain},
    page \pageref{mod:domain}

    Assigns the name \code{name} to the domain.
\end{methoddesc}

\begin{methoddesc} {get\_name}{}
    Module: \module{abstract\_2d\_finite\_volumes.domain}

    Returns the name assigned to the domain by \code{set\_name}. If no name has been
    assigned, returns \code{`domain'}.
\end{methoddesc}

\begin{methoddesc} {set\_datadir}{name}
    Module: \module{abstract\_2d\_finite\_volumes.domain}

    Specifies the directory used for SWW files, assigning it to the
    pathname \code{name}. The default value, before
    \code{set\_datadir} has been run, is the value \code{default\_datadir}
    specified in \code{config.py}.

    Since different operating systems use different formats for specifying pathnames,
    it is necessary to specify path separators using the Python code \code{os.sep}, rather than
    the operating-specific ones such as `$\slash$' or `$\backslash$'.
    For this to work you will need to include the statement \code{import os}
    in your code, before the first appearance of \code{set\_datadir}.

    For example, to set the data directory to a subdirectory
    \code{data} of the directory \code{project}, you could use
    the statements:

    {\small \begin{verbatim}
        import os
        domain.set_datadir{'project' + os.sep + 'data'}
    \end{verbatim}}
\end{methoddesc}

\begin{methoddesc} {get\_datadir}{}
    Module: \module{abstract\_2d\_finite\_volumes.domain}

    Returns the data directory set by \code{set\_datadir} or,
    if \code{set\_datadir} has not
    been run, returns the value \code{default\_datadir} specified in
    \code{config.py}.
\end{methoddesc}


\begin{methoddesc} {set\_minimum_allowed_height}{}
    Module: \module{shallow\_water.shallow\_water\_domain}

    Set the minimum depth (in meters) that will be recognised in
    the numerical scheme (including limiters and flux computations)

    Default value is $10^{-3}$ m, but by setting this to a greater value,
    e.g.\ for large scale simulations, the computation time can be
    significantly reduced.
\end{methoddesc}


\begin{methoddesc} {set\_minimum_storable_height}{}
    Module: \module{shallow\_water.shallow\_water\_domain}

    Sets the minimum depth that will be recognised when writing
    to an sww file. This is useful for removing thin water layers
    that seems to be caused by friction creep.
\end{methoddesc}


\begin{methoddesc} {set\_maximum_allowed_speed}{}
    Module: \module{shallow\_water.shallow\_water\_domain}

    Set the maximum particle speed that is allowed in water
    shallower than minimum_allowed_height. This is useful for
    controlling speeds in very thin layers of water and at the same time
    allow some movement avoiding pooling of water.
\end{methoddesc}


\begin{methoddesc} {set\_time}{time=0.0}
    Module: \module{abstract\_2d\_finite\_volumes.domain}

    Sets the initial time, in seconds, for the simulation. The
    default is 0.0.
\end{methoddesc}

\begin{methoddesc} {set\_default\_order}{n}
    Sets the default (spatial) order to the value specified by
    \code{n}, which must be either 1 or 2. (Assigning any other value
    to \code{n} will cause an error.)
\end{methoddesc}


\begin{methoddesc} {set\_store\_vertices\_uniquely}{flag}
Decide whether vertex values should be stored uniquely as
computed in the model or whether they should be reduced to one
value per vertex using averaging.

Triangles stored in the sww file can be discontinuous reflecting
the internal representation of the finite-volume scheme
(this is a feature allowing for arbitrary steepness).
However, for visual purposes and also for use with \code{Field\_boundary}
(and \code{File\_boundary}) it is often desirable to store triangles
with values at each vertex point as the average of the potentially
discontinuous numbers found at vertices of different triangles sharing the
same vertex location.

Storing one way or the other is controlled in ANUGA through the method
\code{domain.store\_vertices\_uniquely}. Options are
\begin{itemize}
  \item \code{domain.store\_vertices\_uniquely(True)}: Allow discontinuities in the sww file
  \item \code{domain.store\_vertices\_uniquely(False)}: (Default).
  Average values
  to ensure continuity in sww file. The latter also makes for smaller
  sww files.
\end{itemize}

\end{methoddesc}


% Structural methods
\begin{methoddesc}{get\_nodes}{absolute=False}
    Return x,y coordinates of all nodes in mesh.

    The nodes are ordered in an Nx2 array where N is the number of nodes.
    This is the same format they were provided in the constructor
    i.e. without any duplication.

    Boolean keyword argument absolute determines whether coordinates
    are to be made absolute by taking georeference into account
    Default is False as many parts of ANUGA expects relative coordinates.
\end{methoddesc}


\begin{methoddesc}{get\_vertex_coordinates}{absolute=False}

    Return vertex coordinates for all triangles.

    Return all vertex coordinates for all triangles as a 3*M x 2 array
    where the jth vertex of the ith triangle is located in row 3*i+j and
    M the number of triangles in the mesh.

    Boolean keyword argument absolute determines whether coordinates
    are to be made absolute by taking georeference into account
    Default is False as many parts of ANUGA expects relative coordinates.
\end{methoddesc}


\begin{methoddesc}{get\_triangles}{indices=None}

        Return Mx3 integer array where M is the number of triangles.
        Each row corresponds to one triangle and the three entries are
        indices into the mesh nodes which can be obtained using the method
        get\_nodes()

        Optional argument, indices is the set of triangle ids of interest.
\end{methoddesc}

\begin{methoddesc}{get\_disconnected\_triangles}{}

Get mesh based on nodes obtained from get_vertex_coordinates.

        Return array Mx3 array of integers where each row corresponds to
        a triangle. A triangle is a triplet of indices into
        point coordinates obtained from get_vertex_coordinates and each
        index appears only once.\\

        This provides a mesh where no triangles share nodes
        (hence the name disconnected triangles) and different
        nodes may have the same coordinates.\\

        This version of the mesh is useful for storing meshes with
        discontinuities at each node and is e.g. used for storing
        data in sww files.\\

        The triangles created will have the format

    {\small \begin{verbatim}
        [[0,1,2],
         [3,4,5],
         [6,7,8],
         ...
         [3*M-3 3*M-2 3*M-1]]
     \end{verbatim}}
\end{methoddesc}



%%%%%%
\section{Initial Conditions}
\label{sec:Initial Conditions}
In standard usage of partial differential equations, initial conditions
refers to the values associated to the system variables (the conserved
quantities here) for \code{time = 0}. In setting up a scenario script
as described in Sections \ref{sec:simpleexample} and \ref{sec:realdataexample},
\code{set_quantity} is used to define the initial conditions of variables
other than the conserved quantities, such as friction. Here, we use the terminology
of initial conditions to refer to initial values for variables which need
prescription to solve the shallow water wave equation. Further, it must be noted
that \code{set_quantity} does not necessarily have to be used in the initial
condition setting; it can be used at any time throughout the simulation.

\begin{methoddesc}{set\_quantity}{name,
    numeric = None,
    quantity = None,
    function = None,
    geospatial_data = None,
    filename = None,
    attribute_name = None,
    alpha = None,
    location = 'vertices',
    indices = None,
    verbose = False,
    use_cache = False}
  Module: \module{abstract\_2d\_finite\_volumes.domain}
  (see also \module{abstract\_2d\_finite\_volumes.quantity.set\_values})

This function is used to assign values to individual quantities for a
domain. It is very flexible and can be used with many data types: a
statement of the form \code{domain.set\_quantity(name, x)} can be used
to define a quantity having the name \code{name}, where the other
argument \code{x} can be any of the following:

\begin{itemize}
\item a number, in which case all vertices in the mesh gets that for
the quantity in question.
\item a list of numbers or a Numeric array ordered the same way as the mesh vertices.
\item a function (e.g.\ see the samples introduced in Chapter 2)
\item an expression composed of other quantities and numbers, arrays, lists (for
example, a linear combination of quantities, such as
\code{domain.set\_quantity('stage','elevation'+x))}
\item the name of a file from which the data can be read. In this case, the optional argument attribute\_name will select which attribute to use from the file. If left out, set\_quantity will pick one. This is useful in cases where there is only one attribute.
\item a geospatial dataset (See Section \ref{sec:geospatial}).
Optional argument attribute\_name applies here as with files.
\end{itemize}


Exactly one of the arguments
  numeric, quantity, function, points, filename
must be present.


Set quantity will look at the type of the second argument (\code{numeric}) and
determine what action to take.

Values can also be set using the appropriate keyword arguments.
If x is a function, for example, \code{domain.set\_quantity(name, x)}, \code{domain.set\_quantity(name, numeric=x)}, and \code{domain.set\_quantity(name, function=x)}
are all equivalent.


Other optional arguments are
\begin{itemize}
\item \code{indices} which is a list of ids of triangles to which set\_quantity should apply its assignment of values.
\item \code{location} determines which part of the triangles to assign
  to. Options are 'vertices' (default), 'edges', 'unique vertices', and 'centroids'.
\end{itemize}

%%%
\anuga provides a number of predefined initial conditions to be used
with \code{set\_quantity}. See for example callable object
\code{slump\_tsunami} below.

\end{methoddesc}




\begin{funcdesc}{set_region}{tag, quantity, X, location='vertices'}
  Module: \module{abstract\_2d\_finite\_volumes.domain}

  (see also \module{abstract\_2d\_finite\_volumes.quantity.set\_values})

This function is used to assign values to individual quantities given
a regional tag.   It is similar to \code{set\_quantity}.
For example, if in the mesh-generator a regional tag of 'ditch' was
used, set\_region can be used to set elevation of this region to
-10m. X is the constant or function to be applied to the quantity,
over the tagged region.  Location describes how the values will be
applied.  Options are 'vertices' (default), 'edges', 'unique
vertices', and 'centroids'.

This method can also be called with a list of region objects.  This is
useful for adding quantities in regions, and having one quantity
value based on another quantity. See  \module{abstract\_2d\_finite\_volumes.region} for
more details.
\end{funcdesc}




\begin{funcdesc}{slump_tsunami}{length, depth, slope, width=None, thickness=None,
                x0=0.0, y0=0.0, alpha=0.0,
                gravity=9.8, gamma=1.85,
                massco=1, dragco=1, frictionco=0, psi=0,
                dx=None, kappa=3.0, kappad=0.8, zsmall=0.01,
                domain=None,
                verbose=False}
Module: \module{shallow\_water.smf}

This function returns a callable object representing an initial water
displacement generated by a submarine sediment failure. These failures can take the form of
a submarine slump or slide. In the case of a slide, use \code{slide_tsunami} instead.

The arguments include as a minimum, the slump or slide length, the water depth to the centre of sediment
mass, and the bathymetric slope. Other slump or slide parameters can be included if they are known.
\end{funcdesc}


%%%
\begin{funcdesc}{file\_function}{filename,
    domain = None,
    quantities = None,
    interpolation_points = None,
    verbose = False,
    use_cache = False}
Module: \module{abstract\_2d\_finite\_volumes.util}

Reads the time history of spatial data for
specified interpolation points from a NetCDF file (\code{filename})
and returns
a callable object. \code{filename} could be a \code{sww} file.
Returns interpolated values based on the input
file using the underlying \code{interpolation\_function}.

\code{quantities} is either the name of a single quantity to be
interpolated or a list of such quantity names. In the second case, the resulting
function will return a tuple of values---one for each quantity.

\code{interpolation\_points} is a list of absolute coordinates or a
geospatial object
for points at which values are sought.

The model time stored within the file function can be accessed using
the method \code{f.get\_time()}


The underlying algorithm used is as follows:\\
Given a time series (i.e.\ a series of values associated with
different times), whose values are either just numbers or a set of
 numbers defined at the vertices of a triangular mesh (such as those
 stored in SWW files), \code{Interpolation\_function} is used to
 create a callable object that interpolates a value for an arbitrary
 time \code{t} within the model limits and possibly a point \code{(x,
 y)} within a mesh region.

 The actual time series at which data is available is specified by
 means of an array \code{time} of monotonically increasing times. The
 quantities containing the values to be interpolated are specified in
 an array---or dictionary of arrays (used in conjunction with the
 optional argument \code{quantity\_names}) --- called
 \code{quantities}. The optional arguments \code{vertex\_coordinates}
 and \code{triangles} represent the spatial mesh associated with the
 quantity arrays. If omitted the function created by
 \code{Interpolation\_function} will be a function of \code{t} only.

 Since, in practice, values need to be computed at specified points,
 the syntax allows the user to specify, once and for all, a list
 \code{interpolation\_points} of points at which values are required.
 In this case, the function may be called using the form \code{f(t,
 id)}, where \code{id} is an index for the list
 \code{interpolation\_points}.


\end{funcdesc}

%%%
%% \begin{classdesc}{Interpolation\_function}{self,
%%     time,
%%     quantities,
%%     quantity_names = None,
%%     vertex_coordinates = None,
%%     triangles = None,
%%     interpolation_points = None,
%%     verbose = False}
%% Module: \module{abstract\_2d\_finite\_volumes.least\_squares}

%% Given a time series (i.e.\ a series of values associated with
%% different times), whose values are either just numbers or a set of
%% numbers defined at the vertices of a triangular mesh (such as those
%% stored in SWW files), \code{Interpolation\_function} is used to
%% create a callable object that interpolates a value for an arbitrary
%% time \code{t} within the model limits and possibly a point \code{(x,
%% y)} within a mesh region.

%% The actual time series at which data is available is specified by
%% means of an array \code{time} of monotonically increasing times. The
%% quantities containing the values to be interpolated are specified in
%% an array---or dictionary of arrays (used in conjunction with the
%% optional argument \code{quantity\_names}) --- called
%% \code{quantities}. The optional arguments \code{vertex\_coordinates}
%% and \code{triangles} represent the spatial mesh associated with the
%% quantity arrays. If omitted the function created by
%% \code{Interpolation\_function} will be a function of \code{t} only.

%% Since, in practice, values need to be computed at specified points,
%% the syntax allows the user to specify, once and for all, a list
%% \code{interpolation\_points} of points at which values are required.
%% In this case, the function may be called using the form \code{f(t,
%% id)}, where \code{id} is an index for the list
%% \code{interpolation\_points}.

%% \end{classdesc}

%%%
%\begin{funcdesc}{set\_region}{functions}
%[Low priority. Will be merged into set\_quantity]

%Module:\module{abstract\_2d\_finite\_volumes.domain}
%\end{funcdesc}



%%%%%%
\section{Boundary Conditions}\index{boundary conditions}

\anuga provides a large number of predefined boundary conditions,
represented by objects such as \code{Reflective\_boundary(domain)} and
\code{Dirichlet\_boundary([0.2, 0.0, 0.0])}, described in the examples
in Chapter 2. Alternatively, you may prefer to ``roll your own'',
following the method explained in Section \ref{sec:roll your own}.

These boundary objects may be used with the function \code{set\_boundary} described below
to assign boundary conditions according to the tags used to label boundary segments.

\begin{methoddesc}{set\_boundary}{boundary_map}
Module: \module{abstract\_2d\_finite\_volumes.domain}

This function allows you to assign a boundary object (corresponding to a
pre-defined or user-specified boundary condition) to every boundary segment that
has been assigned a particular tag.

This is done by specifying a dictionary \code{boundary\_map}, whose values are the boundary objects
and whose keys are the symbolic tags.

\end{methoddesc}

\begin{methoddesc} {get\_boundary\_tags}{}
Module: \module{abstract\_2d\_finite\_volumes.domain}

Returns a list of the available boundary tags.
\end{methoddesc}

%%%
\subsection{Predefined boundary conditions}

\begin{classdesc}{Reflective\_boundary}{Boundary}
Module: \module{shallow\_water}

Reflective boundary returns same conserved quantities as those present in
the neighbouring volume but reflected.

This class is specific to the shallow water equation as it works with the
momentum quantities assumed to be the second and third conserved quantities.
\end{classdesc}

%%%
\begin{classdesc}{Transmissive\_boundary}{domain = None}
Module: \module{abstract\_2d\_finite\_volumes.generic\_boundary\_conditions}

A transmissive boundary returns the same conserved quantities as
those present in the neighbouring volume.

The underlying domain must be specified when the boundary is instantiated.
\end{classdesc}

%%%
\begin{classdesc}{Dirichlet\_boundary}{conserved_quantities=None}
Module: \module{abstract\_2d\_finite\_volumes.generic\_boundary\_conditions}

A Dirichlet boundary returns constant values for each of conserved
quantities. In the example of \code{Dirichlet\_boundary([0.2, 0.0, 0.0])},
the \code{stage} value at the boundary is 0.2 and the \code{xmomentum} and
\code{ymomentum} at the boundary are set to 0.0. The list must contain
a value for each conserved quantity.
\end{classdesc}

%%%
\begin{classdesc}{Time\_boundary}{domain = None, f = None}
Module: \module{abstract\_2d\_finite\_volumes.generic\_boundary\_conditions}

A time-dependent boundary returns values for the conserved
quantities as a function \code{f(t)} of time. The user must specify
the domain to get access to the model time.
\end{classdesc}

%%%
\begin{classdesc}{File\_boundary}{Boundary}
Module: \module{abstract\_2d\_finite\_volumes.generic\_boundary\_conditions}

This method may be used if the user wishes to apply a SWW file or
a time series file to a boundary segment or segments.
The boundary values are obtained from a file and interpolated to the
appropriate segments for each conserved quantity.
\end{classdesc}



%%%
\begin{classdesc}{Transmissive\_Momentum\_Set\_Stage\_boundary}{Boundary}
Module: \module{shallow\_water}

This boundary returns same momentum conserved quantities as
those present in its neighbour volume but sets stage as in a Time\_boundary.
The underlying domain must be specified when boundary is instantiated

This type of boundary is useful when stage is known at the boundary as a
function of time, but momenta (or speeds) aren't.

This class is specific to the shallow water equation as it works with the
momentum quantities assumed to be the second and third conserved quantities.
\end{classdesc}


\begin{classdesc}{Dirichlet\_Discharge\_boundary}{Boundary}
Module: \module{shallow\_water}

Sets stage (stage0)
Sets momentum (wh0) in the inward normal direction.
\end{classdesc}



\subsection{User-defined boundary conditions}
\label{sec:roll your own}

All boundary classes must inherit from the generic boundary class
\code{Boundary} and have a method called \code{evaluate} which must
take as inputs \code{self, vol\_id, edge\_id} where self refers to the
object itself and vol\_id and edge\_id are integers referring to
particular edges. The method must return a list of three floating point
numbers representing values for \code{stage},
\code{xmomentum} and \code{ymomentum}, respectively.

The constructor of a particular boundary class may be used to specify
particular values or flags to be used by the \code{evaluate} method.
Please refer to the source code for the existing boundary conditions
for examples of how to implement boundary conditions.



%\section{Forcing Functions}
%
%\anuga provides a number of predefined forcing functions to be used with .....




\section{Evolution}\index{evolution}

  \begin{methoddesc}{evolve}{yieldstep = None, finaltime = None, duration = None, skip_initial_step = False}

  Module: \module{abstract\_2d\_finite\_volumes.domain}

  This function (a method of \class{domain}) is invoked once all the
  preliminaries have been completed, and causes the model to progress
  through successive steps in its evolution, storing results and
  outputting statistics whenever a user-specified period
  \code{yieldstep} is completed (generally during this period the
  model will evolve through several steps internally
  as the method forces the water speed to be calculated
  on successive new cells). The user
  specifies the total time period over which the evolution is to take
  place, by specifying values (in seconds) for either \code{duration}
  or \code{finaltime}, as well as the interval in seconds after which
  results are to be stored and statistics output.

  You can include \method{evolve} in a statement of the type:

  {\small \begin{verbatim}
      for t in domain.evolve(yieldstep, finaltime):
          <Do something with domain and t>
  \end{verbatim}}

  \end{methoddesc}



\subsection{Diagnostics}
\label{sec:diagnostics}


  \begin{funcdesc}{statistics}{}
  Module: \module{abstract\_2d\_finite\_volumes.domain}

  \end{funcdesc}

  \begin{funcdesc}{timestepping\_statistics}{}
  Module: \module{abstract\_2d\_finite\_volumes.domain}

  Returns a string of the following type for each
  timestep:

  \code{Time = 0.9000, delta t in [0.00598964, 0.01177388], steps=12
  (12)}

  Here the numbers in \code{steps=12 (12)} indicate the number of steps taken and
  the number of first-order steps, respectively.\\

  The optional keyword argument \code{track_speeds=True} will
  generate a histogram of speeds generated by each triangle. The
  speeds relate to the size of the timesteps used by ANUGA and
  this diagnostics may help pinpoint problem areas where excessive speeds
  are generated.

  \end{funcdesc}


  \begin{funcdesc}{boundary\_statistics}{quantities = None, tags = None}
  Module: \module{abstract\_2d\_finite\_volumes.domain}

  Returns a string of the following type when \code{quantities = 'stage'} and \code{tags = ['top', 'bottom']}:

  {\small \begin{verbatim}
 Boundary values at time 0.5000:
    top:
        stage in [ -0.25821218,  -0.02499998]
    bottom:
        stage in [ -0.27098821,  -0.02499974]
  \end{verbatim}}

  \end{funcdesc}


  \begin{funcdesc}{get\_quantity}{name, location='vertices', indices = None}
  Module: \module{abstract\_2d\_finite\_volumes.domain}

  Allow access to individual quantities and their methods

  \end{funcdesc}


  \begin{funcdesc}{set\_quantities\_to\_be\_monitored}{}
  Module: \module{abstract\_2d\_finite\_volumes.domain}

  Selects quantities and derived quantities for which extrema attained at internal timesteps
  will be collected.

  Information can be tracked in the evolve loop by printing \code{quantity\_statistics} and
  collected data will be stored in the sww file.

  Optional parameters \code{polygon} and \code{time\_interval} may be specified to restrict the
  extremum computation.
  \end{funcdesc}

  \begin{funcdesc}{quantity\_statistics}{}
  Module: \module{abstract\_2d\_finite\_volumes.domain}

  Reports on extrema attained by selected quantities.

  Returns a string of the following type for each
  timestep:

  \begin{verbatim}
  Monitored quantities at time 1.0000:
    stage-elevation:
      values since time = 0.00 in [0.00000000, 0.30000000]
      minimum attained at time = 0.00000000, location = (0.16666667, 0.33333333)
      maximum attained at time = 0.00000000, location = (0.83333333, 0.16666667)
    ymomentum:
      values since time = 0.00 in [0.00000000, 0.06241221]
      minimum attained at time = 0.00000000, location = (0.33333333, 0.16666667)
      maximum attained at time = 0.22472667, location = (0.83333333, 0.66666667)
    xmomentum:
      values since time = 0.00 in [-0.06062178, 0.47886313]
      minimum attained at time = 0.00000000, location = (0.16666667, 0.33333333)
      maximum attained at time = 0.35103646, location = (0.83333333, 0.16666667)
  \end{verbatim}

  The quantities (and derived quantities) listed here must be selected at model
  initialisation using the method \code{domain.set_quantities_to_be_monitored}.\\

  The optional keyword argument \code{precision='\%.4f'} will
  determine the precision used for floating point values in the output.
  This diagnostics helps track extrema attained by the selected quantities
  at every internal timestep.

  These values are also stored in the sww file for post processing.

  \end{funcdesc}



  \begin{funcdesc}{get\_values}{location='vertices', indices = None}
  Module: \module{abstract\_2d\_finite\_volumes.quantity}

  Extract values for quantity as an array

  \end{funcdesc}


  \begin{funcdesc}{get\_integral}{}
  Module: \module{abstract\_2d\_finite\_volumes.quantity}

  Return computed integral over entire domain for this quantity

  \end{funcdesc}




  \begin{funcdesc}{get\_maximum\_value}{indices = None}
  Module: \module{abstract\_2d\_finite\_volumes.quantity}

  Return maximum value of quantity (on centroids)

  Optional argument indices is the set of element ids that
  the operation applies to. If omitted all elements are considered.

  We do not seek the maximum at vertices as each vertex can
  have multiple values - one for each triangle sharing it.
  \end{funcdesc}



  \begin{funcdesc}{get\_maximum\_location}{indices = None}
  Module: \module{abstract\_2d\_finite\_volumes.quantity}

  Return location of maximum value of quantity (on centroids)

  Optional argument indices is the set of element ids that
  the operation applies to.

  We do not seek the maximum at vertices as each vertex can
  have multiple values - one for each triangle sharing it.

  If there are multiple cells with same maximum value, the
  first cell encountered in the triangle array is returned.
  \end{funcdesc}



  \begin{funcdesc}{get\_wet\_elements}{indices=None}
  Module: \module{shallow\_water.shallow\_water\_domain}

  Return indices for elements where h $>$ minimum_allowed_height
  Optional argument indices is the set of element ids that the operation applies to.
  \end{funcdesc}


  \begin{funcdesc}{get\_maximum\_inundation\_elevation}{indices=None}
  Module: \module{shallow\_water.shallow\_water\_domain}

  Return highest elevation where h $>$ 0.\\
  Optional argument indices is the set of element ids that the operation applies to.\\

  Example to find maximum runup elevation:\\
     z = domain.get_maximum_inundation_elevation()
  \end{funcdesc}


  \begin{funcdesc}{get\_maximum\_inundation\_location}{indices=None}
  Module: \module{shallow\_water.shallow\_water\_domain}

  Return location (x,y) of highest elevation where h $>$ 0.\\
  Optional argument indices is the set of element ids that the operation applies to.\\

  Example to find maximum runup location:\\
     x, y = domain.get_maximum_inundation_location()
  \end{funcdesc}


\section{Queries of SWW model output files}
After a model has been run, it is often useful to extract various information from the sww
output file (see Section \ref{sec:sww format}). This is typically more convenient than using the
diagnostics described in Section \ref{sec:diagnostics} which rely on the model running - something
that can be very time consuming. The sww files are easy and quick to read and offer much information
about the model results such as runup heights, time histories of selected quantities,
flow through cross sections and much more.

\begin{funcdesc}{get\_maximum\_inundation\_elevation}{filename, polygon=None,
    time_interval=None, verbose=False}
  Module: \module{shallow\_water.data\_manager}

  Return highest elevation where depth is positive ($h > 0$)

  Example to find maximum runup elevation:\\
  max_runup = get_maximum_inundation_elevation(filename,
  polygon=None,
  time_interval=None,
  verbose=False)


  filename is a NetCDF sww file containing ANUGA model output.
  Optional arguments polygon and time_interval restricts the maximum runup calculation
  to a points that lie within the specified polygon and time interval.

  If no inundation is found within polygon and time_interval the return value
  is None signifying "No Runup" or "Everything is dry".

  See doc string for general function get_maximum_inundation_data for details.
\end{funcdesc}


\begin{funcdesc}{get\_maximum\_inundation\_location}{filename, polygon=None,
    time_interval=None, verbose=False}
  Module: \module{shallow\_water.data\_manager}

  Return location (x,y) of highest elevation where depth is positive ($h > 0$)

  Example to find maximum runup location:\\
  max_runup_location = get_maximum_inundation_location(filename,
  polygon=None,
  time_interval=None,
  verbose=False)


  filename is a NetCDF sww file containing ANUGA model output.
  Optional arguments polygon and time_interval restricts the maximum runup calculation
  to a points that lie within the specified polygon and time interval.

  If no inundation is found within polygon and time_interval the return value
  is None signifying "No Runup" or "Everything is dry".

  See doc string for general function get_maximum_inundation_data for details.
\end{funcdesc}


\begin{funcdesc}{sww2timeseries}{swwfiles, gauge_filename, production_dirs, report = None, reportname = None,
plot_quantity = None, generate_fig = False, surface = None, time_min = None, time_max = None, time_thinning = 1,
time_unit = None, title_on = None, use_cache = False, verbose = False}

  Module: \module{anuga.abstract\_2d\_finite\_volumes.util}

  Return csv files for the location in the \code{gauge_filename} and can also return plots of them

  See doc string for general function sww2timeseries for details.

\end{funcdesc}



\section{Other}

  \begin{funcdesc}{domain.create\_quantity\_from\_expression}{string}

  Handy for creating derived quantities on-the-fly as for example
  \begin{verbatim}
  Depth = domain.create_quantity_from_expression('stage-elevation')

  exp = '(xmomentum*xmomentum + ymomentum*ymomentum)**0.5')
  Absolute_momentum = domain.create_quantity_from_expression(exp)
  \end{verbatim}

  %See also \file{Analytical\_solution\_circular\_hydraulic\_jump.py} for an example of use.
  \end{funcdesc}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{\anuga System Architecture}


\section{File Formats}
\label{sec:file formats}

\anuga makes use of a number of different file formats. The
following table lists all these formats, which are described in more
detail in the paragraphs below.

\bigskip

\begin{center}

\begin{tabular}{|ll|}  \hline

\textbf{Extension} & \textbf{Description} \\
\hline\hline

\code{.sww} & NetCDF format for storing model output
\code{f(t,x,y)}\\

\code{.tms} & NetCDF format for storing time series \code{f(t)}\\

\code{.csv/.txt} & ASCII format called points csv for storing
arbitrary points and associated attributes\\

\code{.pts} & NetCDF format for storing arbitrary points and
associated attributes\\

\code{.asc} & ASCII format of regular DEMs as output from ArcView\\

\code{.prj} & Associated ArcView file giving more metadata for
\code{.asc} format\\

\code{.ers} & ERMapper header format of regular DEMs for ArcView\\

\code{.dem} & NetCDF representation of regular DEM data\\

\code{.tsh} & ASCII format for storing meshes and associated
boundary and region info\\

\code{.msh} & NetCDF format for storing meshes and associated
boundary and region info\\

\code{.nc} & Native ferret NetCDF format\\

\code{.geo} & Houdinis ASCII geometry format (?) \\  \par \hline
%\caption{File formats used by \anuga}
\end{tabular}


\end{center}

The above table shows the file extensions used to identify the
formats of files. However, typically, in referring to a format we
capitalise the extension and omit the initial full stop---thus, we
refer, for example, to `SWW files' or `PRJ files'.

\bigskip

A typical dataflow can be described as follows:

\subsection{Manually Created Files}

\begin{tabular}{ll}
ASC, PRJ & Digital elevation models (gridded)\\
NC & Model outputs for use as boundary conditions (e.g. from MOST)
\end{tabular}

\subsection{Automatically Created Files}

\begin{tabular}{ll}
ASC, PRJ  $\rightarrow$  DEM  $\rightarrow$  PTS & Convert
DEMs to native \code{.pts} file\\

NC $\rightarrow$ SWW & Convert MOST boundary files to
boundary \code{.sww}\\

PTS + TSH $\rightarrow$ TSH with elevation & Least squares fit\\

TSH $\rightarrow$ SWW & Convert TSH to \code{.sww}-viewable using
\code{animate}\\

TSH + Boundary SWW $\rightarrow$ SWW & Simulation using
\code{\anuga}\\

Polygonal mesh outline $\rightarrow$ & TSH or MSH
\end{tabular}




\bigskip

\subsection{SWW and TMS Formats}
\label{sec:sww format}

The SWW and TMS formats are both NetCDF formats, and are of key
importance for \anuga.

An SWW file is used for storing \anuga output and therefore pertains
to a set of points and a set of times at which a model is evaluated.
It contains, in addition to dimension information, the following
variables:

\begin{itemize}
    \item \code{x} and \code{y}: coordinates of the points, represented as Numeric arrays
    \item \code{elevation}, a Numeric array storing bed-elevations
    \item \code{volumes}, a list specifying the points at the vertices of each of the
    triangles
    % Refer here to the example to be provided in describing the simple example
    \item \code{time}, a Numeric array containing times for model
    evaluation
\end{itemize}


The contents of an SWW file may be viewed using the anuga viewer
\code{animate}, which creates an on-screen geometric
representation. See section \ref{sec:animate} (page
\pageref{sec:animate}) in Appendix \ref{ch:supportingtools} for more
on \code{animate}.

Alternatively, there are tools, such as \code{ncdump}, that allow
you to convert an NetCDF file into a readable format such as the
Class Definition Language (CDL). The following is an excerpt from a
CDL representation of the output file \file{runup.sww} generated
from running the simple example \file{runup.py} of
Chapter \ref{ch:getstarted}:

\verbatiminput{examples/bedslopeexcerpt.cdl}

The SWW format is used not only for output but also serves as input
for functions such as \function{file\_boundary} and
\function{file\_function}, described in Chapter \ref{ch:interface}.

A TMS file is used to store time series data that is independent of
position.


\subsection{Mesh File Formats}

A mesh file is a file that has a specific format suited to
triangular meshes and their outlines. A mesh file can have one of
two formats: it can be either a TSH file, which is an ASCII file, or
an MSH file, which is a NetCDF file. A mesh file can be generated
from the function \function{create\_mesh\_from\_regions} (see
Section \ref{sec:meshgeneration}) and used to initialise a domain.

A mesh file can define the outline of the mesh---the vertices and
line segments that enclose the region in which the mesh is
created---and the triangular mesh itself, which is specified by
listing the triangles and their vertices, and the segments, which
are those sides of the triangles that are associated with boundary
conditions.

In addition, a mesh file may contain `holes' and/or `regions'. A
hole represents an area where no mesh is to be created, while a
region is a labelled area used for defining properties of a mesh,
such as friction values.  A hole or region is specified by a point
and bounded by a number of segments that enclose that point.

A mesh file can also contain a georeference, which describes an
offset to be applied to $x$ and $y$ values---eg to the vertices.


\subsection{Formats for Storing Arbitrary Points and Attributes}


A CSV/TXT file is used to store data representing
arbitrary numerical attributes associated with a set of points.

The format for an CSV/TXT file is:\\
%\begin{verbatim}

            first line:     \code{[column names]}\\
            other lines:  \code{[x value], [y value], [attributes]}\\

            for example:\\
            \code{x, y, elevation, friction}\\
            \code{0.6, 0.7, 4.9, 0.3}\\
            \code{1.9, 2.8, 5, 0.3}\\
            \code{2.7, 2.4, 5.2, 0.3}

        The delimiter is a comma. The first two columns are assumed to
        be x, y coordinates.
        

A PTS file is a NetCDF representation of the data held in an points CSV
file. If the data is associated with a set of $N$ points, then the
data is stored using an $N \times 2$ Numeric array of float
variables for the points and an $N \times 1$ Numeric array for each
attribute.

%\end{verbatim}

\subsection{ArcView Formats}

Files of the three formats ASC, PRJ and ERS are all associated with
data from ArcView.

An ASC file is an ASCII representation of DEM output from ArcView.
It contains a header with the following format:

\begin{tabular}{l l}
\code{ncols}      &   \code{753}\\
\code{nrows}      &   \code{766}\\
\code{xllcorner}  &   \code{314036.58727982}\\
\code{yllcorner}  & \code{6224951.2960092}\\
\code{cellsize}   & \code{100}\\
\code{NODATA_value} & \code{-9999}
\end{tabular}

The remainder of the file contains the elevation data for each grid point
in the grid defined by the above information.

A PRJ file is an ArcView file used in conjunction with an ASC file
to represent metadata for a DEM.


\subsection{DEM Format}

A DEM file is a NetCDF representation of regular DEM data.


\subsection{Other Formats}




\subsection{Basic File Conversions}
\label{sec:basicfileconversions}

  \begin{funcdesc}{sww2dem}{basename_in, basename_out = None,
            quantity = None,
            timestep = None,
            reduction = None,
            cellsize = 10,
            NODATA_value = -9999,
            easting_min = None,
            easting_max = None,
            northing_min = None,
            northing_max = None,
            expand_search = False,
            verbose = False,
            origin = None,
            datum = 'WGS84',
            format = 'ers'}
  Module: \module{shallow\_water.data\_manager}

  Takes data from an SWW file \code{basename_in} and converts it to DEM format (ASC or
  ERS) of a desired grid size \code{cellsize} in metres.
  The easting and northing values are used if the user wished to clip the output
  file to a specified rectangular area. The \code{reduction} input refers to a function
  to reduce the quantities over all time step of the SWW file, example, maximum.
  \end{funcdesc}


  \begin{funcdesc}{dem2pts}{basename_in, basename_out=None,
            easting_min=None, easting_max=None,
            northing_min=None, northing_max=None,
            use_cache=False, verbose=False}
  Module: \module{shallow\_water.data\_manager}

  Takes DEM data (a NetCDF file representation of data from a regular Digital
  Elevation Model) and converts it to PTS format.
  \end{funcdesc}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{\anuga mathematical background}
\label{cd:mathematical background}

\section{Introduction}

This chapter outlines the mathematics underpinning \anuga.



\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{ZR1999,nielsen2005} 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{ZR1999}. 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=8.0cm,keepaspectratio=true]{graphics/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{Toro1992}) 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=8.0cm,keepaspectratio=true]{graphics/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{Flux limiting}

The shallow water equations are solved numerically using a
finite volume method on unstructured triangular grid.
The upwind central scheme due to Kurganov and Petrova is used as an
approximate Riemann solver for the computation of inviscid flux functions.
This makes it possible to handle discontinuous solutions.

To alleviate the problems associated with numerical instabilities due to
small water depths near a wet/dry boundary we employ a new flux limiter that
ensures that unphysical fluxes are never encounted.


Let $u$ and $v$ be the velocity components in the $x$ and $y$ direction,
$w$ the absolute water level (stage) and
$z$ the bed elevation. The latter are assumed to be relative to the
same height datum.
The conserved quantities tracked by ANUGA are momentum in the
$x$-direction ($\mu = uh$), momentum in the $y$-direction ($\nu = vh$)
and depth ($h = w-z$).

The flux calculation requires access to the velocity vector $(u, v)$
where each component is obtained as $u = \mu/h$ and $v = \nu/h$ respectively.
In the presence of very small water depths, these calculations become
numerically unreliable and will typically cause unphysical speeds.

We have employed a flux limiter which replaces the calculations above with
the limited approximations.
\begin{equation}
  \hat{u} = \frac{\mu}{h + h_0/h}, \bigskip \hat{v} = \frac{\nu}{h + h_0/h},
\end{equation}
where $h_0$ is a regularisation parameter that controls the minimal
magnitude of the denominator. Taking the limits we have for $\hat{u}$
\[
  \lim_{h \rightarrow 0} \hat{u} =
  \lim_{h \rightarrow 0} \frac{\mu}{h + h_0/h} = 0
\]
and
\[
  \lim_{h \rightarrow \infty} \hat{u} =
  \lim_{h \rightarrow \infty} \frac{\mu}{h + h_0/h} = \frac{\mu}{h} = u
\]
with similar results for $\hat{v}$.

The maximal value of $\hat{u}$ is attained when $h+h_0/h$ is minimal or (by differentiating the denominator)
\[
  1 - h_0/h^2 = 0
\]
or
\[
  h_0 = h^2
\]


ANUGA has a global parameter $H_0$ that controls the minimal depth which
is considered in the various equations. This parameter is typically set to
$10^{-3}$. Setting
\[
  h_0 = H_0^2
\]
provides a reasonable balance between accurracy and stability. In fact,
setting $h=H_0$ will scale the predicted speed by a factor of $0.5$:
\[
  \left[ \frac{\mu}{h + h_0/h} \right]_{h = H_0} = \frac{\mu}{2 H_0}
\]
In general, for multiples of the minimal depth $N H_0$ one obtains
\[
  \left[ \frac{\mu}{h + h_0/h} \right]_{h = N H_0} =
  \frac{\mu}{H_0 (1 + 1/N^2)}
\]
which converges quadratically to the true value with the multiple N.


%The developed numerical model has been applied to several test cases as well as to real flows. Numerical tests prove the robustness and accuracy of the model.





\section{Slope limiting}
A multidimensional slope-limiting technique is employed to achieve second-order spatial accuracy and to prevent spurious oscillations. This is using the MinMod limiter and is documented elsewhere.

However close to the bed, the limiter must ensure that no negative depths occur. On the other hand, in deep water, the bed topography should be ignored for the purpose of the limiter.


Let $w, z, h$  be the stage, bed elevation and depth at the centroid and
let $w_i, z_i, h_i$ be the stage, bed elevation and depth at vertex $i$.
Define the minimal depth across all vertices as $\hmin$ as
\[
  \hmin = \min_i h_i
\]

Let $\tilde{w_i}$ be the stage obtained from a gradient limiter
limiting on stage only. The corresponding depth is then defined as
\[
  \tilde{h_i} = \tilde{w_i} - z_i
\]
We would use this limiter in deep water which we will define (somewhat boldly)
as
\[
  \hmin \ge \epsilon
\]


Similarly, let $\bar{w_i}$ be the stage obtained from a gradient
limiter limiting on depth respecting the bed slope.
The corresponding depth is defined as
\[
  \bar{h_i} = \bar{w_i} - z_i
\]


We introduce the concept of a balanced stage $w_i$ which is obtained as
the linear combination

\[
  w_i = \alpha \tilde{w_i} + (1-\alpha) \bar{w_i}
\]
or
\[
  w_i = z_i + \alpha \tilde{h_i} + (1-\alpha) \bar{h_i}
\]
where $\alpha \in [0, 1]$.

Since $\tilde{w_i}$ is obtained in 'deep' water where the bedslope
is ignored we have immediately that
\[
  \alpha = 1 \mbox{ for } \hmin \ge \epsilon %or dz=0
\]
%where the maximal bed elevation range $dz$ is defined as
%\[
%  dz = \max_i |z_i - z|
%\]

If $\hmin < \epsilon$ we want to use the 'shallow' limiter just enough that
no negative depths occur. Formally, we will require that
\[
  \alpha \tilde{h_i} + (1-\alpha) \bar{h_i} > \epsilon, \forall i
\]
or
\begin{equation}
  \alpha(\tilde{h_i} - \bar{h_i}) > \epsilon - \bar{h_i}, \forall i
  \label{eq:limiter bound}
\end{equation}

There are two cases:
\begin{enumerate}
  \item $\bar{h_i} \le \tilde{h_i}$: The deep water (limited using stage)
  vertex is at least as far away from the bed than the shallow water
  (limited using depth). In this case we won't need any contribution from
  $\bar{h_i}$ and can accept any $\alpha$.

  E.g.\ $\alpha=1$ reduces Equation \ref{eq:limiter bound} to
  \[
    \tilde{h_i} > \epsilon
  \]
  whereas $\alpha=0$ yields
  \[
    \bar{h_i} > \epsilon
  \]
  all well and good.
  \item $\bar{h_i} > \tilde{h_i}$: In this case the the deep water vertex is
  closer to the bed than the shallow water vertex or even below the bed.
  In this case we need to find an $\alpha$ that will ensure a positive depth.
  Rearranging Equation \ref{eq:limiter bound} and solving for $\alpha$ one
  obtains the bound
  \[
    \alpha < \frac{\epsilon - \bar{h_i}}{\tilde{h_i} - \bar{h_i}}, \forall i
  \]
\end{enumerate}

Ensuring Equation \ref{eq:limiter bound} holds true for all vertices one
arrives at the definition
\[
  \alpha = \min_{i} \frac{\bar{h_i} - \epsilon}{\bar{h_i} - \tilde{h_i}}
\]
which will guarantee that no vertex 'cuts' through the bed. Finally, should
$\bar{h_i} < \epsilon$ and therefore $\alpha < 0$, we suggest setting
$\alpha=0$ and similarly capping $\alpha$ at 1 just in case.

%Furthermore,
%dropping the $\epsilon$ ensures that alpha is always positive and also
%provides a numerical safety {??)





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Basic \anuga Assumptions}


Physical model time cannot be earlier than 1 Jan 1970 00:00:00.
If one wished to recreate scenarios prior to that date it must be done
using some relative time (e.g. 0).


All spatial data relates to the WGS84 datum (or GDA94) and has been
projected into UTM with false easting of 500000 and false northing of
1000000 on the southern hemisphere (0 on the northern).

It is assumed that all computations take place within one UTM zone and
all locations must consequently be specified in Cartesian coordinates
(eastings, northings) or (x,y) where the unit is metres.

DEMs, meshes and boundary conditions can have different origins within
one UTM zone. However, the computation will use that of the mesh for
numerical stability.

When generating a mesh it is assumed that polygons do not cross.
Having polygons tht cross can cause the mesh generation to fail or bad
meshes being produced.


%OLD
%The dataflow is: (See data_manager.py and from scenarios)
%
%
%Simulation scenarios
%--------------------%
%%
%
%Sub directories contain scrips and derived files for each simulation.
%The directory ../source_data contains large source files such as
%DEMs provided externally as well as MOST tsunami simulations to be used
%as boundary conditions.
%
%Manual steps are:
%  Creation of DEMs from argcview (.asc + .prj)
%  Creation of mesh from pmesh (.tsh)
%  Creation of tsunami simulations from MOST (.nc)
%%
%
%Typical scripted steps are%
%
%  prepare_dem.py:  Convert asc and prj files supplied by arcview to
%                   native dem and pts formats%
%
%  prepare_pts.py: Convert netcdf output from MOST to an sww file suitable
%                  as boundary condition%
%
%  prepare_mesh.py: Merge DEM (pts) and mesh (tsh) using least squares
%                   smoothing. The outputs are tsh files with elevation data.%
%
%  run_simulation.py: Use the above together with various parameters to
%                     run inundation simulation.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\appendix

\chapter{Supporting Tools}
\label{ch:supportingtools}

This section describes a number of supporting tools, supplied with \anuga, that offer a
variety of types of functionality and enhance the basic capabilities of \anuga.

\section{caching}
\label{sec:caching}

The \code{cache} function is used to provide supervised caching of function
results. A Python function call of the form

      {\small \begin{verbatim}
      result = func(arg1,...,argn)
      \end{verbatim}}

  can be replaced by

      {\small \begin{verbatim}
      from caching import cache
      result = cache(func,(arg1,...,argn))
      \end{verbatim}}

  which returns the same output but reuses cached
  results if the function has been computed previously in the same context.
  \code{result} and the arguments can be simple types, tuples, list, dictionaries or
  objects, but not unhashable types such as functions or open file objects.
  The function \code{func} may be a member function of an object or a module.

  This type of caching is particularly useful for computationally intensive
  functions with few frequently used combinations of input arguments. Note that
  if the inputs or output are very large caching may not save time because
  disc access may dominate the execution time.

  If the function definition changes after a result has been cached, this will be
  detected by examining the functions \code{bytecode (co\_code, co\_consts,
  func\_defaults, co\_argcount)} and the function will be recomputed.
  However, caching will not detect changes in modules used by \code{func}.
  In this case cache must be cleared manually.

  Options are set by means of the function \code{set\_option(key, value)},
  where \code{key} is a key associated with a
  Python dictionary \code{options}. This dictionary stores settings such as the name of
  the directory used, the maximum
  number of cached files allowed, and so on.

  The \code{cache} function allows the user also to specify a list of dependent files. If any of these
  have been changed, the function is recomputed and the results stored again.

  %Other features include support for compression and a capability to \ldots


   \textbf{USAGE:} \nopagebreak

    {\small \begin{verbatim}
    result = cache(func, args, kwargs, dependencies, cachedir, verbose,
                   compression, evaluate, test, return_filename)
    \end{verbatim}}


\section{ANUGA viewer - animate}
\label{sec:animate}
 The output generated by \anuga may be viewed by
means of the visualisation tool \code{animate}, which takes the
\code{SWW} file output by \anuga and creates a visual representation
of the data. Examples may be seen in Figures \ref{fig:runupstart}
and \ref{fig:runup2}. To view an \code{SWW} file with
\code{animate} in the Windows environment, you can simply drag the
icon representing the file over an icon on the desktop for the
\code{animate} executable file (or a shortcut to it), or set up a
file association to make files with the extension \code{.sww} open
with \code{animate}. Alternatively, you can operate \code{animate}
from the command line, in both Windows and Linux environments.

On successful operation, you will see an interactive moving-picture
display. You can use keys and the mouse to slow down, speed up or
stop the display, change the viewing position or carry out a number
of other simple operations. Help is also displayed when you press
the \code{h} key.

The main keys operating the interactive screen are:\\

\begin{center}
\begin{tabular}{|ll|}   \hline

\code{w} & toggle wireframe \\

space bar & start/stop\\

up/down arrows & increase/decrease speed\\

left/right arrows & direction in time \emph{(when running)}\\
& step through simulation \emph{(when stopped)}\\

left mouse button & rotate\\

middle mouse button & pan\\

right mouse button & zoom\\  \hline

\end{tabular}
\end{center}

\vfill

The following table describes how to operate animate from the command line:

Usage: \code{animate [options] swwfile \ldots}\\  \nopagebreak
Options:\\  \nopagebreak
\begin{tabular}{ll}
  \code{--display <type>} & \code{MONITOR | POWERWALL | REALITY\_CENTER |}\\
                                    & \code{HEAD\_MOUNTED\_DISPLAY}\\
  \code{--rgba} & Request a RGBA colour buffer visual\\
  \code{--stencil} & Request a stencil buffer visual\\
  \code{--stereo} & Use default stereo mode which is \code{ANAGLYPHIC} if not \\
                                    & overridden by environmental variable\\
  \code{--stereo <mode>} & \code{ANAGLYPHIC | QUAD\_BUFFER | HORIZONTAL\_SPLIT |}\\
                                    & \code{VERTICAL\_SPLIT | LEFT\_EYE | RIGHT\_EYE |}\\
                                     & \code{ON | OFF} \\
  \code{-alphamax <float 0-1>} & Maximum transparency clamp value\\
  \code{-alphamin <float 0-1>} & Transparency value at \code{hmin}\\
\end{tabular}

\begin{tabular}{ll}
  \code{-cullangle <float angle 0-90>} & Cull triangles steeper than this value\\
  \code{-help} & Display this information\\
  \code{-hmax <float>} & Height above which transparency is set to
                                     \code{alphamax}\\
\end{tabular}

\begin{tabular}{ll}

  \code{-hmin <float>} & Height below which transparency is set to
                                     zero\\
\end{tabular}

\begin{tabular}{ll}
  \code{-lightpos <float>,<float>,<float>} & $x,y,z$ of bedslope directional light ($z$ is
                                     up, default is overhead)\\
\end{tabular}

\begin{tabular}{ll}
  \code{-loop}  & Repeated (looped) playback of \code{.swm} files\\

\end{tabular}

\begin{tabular}{ll}
  \code{-movie <dirname>} & Save numbered images to named directory and
                                     quit\\

  \code{-nosky} & Omit background sky\\


  \code{-scale <float>} & Vertical scale factor\\
  \code{-texture <file>} & Image to use for bedslope topography\\
  \code{-tps <rate>} & Timesteps per second\\
  \code{-version} & Revision number and creation (not compile)
                                     date\\
\end{tabular}

\section{utilities/polygons}

  \declaremodule{standard}{utilities.polygon}
  \refmodindex{utilities.polygon}

  \begin{classdesc}{Polygon\_function}{regions, default=0.0, geo_reference=None}
  Module: \code{utilities.polygon}

  Creates a callable object that returns one of a specified list of values when
  evaluated at a point \code{x, y}, depending on which polygon, from a specified list of polygons, the
  point belongs to. The parameter \code{regions} is a list of pairs
  \code{(P, v)}, where each \code{P} is a polygon and each \code{v}
  is either a constant value or a function of coordinates \code{x}
  and \code{y}, specifying the return value for a point inside \code{P}. The
  optional parameter \code{default} may be used to specify a value 
  (or a function)
  for a point not lying inside any of the specified polygons. When a
  point lies in more than one polygon, the return value is taken to
  be the value for whichever of these polygon appears later in the
  list.
  %FIXME (Howard): CAN x, y BE VECTORS?
  The optional parameter geo\_reference refers to the status of points
  that are passed into the function. Typically they will be relative to 
  some origin. In ANUGA, a typical call will take the form:
  {\small \begin{verbatim}
     set_quantity('elevation', 
                  Polygon_function([(P1, v1), (P2, v2)],
                                   default=v3,
                                   geo_reference=domain.geo_reference))
  \end{verbatim}}
  

  \end{classdesc}

  \begin{funcdesc}{read\_polygon}{filename}
  Module: \code{utilities.polygon}

  Reads the specified file and returns a polygon. Each
  line of the file must contain exactly two numbers, separated by a comma, which are interpreted
  as coordinates of one vertex of the polygon.
  \end{funcdesc}

  \begin{funcdesc}{populate\_polygon}{polygon, number_of_points, seed = None, exclude = None}
  Module: \code{utilities.polygon}

  Populates the interior of the specified polygon with the specified number of points,
  selected by means of a uniform distribution function.
  \end{funcdesc}

  \begin{funcdesc}{point\_in\_polygon}{polygon, delta=1e-8}
  Module: \code{utilities.polygon}

  Returns a point inside the specified polygon and close to the edge. The distance between
  the returned point and the nearest point of the polygon is less than $\sqrt{2}$ times the
  second argument \code{delta}, which is taken as $10^{-8}$ by default.
  \end{funcdesc}

  \begin{funcdesc}{inside\_polygon}{points, polygon, closed = True, verbose = False}
  Module: \code{utilities.polygon}

  Used to test whether the members of a list of points
  are inside the specified polygon. Returns a Numeric
  array comprising the indices of the points in the list that lie inside the polygon.
  (If none of the points are inside, returns \code{zeros((0,), 'l')}.)
  Points on the edges of the polygon are regarded as inside if
  \code{closed} is set to \code{True} or omitted; otherwise they are regarded as outside.
  \end{funcdesc}

  \begin{funcdesc}{outside\_polygon}{points, polygon, closed = True, verbose = False}
  Module: \code{utilities.polygon}

  Exactly like \code{inside\_polygon}, but with the words `inside' and `outside' interchanged.
  \end{funcdesc}

  \begin{funcdesc}{is\_inside\_polygon}{point, polygon, closed=True, verbose=False}
  Module: \code{utilities.polygon}

  Returns \code{True} if \code{point} is inside \code{polygon} or
  \code{False} otherwise. Points on the edges of the polygon are regarded as inside if
  \code{closed} is set to \code{True} or omitted; otherwise they are regarded as outside.
  \end{funcdesc}

  \begin{funcdesc}{is\_outside\_polygon}{point, polygon, closed=True, verbose=False}
  Module: \code{utilities.polygon}

  Exactly like \code{is\_outside\_polygon}, but with the words `inside' and `outside' interchanged.
  \end{funcdesc}

  \begin{funcdesc}{point\_on\_line}{x, y, x0, y0, x1, y1}
  Module: \code{utilities.polygon}

  Returns \code{True} or \code{False}, depending on whether the point with coordinates
  \code{x, y} is on the line passing through the points with coordinates \code{x0, y0}
  and \code{x1, y1} (extended if necessary at either end).
  \end{funcdesc}

  \begin{funcdesc}{separate\_points\_by\_polygon}{points, polygon, closed = True, verbose = False}
    \indexedcode{separate\_points\_by\_polygon}
  Module: \code{utilities.polygon}

  \end{funcdesc}

  \begin{funcdesc}{polygon\_area}{polygon}
  Module: \code{utilities.polygon}

  Returns area of arbitrary polygon (reference http://mathworld.wolfram.com/PolygonArea.html)
  \end{funcdesc}

  \begin{funcdesc}{plot\_polygons}{polygons, figname, verbose = False}
  Module: \code{utilities.polygon}

  Plots each polygon contained in input polygon list, e.g.
 \code{polygons = [poly1, poly2, poly3]} where \code{poly1 = [[x11,y11],[x12,y12],[x13,y13]]}
 etc.  Each polygon is closed for plotting purposes and subsequent plot saved to \code{figname}.
  Returns list containing the minimum and maximum of \code{x} and \code{y},
  i.e. \code{[x_{min}, x_{max}, y_{min}, y_{max}]}.
  \end{funcdesc}

\section{coordinate\_transforms}

\section{geospatial\_data}
\label{sec:geospatial}

This describes a class that represents arbitrary point data in UTM
coordinates along with named attribute values.

%FIXME (Ole): This gives a LaTeX error
%\declaremodule{standard}{geospatial_data}
%\refmodindex{geospatial_data}



\begin{classdesc}{Geospatial\_data}
  {data_points = None,
    attributes = None,
    geo_reference = None,
    default_attribute_name = None,
    file_name = None}
Module: \code{geospatial\_data}

This class is used to store a set of data points and associated
attributes, allowing these to be manipulated by methods defined for
the class.

The data points are specified either by reading them from a NetCDF
or CSV file, identified through the parameter \code{file\_name}, or
by providing their \code{x}- and \code{y}-coordinates in metres,
either as a sequence of 2-tuples of floats or as an $M \times 2$
Numeric array of floats, where $M$ is the number of points.
Coordinates are interpreted relative to the origin specified by the
object \code{geo\_reference}, which contains data indicating the UTM
zone, easting and northing. If \code{geo\_reference} is not
specified, a default is used.

Attributes are specified through the parameter \code{attributes},
set either to a list or array of length $M$ or to a dictionary whose
keys are the attribute names and whose values are lists or arrays of
length $M$. One of the attributes may be specified as the default
attribute, by assigning its name to \code{default\_attribute\_name}.
If no value is specified, the default attribute is taken to be the
first one.
\end{classdesc}


\begin{methoddesc}{import\_points\_file}{delimiter = None, verbose = False}

\end{methoddesc}


\begin{methoddesc}{export\_points\_file}{ofile, absolute=False}

\end{methoddesc}


\begin{methoddesc}{get\_data\_points}{absolute = True, as\_lat\_long =
    False}
    If \code{as\_lat\_long} is\code{True} the point information
    returned will be in Latitudes and Longitudes.

\end{methoddesc}


\begin{methoddesc}{set\_attributes}{attributes}

\end{methoddesc}


\begin{methoddesc}{get\_attributes}{attribute_name = None}

\end{methoddesc}


\begin{methoddesc}{get\_all\_attributes}{}

\end{methoddesc}


\begin{methoddesc}{set\_default\_attribute\_name}{default_attribute_name}

\end{methoddesc}


\begin{methoddesc}{set\_geo\_reference}{geo_reference}

\end{methoddesc}


\begin{methoddesc}{add}{}

\end{methoddesc}


\begin{methoddesc}{clip}{}
Clip geospatial data by a polygon

Inputs are \code{polygon} which is either a list of points, an Nx2 array or
a Geospatial data object and \code{closed}(optional) which determines
whether points on boundary should be regarded as belonging to the polygon
(\code{closed=True}) or not (\code{closed=False}).
Default is \code{closed=True}.

Returns new Geospatial data object representing points
inside specified polygon.
\end{methoddesc}


\begin{methoddesc}{clip_outside}{}
Clip geospatial data by a polygon

Inputs are \code{polygon} which is either a list of points, an Nx2 array or
a Geospatial data object and \code{closed}(optional) which determines
whether points on boundary should be regarded as belonging to the polygon
(\code{closed=True}) or not (\code{closed=False}).
Default is \code{closed=True}.

Returns new Geospatial data object representing points
\emph{out}side specified polygon.
\end{methoddesc}


\section{Graphical Mesh Generator GUI}
The program \code{graphical\_mesh\_generator.py} in the pmesh module
allows the user to set up the mesh of the problem interactively.
It can be used to build the outline of a mesh or to visualise a mesh
automatically generated.

Graphical Mesh Generator will let the user select various modes. The
current allowable modes are vertex, segment, hole or region.  The mode
describes what sort of object is added or selected in response to
mouse clicks.  When changing modes any prior selected objects become
deselected.

In general the left mouse button will add an object and the right
mouse button will select an object.  A selected object can de deleted
by pressing the the middle mouse button (scroll bar).

\section{alpha\_shape}
\emph{Alpha shapes} are used to generate close-fitting boundaries
around sets of points. The alpha shape algorithm produces a shape
that approximates to the `shape formed by the points'---or the shape
that would be seen by viewing the points from a coarse enough
resolution. For the simplest types of point sets, the alpha shape
reduces to the more precise notion of the convex hull. However, for
many sets of points the convex hull does not provide a close fit and
the alpha shape usually fits more closely to the original point set,
offering a better approximation to the shape being sought.

In \anuga, an alpha shape is used to generate a polygonal boundary
around a set of points before mesh generation. The algorithm uses a
parameter $\alpha$ that can be adjusted to make the resultant shape
resemble the shape suggested by intuition more closely. An alpha
shape can serve as an initial boundary approximation that the user
can adjust as needed.

The following paragraphs describe the class used to model an alpha
shape and some of the important methods and attributes associated
with instances of this class.

\begin{classdesc}{Alpha\_Shape}{points, alpha = None}
Module: \code{alpha\_shape}

To instantiate this class the user supplies the points from which
the alpha shape is to be created (in the form of a list of 2-tuples
\code{[[x1, y1],[x2, y2]}\ldots\code{]}, assigned to the parameter
\code{points}) and, optionally, a value for the parameter
\code{alpha}. The alpha shape is then computed and the user can then
retrieve details of the boundary through the attributes defined for
the class.
\end{classdesc}


\begin{funcdesc}{alpha\_shape\_via\_files}{point_file, boundary_file, alpha= None}
Module: \code{alpha\_shape}

This function reads points from the specified point file
\code{point\_file}, computes the associated alpha shape (either
using the specified value for \code{alpha} or, if no value is
specified, automatically setting it to an optimal value) and outputs
the boundary to a file named \code{boundary\_file}. This output file
lists the coordinates \code{x, y} of each point in the boundary,
using one line per point.
\end{funcdesc}


\begin{methoddesc}{set\_boundary\_type}{self,raw_boundary=True,
                          remove_holes=False,
                          smooth_indents=False,
                          expand_pinch=False,
                          boundary_points_fraction=0.2}
Module: \code{alpha\_shape},  Class: \class{Alpha\_Shape}

This function sets flags that govern the operation of the algorithm
that computes the boundary, as follows:

\code{raw\_boundary = True} returns raw boundary, i.e. the regular edges of the
                alpha shape.\\
\code{remove\_holes = True} removes small holes (`small' is defined by
\code{boundary\_points\_fraction})\\
\code{smooth\_indents = True} removes sharp triangular indents in
boundary\\
\code{expand\_pinch = True} tests for pinch-off and
corrects---preventing a boundary vertex from having more than two edges.
\end{methoddesc}


\begin{methoddesc}{get\_boundary}{}
Module: \code{alpha\_shape},  Class: \class{Alpha\_Shape}

Returns a list of tuples representing the boundary of the alpha
shape. Each tuple represents a segment in the boundary by providing
the indices of its two endpoints.
\end{methoddesc}


\begin{methoddesc}{write\_boundary}{file_name}
Module: \code{alpha\_shape},  Class: \class{Alpha\_Shape}

Writes the list of 2-tuples returned by \code{get\_boundary} to the
file \code{file\_name}, using one line per tuple.
\end{methoddesc}

\section{Numerical Tools}

The following table describes some useful numerical functions that
may be found in the module \module{utilities.numerical\_tools}:

\begin{tabular}{|p{8cm} p{8cm}|}  \hline
\code{angle(v1, v2=None)} & Angle between two-dimensional vectors
\code{v1} and \code{v2}, or between \code{v1} and the $x$-axis if
\code{v2} is \code{None}. Value is in range $0$ to $2\pi$. \\

\code{normal\_vector(v)} & Normal vector to \code{v}.\\

\code{mean(x)} & Mean value of a vector \code{x}.\\

\code{cov(x, y=None)} & Covariance of vectors \code{x} and \code{y}.
If \code{y} is \code{None}, returns \code{cov(x, x)}.\\

\code{err(x, y=0, n=2, relative=True)} & Relative error of
$\parallel$\code{x}$-$\code{y}$\parallel$ to
$\parallel$\code{y}$\parallel$ (2-norm if \code{n} = 2 or Max norm
if \code{n} = \code{None}). If denominator evaluates to zero or if
\code{y}
is omitted or if \code{relative = False}, absolute error is returned.\\

\code{norm(x)} & 2-norm of \code{x}.\\

\code{corr(x, y=None)} & Correlation of \code{x} and \code{y}. If
\code{y} is \code{None} returns autocorrelation of \code{x}.\\

\code{ensure\_numeric(A, typecode = None)} & Returns a Numeric array
for any sequence \code{A}. If \code{A} is already a Numeric array it
will be returned unaltered. Otherwise, an attempt is made to convert
it to a Numeric array. (Needed because \code{array(A)} can
cause memory overflow.)\\

\code{histogram(a, bins, relative=False)} & Standard histogram. If
\code{relative} is \code{True}, values will be normalised against
the total and thus represent frequencies rather than counts.\\

\code{create\_bins(data, number\_of\_bins = None)} & Safely create
bins for use with histogram. If \code{data} contains only one point
or is constant, one bin will be created. If \code{number\_of\_bins}
is omitted, 10 bins will be created.\\  \hline

\end{tabular}


\chapter{Modules available in \anuga}


\section{\module{abstract\_2d\_finite\_volumes.general\_mesh} }
\declaremodule[generalmesh]{}{general\_mesh}
\label{mod:generalmesh}

\section{\module{abstract\_2d\_finite\_volumes.neighbour\_mesh} }
\declaremodule[neighbourmesh]{}{neighbour\_mesh}
\label{mod:neighbourmesh}

\section{\module{abstract\_2d\_finite\_volumes.domain} --- Generic module for 2D triangular domains for finite-volume computations of conservation laws}
\declaremodule{}{domain}
\label{mod:domain}


\section{\module{abstract\_2d\_finite\_volumes.quantity}}
\declaremodule{}{quantity}
\label{mod:quantity}

\begin{verbatim}
Class Quantity - Implements values at each triangular element

To create:

   Quantity(domain, vertex_values)

   domain: Associated domain structure. Required.

   vertex_values: N x 3 array of values at each vertex for each element.
                  Default None

   If vertex_values are None Create array of zeros compatible with domain.
   Otherwise check that it is compatible with dimenions of domain.
   Otherwise raise an exception

\end{verbatim}




\section{\module{shallow\_water} --- 2D triangular domains for finite-volume
computations of the shallow water wave equation. This module contains a specialisation
of class Domain from module domain.py consisting of methods specific to the Shallow Water
Wave Equation
}
\declaremodule[shallowwater]{}{shallow\_water}
\label{mod:shallowwater}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Frequently Asked Questions}


\section{General Questions}

\subsubsection{What is \anuga?}
It is a software package suitable for simulating 2D water flows in
complex geometries.

\subsubsection{Why is it called \anuga?}
The software was developed in collaboration between the
Australian National University (ANU) and Geoscience Australia (GA).

\subsubsection{How do I obtain a copy of \anuga?}
See \url{https://datamining.anu.edu.au/anuga} for all things ANUGA.

%\subsubsection{What developments are expected for \anuga in the future?}
%This

\subsubsection{Are there any published articles about \anuga that I can reference?}
See \url{https://datamining.anu.edu.au/anuga} for links.


\section{Modelling Questions}

\subsubsection{Which type of problems are \anuga good for?}
General 2D waterflows in complex geometries such as
dam breaks, flows amoung structurs, coastal inundation etc.

\subsubsection{Which type of problems are beyond the scope of \anuga?}
See Chapter \ref{ch:limitations}.

\subsubsection{Can I start the simulation at an arbitrary time?}
Yes, using \code{domain.set\_time()} you can specify an arbitrary
starting time. This is for example useful in conjunction with a
file\_boundary, which may start hours before anything hits the model
boundary. By assigning a later time for the model to start,
computational resources aren't wasted.

\subsubsection{Can I change values for any quantity during the simulation?}
Yes, using \code{domain.set\_quantity()} inside the domain.evolve
loop you can change values of any quantity. This is for example
useful if you wish to let the system settle for a while before
assigning an initial condition. Another example would be changing
the values for elevation to model e.g. erosion.

\subsubsection{Can I change boundary conditions during the simulation?}
Yes - see example on page \pageref{sec:change boundary code} in Section
\ref{sec:change boundary}.

\subsubsection{How do I access model time during the simulation?}
The variable \code{t} in the evolve for loop is the model time.
For example to change the boundary at a particular time (instead of basing this on the state of the system as in Section \ref{sec:change boundary})
one would write something like
{\small \begin{verbatim}
    for t in domain.evolve(yieldstep = 0.2, duration = 40.0):

        if Numeric.allclose(t, 15):
            print 'Changing boundary to outflow'
            domain.set_boundary({'right': Bo})

\end{verbatim}}
The model time can also be accessed through the public interface \code{domain.get\_time()}, or changed (at your own peril) through \code{domain.set\_time()}.


\subsubsection{Why does a file\_function return a list of numbers when evaluated?}
Currently, file\_function works by returning values for the conserved
quantities \code{stage}, \code{xmomentum} and \code{ymomentum} at a given point in time
and space as a triplet. To access e.g.\ \code{stage} one must specify element 0 of the
triplet returned by file\_function.

\subsubsection{Which diagnostics are available to troubleshoot a simulation?}

\subsubsection{How do I use a DEM in my simulation?}
You use \code{dem2pts} to convert your DEM to the required .pts format. This .pts file is then called
when setting the elevation data to the mesh in \code{domain.set_quantity}

\subsubsection{What sort of DEM resolution should I use?}
Try and work with the \emph{best} you have available. Onshore DEMs
are typically available in 25m, 100m and 250m grids. Note, offshore
data is often sparse, or non-existent.

\subsubsection{What sort of mesh resolution should I use?}
The mesh resolution should be commensurate with your DEM - it does not make sense to put in place a mesh which is finer than your DEM. As an example,
if your DEM is on a 25m grid, then the cell resolution should be of the order of 315$m^2$ (this represents half the area of the square grid). Ideally,
you need a fine mesh over regions where the DEM changes rapidly, and other areas of significant interest, such as the coast.
If meshes are too coarse, discretisation errors in both stage and momentum may lead to unphysical results. All studies should include sensitivity and convergence studies based on different resolutions.


\subsubsection{How do I tag interior polygons?}
At the moment create_mesh_from_regions does not allow interior
polygons with symbolic tags. If tags are needed, the interior
polygons must be created subsequently. For example, given a filename
of polygons representing solid walls (in Arc Ungenerate format) can
be tagged as such using the code snippet:
\begin{verbatim}
  # Create mesh outline with tags
  mesh = create_mesh_from_regions(bounding_polygon,
                                  boundary_tags=boundary_tags)
  # Add buildings outlines with tags set to 'wall'. This would typically
  # bind to a Reflective boundary
  mesh.import_ungenerate_file(buildings_filename, tag='wall')

  # Generate and write mesh to file
  mesh.generate_mesh(maximum_triangle_area=max_area)
  mesh.export_mesh_file(mesh_filename)
\end{verbatim}

Note that a mesh object is returned from \code{create_mesh_from_regions}
when file name is omitted.

\subsubsection{How often should I store the output?}
This will depend on what you are trying to answer with your model and how much memory you have available on your machine. If you need
to look in detail at the evolution, then you will need to balance your storage requirements and the duration of the simulation.
If the SWW file exceeds 1Gb, another SWW file will be created until the end of the simulation. As an example, to store all the conserved
quantities on a mesh with approximately 300000 triangles on a 2 min interval for 5 hours will result in approximately 350Mb SWW file
(as for the \file{run\_sydney\_smf.py} example).

\subsubsection{How can I set the friction in different areas in the domain?}
The model area will typically be estimating the water height and momentum over varying
topographies which will have different friction values. One way of assigning
different friction values is to create polygons (say \code{poly1, poly2 and poly3}) describing each
area and then set the corresponding friction values in the following way

\code{domain.set_quantity('friction',Polygon_function([(poly1,f1),(poly2,f2),
(poly3,f3))]))}

The values of \code{f1,f2} and \code{f3} could be constant or functions
as determined by the user.

\subsubsection{How can I combine data sets?}

A user may have access to a range of different resolution DEMs and raw data points (such
as beach profiles, spot heights, single or multi-beam data etc) and will need
to combine them to create an overall elevation data set.

If there are multiple DEMs, say of 10m and 25m resolution, then the technique is similar to
that defined in the Cairns example described earlier, that is

{\small \begin{verbatim}
convert_dem_from_ascii2netcdf(10m_dem_name, use_cache=True, verbose=True)
convert_dem_from_ascii2netcdf(25m_dem_name, use_cache=True, verbose=True)
\end{verbatim}}
followed by
{\small \begin{verbatim}
dem2pts(10m_dem_name, use_cache=True, verbose=True)
dem2pts(25m_dem_name, use_cache=True, verbose=True)
\end{verbatim}}
These data sets can now be combined by
{\small \begin{verbatim}
from anuga.geospatial_data.geospatial_data import *
G1 = Geospatial_data(file_name = 10m_dem_name + '.pts')
G2 = Geospatial_data(file_name = 25m_dem_name + '.pts')
G = G1 + G2
G.export_points_file(combined_dem_name + ‘.pts’)
\end{verbatim}}
this is the basic way of combining data sets, however, the user will need to
assess the boundaries of each data set and whether they overlap. For example, consider
if the 10m DEM is describing by \code{poly1} and the 25m DEM is described by \code{poly2}
with \code{poly1} completely enclosed in \code{poly2} as shown in Figure \ref{fig:polydata}
\begin{figure}[hbt]
  \centerline{\includegraphics{graphics/polyanddata.jpg}}
  \caption{Polygons describing the extent of the 10m and 25m DEM.}
  \label{fig:polydata}
\end{figure}
To combine the data sets, the geospatial addition is updated to
{\small \begin{verbatim}
G = G1 + G2.clip_outside(Geospatial_data(poly1))
\end{verbatim}}
For this example, we assume that \code{poly2} is the domain, otherwise an additional dataset
would be required for the remainder of the domain.

This technique can be expanded to handle point data sets as well. In the case
of a bathymetry data set available in text format in an \code{.csv} file, then
the geospatial addition is updated to
{\small \begin{verbatim}
G3 = Geospatial_data(file_name = bathy_data_name + '.csv')
G = G1 + G2.clip_outside(Geospatial_data(poly1)) + G3
\end{verbatim}}
The \code{.csv} file has the data stored as \code{x,y,elevation} with the text \code{elevation}
on the first line.

The coastline could be included
as part of the clipping polygon to separate the offshore and onshore datasets if required.
Assume that \code{poly1} crosses the coastline
In this case, two new polygons could be created out of \code{poly1} which uses the coastline
as the divider. As shown in Figure \ref{fig:polycoast}, \code{poly3} describes the
onshore data and \code{poly4} describes the offshore data.
\begin{figure}[hbt]
  \centerline{\includegraphics{graphics/polyanddata2.jpg}}
  \caption{Inclusion of new polygons separating the 10m DEM area into an
  onshore (poly3) and offshore (poly4) data set.}
  \label{fig:polycoast}
\end{figure}
Let's include the bathymetry
data described above, so to combine the datasets in this case,
{\small \begin{verbatim}
G = G1.clip(Geospatial_data(poly3)) + G2.clip_outside(Geospatial_data(poly1)) + G3
\end{verbatim}}

Finally, to fit the elevation data to the mesh, the script is adjusted in this way
{\small \begin{verbatim}
    domain.set_quantity('elevation',
                        filename = combined_dem_name + '.pts',
                        use_cache = True,
                        verbose = True)
\end{verbatim}}
\subsection{Boundary Conditions}

\subsubsection{How do I create a Dirichlet boundary condition?}

A Dirichlet boundary condition sets a constant value for the
conserved quantities at the boundaries. A list containing
the constant values for stage, xmomentum and ymomentum is constructed
and used in the function call, e.g. \code{Dirichlet_boundary([0.2,0.,0.])}

\subsubsection{How do I know which boundary tags are available?}
The method \code{domain.get\_boundary\_tags()} will return a list of
available tags for use with
\code{domain.set\_boundary\_condition()}.

\subsubsection{What is the difference between file_boundary and field_boundary?}
The only difference is field_boundary will allow you to change the level of the stage height when you read in the boundary condition.
This is very useful when running different tide heights in the same area as you need only to convert
one boundary condition to a SWW file, ideally for tide height of 0 m (saving disk space). Then you can
use field_boundary to read this SWW file and change the stage height (tide) on the fly depending on the scenario.




\subsection{Analysing Results}

\subsubsection{How do I easily plot "tide gauges" timeseries graphs from a SWW file?}

There is two ways to do this.

1) Create csv files from the sww file using \code{anuga.abstract_2d_finite_volumes.util sww2csv_gauges}
and then use \code{anuga.abstract_2d_finite_volumes.util csv2timeseries_graphs} to
create the plots. This code is newer, has unit tests and might be easier to use. Read doc strings for more information and
review section 4.7 of this manual.

Or

2) Use \code{anuga.abstract_2d_finite_volumes.util sww2timeseries} to do the whole thing
however this doesn't have a much control on the file name and plots. Plus there is no unit tests yet.

Read the doc string for more information.

\chapter{Glossary}

\begin{tabular}{|lp{10cm}|c|}  \hline
%\begin{tabular}{|llll|}  \hline
    \emph{Term} & \emph{Definition} & \emph{Page}\\  \hline

    \indexedbold{\anuga} & Name of software (joint development between ANU and
    GA) & \pageref{def:anuga}\\

    \indexedbold{bathymetry} & offshore elevation &\\

    \indexedbold{conserved quantity} & conserved (stage, x and y
    momentum) & \\

%    \indexedbold{domain} & The domain of a function is the set of all input values to the
%    function.&\\

    \indexedbold{Digital Elevation Model (DEM)} & DEMs are digital files consisting of points of elevations,
sampled systematically at equally spaced intervals.& \\

    \indexedbold{Dirichlet boundary} & A boundary condition imposed on a differential equation
 that specifies the values the solution is to take on the boundary of the
 domain. & \pageref{def:dirichlet boundary}\\

    \indexedbold{edge} & A triangular cell within the computational mesh can be depicted
    as a set of vertices joined by lines (the edges). & \\

    \indexedbold{elevation} & refers to bathymetry and topography &\\

    \indexedbold{evolution} & integration of the shallow water wave equations
    over time &\\

    \indexedbold{finite volume method} & The method evaluates the terms in the shallow water
    wave equation as fluxes at the surfaces of each finite volume. Because the
    flux entering a given volume is identical to that leaving the adjacent volume,
    these methods are conservative. Another advantage of the finite volume method is
    that it is easily formulated to allow for unstructured meshes. The method is used
    in many computational fluid dynamics packages. & \\

    \indexedbold{forcing term} & &\\

    \indexedbold{flux} & the amount of flow through the volume per unit
    time & \\

    \indexedbold{grid} & Evenly spaced mesh & \\

    \indexedbold{latitude} & The angular distance on a mericlear north and south of the
    equator, expressed in degrees and minutes. & \\

    \indexedbold{longitude} & The angular distance east or west, between the meridian
    of a particular place on Earth and that of the Prime Meridian (located in Greenwich,
    England) expressed in degrees or time.& \\

    \indexedbold{Manning friction coefficient} & &\\

    \indexedbold{mesh} & Triangulation of domain &\\

    \indexedbold{mesh file} & A TSH or MSH file & \pageref{def:mesh file}\\

    \indexedbold{NetCDF} & &\\

    \indexedbold{node} & A point at which edges meet & \\

    \indexedbold{northing} & A rectangular (x,y) coordinate measurement of distance
    north from a north-south reference line, usually a meridian used as the axis of
    origin within a map zone or projection. Northing is a UTM (Universal Transverse
    Mercator) coordinate. & \\

    \indexedbold{points file} & A PTS or CSV file & \\  \hline

    \end{tabular}

    \begin{tabular}{|lp{10cm}|c|}  \hline

    \indexedbold{polygon} & A sequence of points in the plane. \anuga represents a polygon
    either as a list consisting of Python tuples or lists of length 2 or as an $N \times 2$
    Numeric array, where $N$ is the number of points.

    The unit square, for example, would be represented either as
    \code{[ [0,0], [1,0], [1,1], [0,1] ]} or as \code{array( [0,0], [1,0], [1,1],
    [0,1] )}.

    NOTE: For details refer to the module \module{utilities/polygon.py}. &
    \\     \indexedbold{resolution} &  The maximal area of a triangular cell in a
    mesh & \\


    \indexedbold{reflective boundary} & Models a solid wall. Returns same conserved
    quantities as those present in the neighbouring volume but reflected. Specific to the
    shallow water equation as it works with the momentum quantities assumed to be the
    second and third conserved quantities. & \pageref{def:reflective boundary}\\

    \indexedbold{stage} & &\\

%    \indexedbold{try this}

    \indexedbold{animate} & visualisation tool used with \anuga &
    \pageref{sec:animate}\\

    \indexedbold{time boundary} & Returns values for the conserved
quantities as a function of time. The user must specify
the domain to get access to the model time. & \pageref{def:time boundary}\\

    \indexedbold{topography} & onshore elevation &\\

    \indexedbold{transmissive boundary} & & \pageref{def:transmissive boundary}\\

    \indexedbold{vertex} & A point at which edges meet. & \\

    \indexedbold{xmomentum} & conserved quantity (note, two-dimensional SWW equations say
    only \code{x} and \code{y} and NOT \code{z}) &\\

    \indexedbold{ymomentum}  & conserved quantity & \\  \hline

    \end{tabular}


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%appendices.


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%  keep LaTeX2HTML quiet during the \renewcommand{} macros; they're
%  not really valuable.
%
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%  until the second \input line.
%

%begin{latexonly}
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\begin{thebibliography}{99}
\bibitem[nielsen2005]{nielsen2005}
{\it Hydrodynamic modelling of coastal inundation}.
Nielsen, O., S. Roberts, D. Gray, A. McPherson and A. Hitchman.
In Zerger, A. and Argent, R.M. (eds) MODSIM 2005 International Congress on
Modelling and Simulation. Modelling and Simulation Society of Australia and
New Zealand, December 2005, pp. 518-523. ISBN: 0-9758400-2-9.\\
http://www.mssanz.org.au/modsim05/papers/nielsen.pdf

\bibitem[grid250]{grid250}
Australian Bathymetry and Topography Grid, June 2005.
Webster, M.A. and Petkovic, P.
Geoscience Australia Record 2005/12. ISBN: 1-920871-46-2.\\
http://www.ga.gov.au/meta/ANZCW0703008022.html

\bibitem[ZR1999]{ZR1999}
\newblock {Catastrophic Collapse of Water Supply Reservoirs in Urban Areas}.
\newblock C.~Zoppou and S.~Roberts.
\newblock {\em ASCE J. Hydraulic Engineering}, 125(7):686--695, 1999.

\bibitem[Toro1999]{Toro1992}
\newblock Riemann problems and the waf method for solving the two-dimensional
  shallow water equations.
\newblock E.~F. Toro.
\newblock {\em Philosophical Transactions of the Royal Society, Series A},
  338:43--68, 1992.

\bibitem{KurNP2001}
\newblock Semidiscrete central-upwind schemes for hyperbolic conservation laws
  and hamilton-jacobi equations.
\newblock A.~Kurganov, S.~Noelle, and G.~Petrova.
\newblock {\em SIAM Journal of Scientific Computing}, 23(3):707--740, 2001.
\end{thebibliography}{99}

\end{document}