1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Lattice systems</title> 5<link rel="stylesheet" href="../../../../../../../doc/src/boostbook.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint"> 8<link rel="up" href="../tutorial.html" title="Tutorial"> 9<link rel="prev" href="complex_state_types.html" title="Complex state types"> 10<link rel="next" href="ensembles_of_oscillators.html" title="Ensembles of oscillators"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../logo.jpg"></td> 15<td align="center"><a href="../../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="complex_state_types.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../tutorial.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ensembles_of_oscillators.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="boost_numeric_odeint.tutorial.lattice_systems"></a><a class="link" href="lattice_systems.html" title="Lattice systems">Lattice 28 systems</a> 29</h3></div></div></div> 30<p> 31 odeint can also be used to solve ordinary differential equations defined 32 on lattices. A prominent example is the Fermi-Pasta-Ulam system <a class="link" href="../literature.html#fpu_scholarpedia">[8] 33 </a>. It is a Hamiltonian system of nonlinear coupled harmonic oscillators. 34 The Hamiltonian is 35 </p> 36<p> 37 <span class="emphasis"><em>H = Σ<sub>i</sub> p<sub>i</sub><sup>2</sup>/2 + 1/2 ( q<sub>i+1</sub> - q<sub>i</sub> )^2 + β / 4 ( q<sub>i+1</sub> - q<sub>i</sub> )^4 </em></span> 38 </p> 39<p> 40 Remarkably, the Fermi-Pasta-Ulam system was the first numerical experiment 41 to be implemented on a computer. It was studied at Los Alamos in 1953 on 42 one of the first computers (a MANIAC I) and it triggered a whole new tree 43 of mathematical and physical science. 44 </p> 45<p> 46 Like the <a class="link" href="solar_system.html" title="Solar system">Solar 47 System</a>, the FPU is solved again by a symplectic solver, but in this 48 case we can speed up the computation because the <span class="emphasis"><em>q</em></span> components 49 trivially reduce to <span class="emphasis"><em>dq<sub>i</sub> / dt = p<sub>i</sub></em></span>. odeint is capable 50 of doing this performance improvement. All you have to do is to call the 51 symplectic solver with an state function for the <span class="emphasis"><em>p</em></span> components. 52 Here is how this function looks like 53 </p> 54<p> 55</p> 56<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">container_type</span><span class="special">;</span> 57 58<span class="keyword">struct</span> <span class="identifier">fpu</span> 59<span class="special">{</span> 60 <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">m_beta</span><span class="special">;</span> 61 62 <span class="identifier">fpu</span><span class="special">(</span> <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">beta</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_beta</span><span class="special">(</span> <span class="identifier">beta</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span> 63 64 <span class="comment">// system function defining the ODE</span> 65 <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">dpdt</span> <span class="special">)</span> <span class="keyword">const</span> 66 <span class="special">{</span> 67 <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span> 68 <span class="keyword">double</span> <span class="identifier">tmp</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="number">0.0</span><span class="special">;</span> 69 <span class="keyword">double</span> <span class="identifier">tmp2</span> <span class="special">=</span> <span class="identifier">tmp</span> <span class="special">+</span> <span class="identifier">m_beta</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span><span class="special">;</span> 70 <span class="identifier">dpdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">tmp2</span><span class="special">;</span> 71 <span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">n</span><span class="special">-</span><span class="number">1</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> 72 <span class="special">{</span> 73 <span class="identifier">tmp</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">+</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span> 74 <span class="identifier">tmp2</span> <span class="special">=</span> <span class="identifier">tmp</span> <span class="special">+</span> <span class="identifier">m_beta</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span><span class="special">;</span> 75 <span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">+=</span> <span class="identifier">tmp2</span><span class="special">;</span> 76 <span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">+</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">tmp2</span><span class="special">;</span> 77 <span class="special">}</span> 78 <span class="identifier">tmp</span> <span class="special">=</span> <span class="special">-</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">n</span><span class="special">-</span><span class="number">1</span><span class="special">];</span> 79 <span class="identifier">tmp2</span> <span class="special">=</span> <span class="identifier">tmp</span> <span class="special">+</span> <span class="identifier">m_beta</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span><span class="special">;</span> 80 <span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">n</span><span class="special">-</span><span class="number">1</span><span class="special">]</span> <span class="special">+=</span> <span class="identifier">tmp2</span><span class="special">;</span> 81 <span class="special">}</span> 82 83 <span class="comment">// calculates the energy of the system</span> 84 <span class="keyword">double</span> <span class="identifier">energy</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">)</span> <span class="keyword">const</span> 85 <span class="special">{</span> 86 <span class="comment">// ...</span> 87 <span class="special">}</span> 88 89 <span class="comment">// calculates the local energy of the system</span> 90 <span class="keyword">void</span> <span class="identifier">local_energy</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">,</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">e</span> <span class="special">)</span> <span class="keyword">const</span> 91 <span class="special">{</span> 92 <span class="comment">// ...</span> 93 <span class="special">}</span> 94<span class="special">};</span> 95</pre> 96<p> 97 </p> 98<p> 99 You can also use <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">N</span> <span class="special">></span></code> 100 for the state type. 101 </p> 102<p> 103 Now, you have to define your initial values and perform the integration: 104 </p> 105<p> 106</p> 107<pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">64</span><span class="special">;</span> 108<span class="identifier">container_type</span> <span class="identifier">q</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">p</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span> 109 110<span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">n</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> 111<span class="special">{</span> 112 <span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span> 113 <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="number">32.0</span> <span class="special">*</span> <span class="identifier">sin</span><span class="special">(</span> <span class="keyword">double</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">+</span> <span class="number">1</span> <span class="special">)</span> <span class="special">/</span> <span class="keyword">double</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">+</span> <span class="number">1</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">M_PI</span> <span class="special">);</span> 114<span class="special">}</span> 115 116 117<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span> 118 119<span class="keyword">typedef</span> <span class="identifier">symplectic_rkn_sb3a_mclachlan</span><span class="special"><</span> <span class="identifier">container_type</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span> 120<span class="identifier">fpu</span> <span class="identifier">fpu_instance</span><span class="special">(</span> <span class="number">8.0</span> <span class="special">);</span> 121 122<span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">fpu_instance</span> <span class="special">,</span> 123 <span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">q</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">p</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span> 124 <span class="number">0.0</span> <span class="special">,</span> <span class="number">1000.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="identifier">streaming_observer</span><span class="special">(</span> <span class="identifier">cout</span> <span class="special">,</span> <span class="identifier">fpu_instance</span> <span class="special">,</span> <span class="number">10</span> <span class="special">)</span> <span class="special">);</span> 125</pre> 126<p> 127 </p> 128<p> 129 The observer uses a reference to the system object to calculate the local 130 energies: 131 </p> 132<p> 133</p> 134<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">streaming_observer</span> 135<span class="special">{</span> 136 <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&</span> <span class="identifier">m_out</span><span class="special">;</span> 137 <span class="keyword">const</span> <span class="identifier">fpu</span> <span class="special">&</span><span class="identifier">m_fpu</span><span class="special">;</span> 138 <span class="identifier">size_t</span> <span class="identifier">m_write_every</span><span class="special">;</span> 139 <span class="identifier">size_t</span> <span class="identifier">m_count</span><span class="special">;</span> 140 141 <span class="identifier">streaming_observer</span><span class="special">(</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span> <span class="special">&</span><span class="identifier">out</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">fpu</span> <span class="special">&</span><span class="identifier">f</span> <span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">write_every</span> <span class="special">=</span> <span class="number">100</span> <span class="special">)</span> 142 <span class="special">:</span> <span class="identifier">m_out</span><span class="special">(</span> <span class="identifier">out</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_fpu</span><span class="special">(</span> <span class="identifier">f</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_write_every</span><span class="special">(</span> <span class="identifier">write_every</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_count</span><span class="special">(</span> <span class="number">0</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span> 143 144 <span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">></span> 145 <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> 146 <span class="special">{</span> 147 <span class="keyword">if</span><span class="special">(</span> <span class="special">(</span> <span class="identifier">m_count</span> <span class="special">%</span> <span class="identifier">m_write_every</span> <span class="special">)</span> <span class="special">==</span> <span class="number">0</span> <span class="special">)</span> 148 <span class="special">{</span> 149 <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">first</span><span class="special">;</span> 150 <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">second</span><span class="special">;</span> 151 <span class="identifier">container_type</span> <span class="identifier">energy</span><span class="special">(</span> <span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">);</span> 152 <span class="identifier">m_fpu</span><span class="special">.</span><span class="identifier">local_energy</span><span class="special">(</span> <span class="identifier">q</span> <span class="special">,</span> <span class="identifier">p</span> <span class="special">,</span> <span class="identifier">energy</span> <span class="special">);</span> 153 <span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> 154 <span class="special">{</span> 155 <span class="identifier">m_out</span> <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">i</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">energy</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> 156 <span class="special">}</span> 157 <span class="identifier">m_out</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> 158 <span class="identifier">clog</span> <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">accumulate</span><span class="special">(</span> <span class="identifier">energy</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">energy</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> 159 <span class="special">}</span> 160 <span class="special">++</span><span class="identifier">m_count</span><span class="special">;</span> 161 <span class="special">}</span> 162<span class="special">};</span> 163</pre> 164<p> 165 </p> 166<p> 167 The full cpp file for this FPU example can be found here <a href="https://github.com/headmyshoulder/odeint-v2/blob/master/examples/fpu.cpp" target="_top">fpu.cpp</a> 168 </p> 169</div> 170<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 171<td align="left"></td> 172<td align="right"><div class="copyright-footer">Copyright © 2009-2015 Karsten Ahnert and Mario Mulansky<p> 173 Distributed under the Boost Software License, Version 1.0. (See accompanying 174 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) 175 </p> 176</div></td> 177</tr></table> 178<hr> 179<div class="spirit-nav"> 180<a accesskey="p" href="complex_state_types.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../tutorial.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ensembles_of_oscillators.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 181</div> 182</body> 183</html> 184
