brusselator.xmds

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

<?xml version="1.0"?>
<simulation>
  
<!-- $Id: brusselator.xmds,v 1.2 2004/08/03 05:29:49 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.            -->
<!--                                                                    -->
<!--  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.                      -->
<!--                                                                    -->
<!--  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.                                              -->

  <name> brusselator </name>      <!-- the name of the simulation -->
  
  <author> Paul Cochrane </author>  <!-- the author of the simulation -->
  <description>
    <!-- a description of what the simulation is supposed to do -->
    Example simulation of the Brusslator model oscillating chemical
    kinetics equations.
    Calculates concentrations of components participating 
    in the autocatalytic Brusselator model.  The reaction scheme is

			A ----> X           (1)
			B + X ----> R + Y   (2)
			Y + 2 X ----> 3 X   (3)
			X ----> S.          (4)

    Rate equations for the intermediates X and Y are 

    d[X]/dt = k1[A] - k2 [B] [X] + 	k3 [X]^2 [Y] - k4 [X]
    d[Y]/dt = k2 [B] [X] - k3 [X]^2 [Y],

    which are transformed in the program to

    d[X]/d(tau) = (k1[A] - k2 [B] [X] + k3 [X]^2 [Y] - k4 [X])/k4
    d[Y]/d(tau) = (k2 [B] [X] - k3[X]^2 [Y])/k4,

    with tau = k4 t a unitless time-related variable.
    Adapted for xmds from "Mathematica computer programs for physical
    chemistry", William H. Cropper, Springer Verlag (1998)
    Equations are:
      d[X]_d(tau) = (k1[A] - k2[B][X] + k3[X]^2 [Y] - k4[X])/k4
      d[Y]_d(tau) = (k2[B][X] - k3[X]^2 [Y])/k4

  </description>
  
  <!-- Global system parameters and functionality -->
  <prop_dim> tau </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 -->
  <use_prefs> yes </use_prefs>       <!-- defaults to yes -->
  
  <!-- Global variables for the simulation -->
  <globals>
  <![CDATA[
    // rate constants
    //   Units are s^-1 for first-order reactions,
    //   L mol^-1 s^-1 for second-order reactions, and L^2
    //   mol^-2 s^-1 for third-order reactions. 
    const double k1 = 0.025;
    const double k2 = 1.0;
    const double k3 = 1.0;
    const double k4 = 0.01;
    
    // the constant concentrations of the components A and B
    // in mol L^-1
    const double a = 0.02;
    const double b = 0.02;

    // the initial concentrations of the components X and Y
    const double CX0 = 0.5;
    const double CY0 = 0.0;
  ]]>
  </globals>
  
  <!-- Field to be integrated over -->
  <field>
    <name> main </name>
    <samples> 1 </samples>       <!-- sample 1st point of dim? -->
    
    <vector>
      <name> main </name>
      <type> double </type>           <!-- data type of vector -->
      <components> CX CY </components>       <!-- names of components -->
      <![CDATA[
        CX = CX0;
	CY = CY0;
      ]]>
    </vector>
  </field>
  
  <!-- The sequence of integrations to perform -->
  <sequence>
    <integrate>
      <algorithm> RK4IP </algorithm> <!-- RK4EX, RK4IP, SIEX, SIIP -->
      <interval> 250 </interval>   <!-- how far in main dim? -->
      <lattice> 5000 </lattice>     <!-- no. points in main dim -->
      <samples> 500 </samples> <!-- no. pts in output moment group -->
      
      <![CDATA[
        dCX_dtau = (k1*a - k2*b*CX + k3*CX*CX*CY - k4*CX)/k4;
	dCY_dtau = (k2*b*CX - k3*CX*CX*CY)/k4;
      ]]>
    </integrate>
  </sequence>
  
  <!-- The output to generate -->
  <output format="ascii">
    <group>
      <sampling>
        <lattice> 500 </lattice>           <!-- no. points to sample -->
        <moments> X Y </moments>           <!-- names of moments -->
        <![CDATA[
          X = CX;
	  Y = CY;
        ]]>
      </sampling>
    </group>
  </output>
  
</simulation>