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<div class="titlepage"><div><div><h3 class="title">
<a name="boost_sandbox_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.
</p>
<p>
what are stiff systems?
</p>
<p>
examples
</p>
<p>
applications
</p>
<p>
To 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="identifier">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="identifier">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="identifier">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. With a little trick
you can simply make this functions valid for other state and matrix types,
just templatize 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="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>
</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">std</span><span class="special">::</span><span class="identifier">tr1</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 so called Rosenbrock method.
It has a step size control and dense output facilities and can be used like
all the other stepper:
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">rosenbrock4</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">rosenbrock4_controller</span><span class="special"><</span> <span class="identifier">stepper_type</span> <span class="special">></span> <span class="identifier">controlled_stepper_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">rosenbrock4_dense_output</span><span class="special"><</span> <span class="identifier">controlled_stepper_type</span> <span class="special">></span> <span class="identifier">dense_output_type</span><span class="special">;</span>
<span class="identifier">vector_type</span> <span class="identifier">x</span><span class="special">(</span> <span class="number">3</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">dense_output_type</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="keyword">typedef</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</span> <span class="identifier">vector_type</span> <span class="special">></span> <span class="identifier">dopri5_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">controlled_error_stepper</span><span class="special"><</span> <span class="identifier">dopri5_type</span> <span class="special">></span> <span class="identifier">controlled_dopri5_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">dense_output_controlled_explicit</span><span class="special"><</span> <span class="identifier">controlled_dopri5_type</span> <span class="special">></span> <span class="identifier">dense_output_dopri5_type</span><span class="special">;</span>
<span class="identifier">vector_type</span> <span class="identifier">x2</span><span class="special">(</span> <span class="number">3</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">dense_output_dopri5_type</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/1_46_1/libs/spirit/phoenix/doc/html/index.html" 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="../../../../examples/stiff_system.cpp" target="_top">../../examples/stiff_system.cpp</a>
</p>
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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>)
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