lotka_volterra.xmds

XMDS is a code generator that integrates equations. You write them down in human readable form in a

<?xml version="1.0"?>
<!-- Lotka-Volterra system simulation -->

<!-- $Id: lotka_volterra.xmds,v 1.1 2004/04/28 05:12:18 paultcochrane Exp $ -->

<!--  Copyright (C) 2000-2004                                           -->
<!--                                                                    -->
<!--  Code contributed by Greg Collecutt, Joseph Hope and Paul Cochrane -->
<!--                                                                    -->
<!--  This file is part of xmds.                                        -->
<!--                                                                    -->
<!--  This program is free software; you can redistribute it and/or     -->
<!--  modify it under the terms of the GNU General Public License       -->
<!--  as published by the Free Software Foundation; either version 2    -->
<!--  of the License, or (at your option) any later version.            -->
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<!--  This program is distributed in the hope that it will be useful,   -->
<!--  but WITHOUT ANY WARRANTY; without even the implied warranty of    -->
<!--  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the     -->
<!--  GNU General Public License for more details.                      -->
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<!--  You should have received a copy of the GNU General Public License -->
<!--  along with this program; if not, write to the Free Software       -->
<!--  Foundation, Inc., 59 Temple Place - Suite 330, Boston,            -->
<!--  MA  02111-1307, USA.                                              -->

<simulation>
  
  <name> lotka_volterra </name>      <!-- the name of the simulation -->

  <author> Paul Cochrane </author>  <!-- the author of the simulation -->
  <description>
    Example simulation: The Lotka-Volterra system, used to model
    undamped oscillations in autocatalytic chemical reactions.
    Adapted from the journal article: "A new transformation for the 
    Lotka-Volterra problem", C.M. Evans and G.L. Findley, J. Math. Chem.
    <b>25</b>, 1, Jan (1999)
  </description>
  
  <!-- Global system parameters and functionality -->
  <prop_dim> t </prop_dim>    <!-- name of main propagation dim -->
  
  <error_check> yes </error_check>   <!-- defaults to yes -->
  <use_wisdom> yes </use_wisdom>     <!-- defaults to no -->
  <benchmark> yes </benchmark>       <!-- defaults to no -->
  
  <!-- Global variables for the simulation -->
  <globals>
  <![CDATA[
    double const a = 0.50;
    double const b = 1.30;
    double const c = 0.67;
    // these variables are the transformation variables in the paper, 
    // we don't do this here, however we use their initial values
    // (it makes converting the paper to xmds easier)
    double const wo = 2.0;
    double const wo_dot = 0.5;
  ]]>
  </globals>
  
  <!-- Field to be integrated over -->
  <field>
    <samples> 1 </samples>       <!-- sample 1st point of dim? -->
    
    <vector>
      <name> main </name>
      <type> double </type>           <!-- data type of vector -->
      <components> x1 x2 </components>       <!-- names of components -->
      <![CDATA[
        x1 = (c*wo + wo_dot)/b;
	x2 = (a*wo - wo_dot)/b;
      ]]>
    </vector>
  </field>
  
  <!-- The sequence of integrations to perform -->
  <sequence>
    <integrate>
      <algorithm> RK4EX </algorithm> <!-- RK4EX, RK4IP, SIEX, SIIP -->
      <interval> 30.0 </interval>   <!-- how far in main dim? -->
      <lattice> 3000 </lattice>     <!-- no. points in main dim -->
      <samples> 100 </samples> <!-- no. pts in output moment group -->
      
      <vectors> </vectors>     <!-- vector names -->
      <![CDATA[
        dx1_dt =  a*x1 - b*x1*x2;
	dx2_dt = -c*x2 + b*x1*x2;
      ]]>
    </integrate>
  </sequence>
  
  <!-- The output to generate -->
  <output format="ascii">
    <group>
      <sampling>
	<moments> x1Out x2Out </moments>           <!-- names of moments -->
	<![CDATA[
	    x1Out = x1;
	    x2Out = x2;
	]]>
      </sampling>
    </group>
  </output>
  
</simulation>