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63 lines
2.4 KiB
Plaintext
63 lines
2.4 KiB
Plaintext
[/============================================================================
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Boost.odeint
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Copyright 2011-2013 Karsten Ahnert
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Copyright 2011-2012 Mario Mulansky
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Copyright 2012 Sylwester Arabas
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Use, modification and distribution is subject to the Boost Software License,
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Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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=============================================================================/]
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[section Stiff systems]
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[import ../examples/stiff_system.cpp]
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An important class of ordinary differential equations are so called stiff
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system which are characterized by two or more time scales of different
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order. Examples of such systems are found in chemical systems where reaction
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rates of individual sub-reaction might differ over large ranges, for example:
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['d S[subl 1] / dt = - 101 S[subl 2] - 100 S[subl 1]]
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['d S[subl 2] / dt = S[subl 1]]
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In order to efficiently solve stiff systems numerically the Jacobian
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['J = d f[subl i] / d x[subl j]]
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is needed. Here is the definition of the above example
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[stiff_system_definition]
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The state type has to be a `ublas::vector` and the matrix type must by a
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`ublas::matrix` since the stiff integrator only accepts these types.
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However, you might want use non-stiff integrators on this system, too - we will
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do so later for demonstration. Therefore we want to use the same function also
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with other state_types, realized by templatizing the `operator()`:
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[stiff_system_alternative_definition]
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Now you can use `stiff_system` in combination with `std::vector` or
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`boost::array`. In the example the explicit time derivative of ['f(x,t)] is
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introduced separately in the Jacobian. If ['df / dt = 0] simply fill `dfdt` with zeros.
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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:
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[integrate_stiff_system]
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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.
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[integrate_stiff_system_alternative]
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Note, that we have used __boost_phoenix, a great functional programming library, to create and compose the observer.
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The full example can be found here: [github_link examples/stiff_system.cpp stiff_system.cpp]
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[endsect]
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