1 | % Use the standard \LaTeXe\ article style in 12pt Computer Modern font |
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12 | |
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13 | % The preamble is to contain your own \LaTeX\ commands and to say |
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14 | % what packages to use. Three useful packages are the following: |
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15 | \usepackage{graphicx} % avoid epsfig or earlier such packages |
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16 | \usepackage{url} % for URLs and DOIs |
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17 | \newcommand{\doi}[1]{\url{http://dx.doi.org/#1}} |
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18 | \usepackage{amsmath} % many want amsmath extensions |
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19 | \usepackage{amsfonts} |
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20 | \usepackage{underscore} |
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21 | % Avoid loading unused packages (as done by some \LaTeX\ editors). |
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22 | |
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23 | % Create title and authors using \verb|\maketitle|. Separate authors by |
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24 | % \verb|\and| and put addresses in \verb|\thanks| command with |
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25 | % \verb|\url| command \verb|\protect|ed. |
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26 | \title{Open Source Software for Computational Modelling of Fluid Flow} |
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27 | |
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28 | \author{ |
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29 | O.~M.~Nielsen\thanks{Risk Assessment Methods Project, Geospatial and |
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30 | Earth Monitoring Division, Geoscience Australia, Symonston, |
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31 | \textsc{Australia}. \protect\url{mailto:Ole.Nielsen@ga.gov.au}} |
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32 | \and |
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33 | S.~G.~Roberts\thanks{Dept. of Maths, Australian National University, |
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34 | Canberra, \textsc{Australia}. \protect\url{mailto:stephen.roberts}}} |
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35 | |
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36 | \date{30 December 2006} |
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37 | |
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38 | \newcommand{\AnuGA}{\textsc{Anuga}} |
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39 | \newcommand{\Python}{\textsc{Python}} |
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40 | \newcommand{\VPython}{\textsc{VPython}} |
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41 | \newcommand{\pypar}{\textsc{mpi}} |
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42 | \newcommand{\Metis}{\textsc{Metis}} |
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43 | \newcommand{\mpi}{\textsc{mpi}} |
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44 | |
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45 | \newcommand{\UU}{\mathbf{U}} |
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46 | \newcommand{\VV}{\mathbf{V}} |
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47 | \newcommand{\EE}{\mathbf{E}} |
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48 | \newcommand{\GG}{\mathbf{G}} |
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49 | \newcommand{\FF}{\mathbf{F}} |
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50 | \newcommand{\HH}{\mathbf{H}} |
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51 | \newcommand{\SSS}{\mathbf{S}} |
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52 | \newcommand{\nn}{\mathbf{n}} |
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53 | |
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54 | \newcommand{\code}[1]{\texttt{#1}} |
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55 | |
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56 | |
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57 | %\graphicspath{{../Figures/}} |
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58 | |
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59 | \begin{document} |
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60 | |
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61 | % Use default \verb|\maketitle|. |
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62 | \maketitle |
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63 | |
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64 | % Use the \verb|abstract| environment. |
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65 | \begin{abstract} |
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66 | Geoscience Australia and the Australian National University are |
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67 | developing a hydrodynamic inundation modelling tool called \AnuGA{} |
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68 | to help simulate the impact of natural inundation hazards such as |
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69 | riverine flooding, storm surges and tsunami. The core of \AnuGA{} is |
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70 | a \Python{} implementation of a finite-volume method for solving the |
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71 | conservative form of the Shallow Water Wave equation. In this paper |
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72 | we describe the parallelisation of the code using a domain |
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73 | decomposition strategy. We describe the use of the the \Metis{} |
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74 | graph partitioning library to decompose our finite volume meshes. |
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75 | The parallel efficiency of our code is tested using a number of mesh |
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76 | partitions, and we verify that the \Metis{} graph partition is |
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77 | particularly efficient. |
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78 | \end{abstract} |
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79 | |
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80 | % By default we include a table of contents in each paper. |
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81 | \tableofcontents |
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82 | |
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83 | % Use \verb|\section|, \verb|\subsection|, \verb|\subsubsection| and |
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84 | % possibly \verb|\paragraph| to structure your document. Make sure |
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85 | % you \verb|\label| them for cross-referencing with \verb|\ref|\,. |
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86 | \section{Introduction} |
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87 | \label{sec:intro} |
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88 | |
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89 | %Floods are the single greatest cause of death due to natural hazards |
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90 | %in Australia, causing almost 40{\%} of the fatalities recorded |
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91 | %between 1788 and 2003~\cite{Blong-2005}. Analysis of buildings |
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92 | %damaged between 1900 and 2003 suggests that 93.6{\%} of damage is |
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93 | %the result of meteorological hazards, of which almost 25{\%} is |
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94 | %directly attributable to flooding~\cite{Blong-2005}. |
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95 | |
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96 | Flooding of coastal communities may result from surges of near-shore |
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97 | waters caused by severe storms. The extent of inundation is |
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98 | critically linked to tidal conditions, bathymetry and topography; as |
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99 | recently exemplified in the United States by Hurricane Katrina. |
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100 | While the scale of the impact from such events is not common, the |
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101 | preferential development of Australian coastal corridors means that |
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102 | storm-surge inundation of even a few hundred metres beyond the |
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103 | shoreline has increased potential to cause significant disruption |
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104 | and loss. |
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105 | |
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106 | Coastal communities also face the small but real risk of tsunami. |
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107 | Fortunately, catastrophic tsunami of the scale of the 26 December |
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108 | 2004 event are exceedingly rare. However, smaller-scale tsunami are |
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109 | more common and regularly threaten coastal communities around the |
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110 | world. Earthquakes which occur in the Java Trench near Indonesia |
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111 | (e.g.~\cite{TsuMIS1995}) and along the Puysegur Ridge to the south |
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112 | of New Zealand (e.g.~\cite{LebKC1998}) have potential to generate |
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113 | tsunami that may threaten Australia's northwestern and southeastern |
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114 | coastlines. |
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115 | |
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116 | Hydrodynamic modelling allows flooding, storm-surge and tsunami |
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117 | hazards to be better understood, their impacts to be anticipated |
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118 | and, with appropriate planning, their effects to be mitigated. |
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119 | Geoscience Australia in collaboration with the Mathematical Sciences |
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120 | Institute, Australian National University, is developing a software |
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121 | application called \AnuGA{} to model the hydrodynamics of floods, |
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122 | storm surges and tsunami. These hazards are modelled using the |
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123 | conservative shallow water equations which are described in |
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124 | section~\ref{sec:model}. In \AnuGA{} these equations are solved |
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125 | using a finite volume method as described in section~\ref{sec:fvm}. |
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126 | A more complete discussion of the method can be found in |
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127 | \cite{modsim2005} where the model and solution technique is |
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128 | validated on a standard tsunami benchmark data set. |
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129 | |
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130 | In this paper we will provide a description of the parallelisation |
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131 | of the code using a domain decomposition strategy and provide the |
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132 | preliminary timing results which verify the scalability of the |
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133 | parallel code. |
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134 | |
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135 | |
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136 | |
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137 | \section{Model} |
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138 | \label{sec:model} |
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139 | |
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140 | The shallow water wave equations are a system of differential |
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141 | conservation equations which describe the flow of a thin layer of |
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142 | fluid over terrain. The form of the equations are: |
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143 | \[ |
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144 | \frac{\partial \UU}{\partial t}+\frac{\partial \EE}{\partial |
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145 | x}+\frac{\partial \GG}{\partial y}=\SSS |
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146 | \] |
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147 | where $\UU=\left[ {{\begin{array}{*{20}c} |
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148 | h & {uh} & {vh} \\ |
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149 | \end{array} }} \right]^T$ is the vector of conserved quantities; water depth |
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150 | $h$, $x$ momentum $uh$ and $y$ momentum $vh$. Other quantities |
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151 | entering the system are bed elevation $z$ and stage (absolute water |
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152 | level) $w$, where the relation $w = z + h$ holds true at all times. |
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153 | The fluxes in the $x$ and $y$ directions, $\EE$ and $\GG$ are given |
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154 | by |
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155 | \[ |
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156 | \EE=\left[ {{\begin{array}{*{20}c} |
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157 | {uh} \hfill \\ |
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158 | {u^2h+gh^2/2} \hfill \\ |
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159 | {uvh} \hfill \\ |
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160 | \end{array} }} \right]\mbox{ and }\GG=\left[ {{\begin{array}{*{20}c} |
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161 | {vh} \hfill \\ |
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162 | {vuh} \hfill \\ |
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163 | {v^2h+gh^2/2} \hfill \\ |
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164 | \end{array} }} \right] |
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165 | \] |
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166 | and the source term (which includes gravity and friction) is given |
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167 | by |
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168 | \[ |
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169 | \SSS=\left[ {{\begin{array}{*{20}c} |
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170 | 0 \hfill \\ |
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171 | -{gh(z_{x} + S_{fx} )} \hfill \\ |
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172 | -{gh(z_{y} + S_{fy} )} \hfill \\ |
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173 | \end{array} }} \right] |
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174 | \] |
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175 | where $S_f$ is the bed friction. The friction term is modelled using |
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176 | Manning's resistance law |
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177 | \[ |
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178 | S_{fx} =\frac{u\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}\mbox{ and }S_{fy} |
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179 | =\frac{v\eta ^2\sqrt {u^2+v^2} }{h^{4/3}} |
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180 | \] |
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181 | in which $\eta$ is the Manning resistance coefficient. |
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182 | |
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183 | As demonstrated in our papers, \cite{modsim2005,Rob99l} these |
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184 | equations provide an excellent model of flows associated with |
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185 | inundation such as dam breaks and tsunamis. |
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186 | |
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187 | \section{Finite Volume Method} |
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188 | \label{sec:fvm} |
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189 | |
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190 | We use a finite-volume method for solving the shallow water wave |
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191 | equations \cite{Rob99l}. The study area is represented by a mesh of |
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192 | triangular cells as in Figure~\ref{fig:mesh} in which the conserved |
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193 | quantities of water depth $h$, and horizontal momentum $(uh, vh)$, |
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194 | in each volume are to be determined. The size of the triangles may |
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195 | be varied within the mesh to allow greater resolution in regions of |
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196 | particular interest. |
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197 | |
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198 | \begin{figure} |
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199 | \begin{center} |
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200 | \includegraphics[width=5.0cm,keepaspectratio=true]{step-five} |
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201 | \caption{Triangular mesh used in our finite volume method. Conserved |
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202 | quantities $h$, $uh$ and $vh$ are associated with the centroid of |
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203 | each triangular cell.} \label{fig:mesh} |
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204 | \end{center} |
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205 | \end{figure} |
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206 | |
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207 | The equations constituting the finite-volume method are obtained by |
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208 | integrating the differential conservation equations over each |
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209 | triangular cell of the mesh. Introducing some notation we use $i$ to |
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210 | refer to the $i$th triangular cell $T_i$, and ${\cal N}(i)$ to the |
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211 | set of indices referring to the cells neighbouring the $i$th cell. |
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212 | Then $A_i$ is the area of the $i$th triangular cell and $l_{ij}$ is |
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213 | the length of the edge between the $i$th and $j$th cells. |
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214 | |
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215 | By applying the divergence theorem we obtain for each volume an |
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216 | equation which describes the rate of change of the average of the |
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217 | conserved quantities within each cell, in terms of the fluxes across |
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218 | the edges of the cells and the effect of the source terms. In |
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219 | particular, rate equations associated with each cell have the form |
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220 | $$ |
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221 | \frac{d\UU_i }{dt}+ \frac1{A_i}\sum\limits_{j\in{\cal N}(i)} \HH_{ij} l_{ij} = \SSS_i |
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222 | $$ |
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223 | where |
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224 | \begin{itemize} |
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225 | \item $\UU_i$ the vector of conserved quantities averaged over the $i$th cell, |
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226 | \item $\SSS_i$ is the source term associated with the $i$th cell, |
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227 | and |
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228 | \item $\HH_{ij}$ is the outward normal flux of |
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229 | material across the \textit{ij}th edge. |
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230 | \end{itemize} |
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231 | |
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232 | |
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233 | %\item $l_{ij}$ is the length of the edge between the $i$th and $j$th |
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234 | %cells |
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235 | %\item $m_{ij}$ is the midpoint of |
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236 | %the \textit{ij}th edge, |
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237 | %\item |
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238 | %$\mathbf{n}_{ij} = (n_{ij,1} , n_{ij,2})$is the outward pointing |
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239 | %normal along the \textit{ij}th edge, and The |
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240 | |
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241 | The flux $\HH_{ij}$ is evaluated using a numerical flux function |
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242 | $\HH(\cdot, \cdot ; \ \cdot)$ which is consistent with the shallow |
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243 | water flux in the sense that for all vectors $\UU$ and $\nn$ |
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244 | $$ |
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245 | H(\UU,\UU;\ \nn) = \EE(\UU) n_1 + \GG(\UU) n_2 . |
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246 | $$ |
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247 | |
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248 | Then |
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249 | $$ |
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250 | \HH_{ij} = \HH(\overline{\UU}_i(m_{ij})), |
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251 | \overline{\UU}_i(m_{ij})), \mathbf{n}_{ij}) |
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252 | $$ |
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253 | where $m_{ij}$ is the midpoint of the \textit{ij}th edge and |
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254 | $\mathbf{n}_{ij}$ is the outward pointing normal on the |
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255 | \textit{ij}th edge. The function $\overline{\UU}_i(x)$ for $x \in |
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256 | T_i$ is obtained from the average values, $\UU_i$, of the $i$th and |
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257 | neighbouring cells. |
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258 | |
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259 | We use a second order reconstruction to produce a piece-wise linear |
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260 | function construction of the conserved quantities for all $x \in |
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261 | T_i$ for each cell (see Figure~\ref{fig:mesh:reconstruct}. This |
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262 | function is allowed to be discontinuous across the edges of the |
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263 | cells, but the slope of this function is limited to avoid |
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264 | artificially introduced oscillations. |
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265 | |
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266 | Godunov's method (see \cite{Toro-92}) involves calculating the |
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267 | numerical flux function $\HH(\cdot, \cdot ; \ \cdot)$ by exactly |
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268 | solving the corresponding one dimensional Riemann problem normal to |
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269 | the edge. We use the central-upwind scheme of \cite{KurNP2001} to |
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270 | calculate an approximation of the flux across each edge. |
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271 | |
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272 | \begin{figure} |
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273 | \begin{center} |
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274 | \includegraphics[width=5.0cm,keepaspectratio=true]{step-reconstruct} |
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275 | \caption{From the values of the conserved quantities at the centroid |
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276 | of the cell and its neighbouring cells, a discontinuous piecewise |
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277 | linear reconstruction of the conserved quantities is obtained.} |
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278 | \label{fig:mesh:reconstruct} |
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279 | \end{center} |
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280 | \end{figure} |
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281 | |
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282 | |
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283 | |
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284 | |
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285 | |
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286 | In the computations presented in this paper we use an explicit Euler |
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287 | time stepping method with variable timestepping adapted to the |
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288 | observed CFL condition. |
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289 | |
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290 | |
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291 | |
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292 | |
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293 | |
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294 | |
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295 | |
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296 | \section{Validation} |
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297 | \label{sec:validation} The process of validating the \AnuGA{} |
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298 | application is in its early stages, however initial indications are |
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299 | encouraging. |
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300 | |
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301 | As part of the Third International Workshop on Long-wave Runup |
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302 | Models in 2004 (\url{http://www.cee.cornell.edu/longwave}), four |
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303 | benchmark problems were specified to allow the comparison of |
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304 | numerical, analytical and physical models with laboratory and field |
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305 | data. One of these problems describes a wave tank simulation of the |
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306 | 1993 Okushiri Island tsunami off Hokkaido, Japan \cite{MatH2001}. A |
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307 | significant feature of this tsunami was a maximum run-up of 32~m |
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308 | observed at the head of the Monai Valley. This run-up was not |
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309 | uniform along the coast and is thought to have resulted from a |
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310 | particular topographic effect. Among other features, simulations of |
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311 | the Hokkaido tsunami should capture this run-up phenomenon. |
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312 | |
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313 | \begin{figure}[htbp] |
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314 | \centerline{\includegraphics[width=4in]{tsunami-fig-3.eps}} |
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315 | \caption{Comparison of wave tank and \AnuGA{} water stages at gauge |
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316 | 5.}\label{fig:val} |
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317 | \end{figure} |
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318 | |
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319 | |
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320 | \begin{figure}[htbp] |
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321 | \centerline{\includegraphics[width=4in]{tsunami-fig-4.eps}} |
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322 | \caption{Complex reflection patterns and run-up into Monai Valley |
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323 | simulated by \AnuGA{} and visualised using our netcdf OSG |
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324 | viewer.}\label{fig:run} |
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325 | \end{figure} |
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326 | |
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327 | |
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328 | The wave tank simulation of the Hokkaido tsunami was used as the |
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329 | first scenario for validating \AnuGA{}. The dataset provided |
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330 | bathymetry and topography along with initial water depth and the |
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331 | wave specifications. The dataset also contained water depth time |
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332 | series from three wave gauges situated offshore from the simulated |
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333 | inundation area. |
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334 | |
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335 | Figure~\ref{fig:val} compares the observed wave tank and modelled |
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336 | \AnuGA{} water depth (stage height) at one of the gauges. The plots |
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337 | show good agreement between the two time series, with \AnuGA{} |
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338 | closely modelling the initial draw down, the wave shoulder and the |
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339 | subsequent reflections. The discrepancy between modelled and |
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340 | simulated data in the first 10 seconds is due to the initial |
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341 | condition in the physical tank not being uniformly zero. Similarly |
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342 | good comparisons are evident with data from the other two gauges. |
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343 | Additionally, \AnuGA{} replicates exceptionally well the 32~m Monai |
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344 | Valley run-up, and demonstrates its occurrence to be due to the |
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345 | interaction of the tsunami wave with two juxtaposed valleys above |
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346 | the coastline. The run-up is depicted in Figure~\ref{fig:run}. |
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347 | |
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348 | This successful replication of the tsunami wave tank simulation on a |
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349 | complex 3D beach is a positive first step in validating the \AnuGA{} |
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350 | modelling capability. Subsequent validation will be conducted as |
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351 | additional datasets become available. |
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352 | |
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353 | |
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354 | |
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355 | \section{Parallel Implementation}\label{sec:parallel} |
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356 | |
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357 | To parallelize our algorithm we use a simple domain decomposition |
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358 | strategy. We suppose we have a global mesh which defines the domain |
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359 | of our problem. We must first subdivide the global mesh into a set |
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360 | of non-overlapping submeshes. This partitioning is done using the |
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361 | \Metis{} partitioning library. We will demonstrate the efficiency of |
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362 | this library in the following subsections. Once this partitioning |
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363 | has been done, and the proper communication patterns set, we can |
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364 | start to evolve our solution. Each timestep consists of |
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365 | independently and then communicate the appropriate ``ghost'' cell |
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366 | information once each timestep. |
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367 | |
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368 | %\begin{figure}[hbtp] |
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369 | % \centerline{ \includegraphics[scale = 0.6]{domain.eps}} |
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370 | % \caption{The first step of the parallel algorithm is to divide |
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371 | % the mesh over the processors.} |
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372 | % \label{fig:subpart} |
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373 | %\end{figure} |
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374 | |
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375 | |
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376 | |
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377 | \begin{figure}[h!] |
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378 | \centerline{ \includegraphics[width=6.5cm]{mermesh.eps}} |
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379 | \caption{The global Merimbula mesh} |
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380 | \label{fig:mergrid} |
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381 | \end{figure} |
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382 | |
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383 | \begin{figure}[h!] |
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384 | \centerline{ \includegraphics[width=6.5cm]{mermesh4c.eps} |
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385 | \includegraphics[width=6.5cm]{mermesh4a.eps}} |
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386 | |
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387 | \centerline{ \includegraphics[width=6.5cm]{mermesh4d.eps} |
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388 | \includegraphics[width=6.5cm]{mermesh4b.eps}} |
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389 | \caption{The global Merimbula mesh partitioned into 4 submeshes using \Metis.} |
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390 | \label{fig:mergrid4} |
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391 | \end{figure} |
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392 | |
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393 | |
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394 | |
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395 | |
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396 | \begin{table}[h!] |
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397 | \caption{4-way and 8-way partition tests of Merimbula mesh showing |
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398 | the percentage distribution of cells between the |
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399 | submeshes.}\label{tbl:mer} |
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400 | \begin{center} |
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401 | \begin{tabular}{|c|c c c c|} |
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402 | \hline |
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403 | CPU & 0 & 1 & 2 & 3\\ |
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404 | \hline |
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405 | Cells & 2757 & 2713 & 2761 & 2554\\ |
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406 | \% & 25.6\% & 25.2\% & 25.6\% & 23.7\%\ \\ |
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407 | \hline |
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408 | \end{tabular} |
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409 | |
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410 | \medskip |
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411 | |
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412 | \begin{tabular}{|c|c c c c c c c c|} |
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413 | \hline |
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414 | CPU & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ |
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415 | \hline |
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416 | Cells & 1229 & 1293 & 1352 & 1341 & 1349 & 1401 & 1413 & 1407\\ |
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417 | \% & 11.4\% & 12.0\% & 12.5\% & 12.4\% & 12.5\% & 13.0\% & 13.1\% & 13.1\%\\ |
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418 | \hline |
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419 | \end{tabular} |
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420 | \end{center} |
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421 | \end{table} |
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422 | |
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423 | |
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424 | \section{Conclusions} |
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425 | \label{sec:6} \AnuGA{} is a flexible and robust modelling system |
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426 | that simulates hydrodynamics by solving the shallow water wave |
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427 | equation in a triangular mesh. It can model the process of wetting |
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428 | and drying as water enters and leaves an area and is capable of |
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429 | capturing hydraulic shocks due to the ability of the finite-volume |
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430 | method to accommodate discontinuities in the solution. |
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431 | |
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432 | \AnuGA{} can take as input bathymetric and topographic datasets and |
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433 | simulate the behaviour of riverine flooding, storm surge and |
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434 | tsunami. Initial validation using wave tank data supports AnuGA's |
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435 | ability to model complex scenarios. Further validation will be |
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436 | pursued as additional datasets become available. |
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437 | |
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438 | \AnuGA{} is already being used to model the behaviour of |
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439 | hydrodynamic natural hazards. This modelling capability is part of |
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440 | Geoscience Australia's ongoing research effort to model and |
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441 | understand the potential impact from natural hazards in order to |
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442 | reduce their impact on Australian communities. |
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443 | |
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444 | |
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445 | \paragraph{Acknowledgements:} |
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446 | The authors are grateful to Belinda Barnes, National Centre for |
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447 | Epidemiology and Population Health, Australian National University, |
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448 | and Matt Hayne and Augusto Sanabria, Risk Research Group, Geoscience |
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449 | Australia, for helpful reviews of a previous version of this paper. |
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450 | Author Nielsen publish with the permission of the CEO, Geoscience |
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451 | Australia. |
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452 | |
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453 | % Preferably provide your bibliography as a separate .bbl file. |
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454 | % Include URLs, DOIs, Math Review numbers or Zentralblatt numbers in your bibliography |
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455 | % so we help connect the web of science and ensure maximum visibility |
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456 | % for your article. |
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457 | \bibliographystyle{plain} |
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458 | \bibliography{database1} |
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459 | |
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460 | |
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461 | |
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462 | |
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463 | \end{document} |
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