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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="boost_numeric_odeint.tutorial.stiff_systems"></a><a class="link" href="stiff_systems.html" title="Stiff systems">Stiff systems</a> |
| </h3></div></div></div> |
| <p> |
| An important class of ordinary differential equations are so called stiff |
| system which are characterized by two or more time scales of different order. |
| Examples of such systems are found in chemical systems where reaction rates |
| of individual sub-reaction might differ over large ranges, for example: |
| </p> |
| <p> |
| <span class="emphasis"><em>d S<sub>​1</sub> / dt = - 101 S<sub>​2</sub> - 100 S<sub>​1</sub></em></span> |
| </p> |
| <p> |
| <span class="emphasis"><em>d S<sub>​2</sub> / dt = S<sub>​1</sub></em></span> |
| </p> |
| <p> |
| In order to efficiently solve stiff systems numerically the Jacobian |
| </p> |
| <p> |
| <span class="emphasis"><em>J = d f<sub>​i</sub> / d x<sub>​j</sub></em></span> |
| </p> |
| <p> |
| is needed. Here is the definition of the above example |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">vector_type</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">matrix_type</span><span class="special">;</span> |
| |
| <span class="keyword">struct</span> <span class="identifier">stiff_system</span> |
| <span class="special">{</span> |
| <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">vector_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">vector_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="comment">/* t */</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">dxdt</span><span class="special">[</span> <span class="number">0</span> <span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="number">101.0</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span> <span class="number">0</span> <span class="special">]</span> <span class="special">-</span> <span class="number">100.0</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span> <span class="number">1</span> <span class="special">];</span> |
| <span class="identifier">dxdt</span><span class="special">[</span> <span class="number">1</span> <span class="special">]</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">[</span> <span class="number">0</span> <span class="special">];</span> |
| <span class="special">}</span> |
| <span class="special">};</span> |
| |
| <span class="keyword">struct</span> <span class="identifier">stiff_system_jacobi</span> |
| <span class="special">{</span> |
| <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">vector_type</span> <span class="special">&</span> <span class="comment">/* x */</span> <span class="special">,</span> <span class="identifier">matrix_type</span> <span class="special">&</span><span class="identifier">J</span> <span class="special">,</span> <span class="keyword">const</span> <span class="keyword">double</span> <span class="special">&</span> <span class="comment">/* t */</span> <span class="special">,</span> <span class="identifier">vector_type</span> <span class="special">&</span><span class="identifier">dfdt</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">J</span><span class="special">(</span> <span class="number">0</span> <span class="special">,</span> <span class="number">0</span> <span class="special">)</span> <span class="special">=</span> <span class="special">-</span><span class="number">101.0</span><span class="special">;</span> |
| <span class="identifier">J</span><span class="special">(</span> <span class="number">0</span> <span class="special">,</span> <span class="number">1</span> <span class="special">)</span> <span class="special">=</span> <span class="special">-</span><span class="number">100.0</span><span class="special">;</span> |
| <span class="identifier">J</span><span class="special">(</span> <span class="number">1</span> <span class="special">,</span> <span class="number">0</span> <span class="special">)</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> |
| <span class="identifier">J</span><span class="special">(</span> <span class="number">1</span> <span class="special">,</span> <span class="number">1</span> <span class="special">)</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span> |
| <span class="identifier">dfdt</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> |
| <span class="identifier">dfdt</span><span class="special">[</span><span class="number">1</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="special">};</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| The state type has to be a <code class="computeroutput"><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span></code> |
| and the matrix type must by a <code class="computeroutput"><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span></code> |
| since the stiff integrator only accepts these types. However, you might want |
| use non-stiff integrators on this system, too - we will do so later for demonstration. |
| Therefore we want to use the same function also with other state_types, realized |
| by templatizing the <code class="computeroutput"><span class="keyword">operator</span><span class="special">()</span></code>: |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">vector_type</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">matrix_type</span><span class="special">;</span> |
| |
| <span class="keyword">struct</span> <span class="identifier">stiff_system</span> |
| <span class="special">{</span> |
| <span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">></span> |
| <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="identifier">State</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="special">...</span> |
| <span class="special">}</span> |
| <span class="special">};</span> |
| |
| <span class="keyword">struct</span> <span class="identifier">stiff_system_jacobi</span> |
| <span class="special">{</span> |
| <span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Matrix</span> <span class="special">></span> |
| <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="identifier">Matrix</span> <span class="special">&</span><span class="identifier">J</span> <span class="special">,</span> <span class="keyword">const</span> <span class="keyword">double</span> <span class="special">&</span><span class="identifier">t</span> <span class="special">,</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">dfdt</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="special">...</span> |
| <span class="special">}</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| Now you can use <code class="computeroutput"><span class="identifier">stiff_system</span></code> |
| in combination with <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code> or <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code>. |
| In the example the explicit time derivative of <span class="emphasis"><em>f(x,t)</em></span> |
| is introduced separately in the Jacobian. If <span class="emphasis"><em>df / dt = 0</em></span> |
| simply fill <code class="computeroutput"><span class="identifier">dfdt</span></code> with zeros. |
| </p> |
| <p> |
| A well know solver for stiff systems is the Rosenbrock method. It has a step |
| size control and dense output facilities and can be used like all the other |
| steppers: |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="identifier">vector_type</span> <span class="identifier">x</span><span class="special">(</span> <span class="number">2</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span> |
| |
| <span class="identifier">size_t</span> <span class="identifier">num_of_steps</span> <span class="special">=</span> <span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">make_dense_output</span><span class="special"><</span> <span class="identifier">rosenbrock4</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="special">>(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">)</span> <span class="special">,</span> |
| <span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">stiff_system</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">stiff_system_jacobi</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span> |
| <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">50.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">,</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">phoenix</span><span class="special">::</span><span class="identifier">arg_names</span><span class="special">::</span><span class="identifier">arg2</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">phoenix</span><span class="special">::</span><span class="identifier">arg_names</span><span class="special">::</span><span class="identifier">arg1</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="special">);</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| During the integration 71 steps have been done. Comparing to a classical |
| Runge-Kutta solver this is a very good result. For example the Dormand-Prince |
| 5 method with step size control and dense output yields 1531 steps. |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="identifier">vector_type</span> <span class="identifier">x2</span><span class="special">(</span> <span class="number">2</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span> |
| |
| <span class="identifier">size_t</span> <span class="identifier">num_of_steps2</span> <span class="special">=</span> <span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">make_dense_output</span><span class="special"><</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">vector_type</span> <span class="special">></span> <span class="special">>(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">)</span> <span class="special">,</span> |
| <span class="identifier">stiff_system</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x2</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">50.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">,</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">phoenix</span><span class="special">::</span><span class="identifier">arg_names</span><span class="special">::</span><span class="identifier">arg2</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">phoenix</span><span class="special">::</span><span class="identifier">arg_names</span><span class="special">::</span><span class="identifier">arg1</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="special">);</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| Note, that we have used <a href="http://www.boost.org/doc/libs/release/libs/phoenix/" target="_top">Boost.Phoenix</a>, |
| a great functional programming library, to create and compose the observer. |
| </p> |
| <p> |
| The full example can be found here: <a href="https://github.com/headmyshoulder/odeint-v2/blob/master/examples/stiff_system.cpp" target="_top">stiff_system.cpp</a> |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2009-2012 Karsten |
| Ahnert and Mario Mulansky<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
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