| [/============================================================================ |
| Boost.odeint |
| |
| Copyright 2011-2013 Karsten Ahnert |
| Copyright 2011-2012 Mario Mulansky |
| Copyright 2012 Sylwester Arabas |
| |
| Use, modification and distribution is subject to the Boost Software License, |
| Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at |
| http://www.boost.org/LICENSE_1_0.txt) |
| =============================================================================/] |
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| [section Stiff systems] |
| |
| [import ../examples/stiff_system.cpp] |
| |
| 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: |
| |
| ['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 |
| `ublas::matrix` 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 `operator()`: |
| |
| [stiff_system_alternative_definition] |
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| Now you can use `stiff_system` in combination with `std::vector` or |
| `boost::array`. In the example the explicit time derivative of ['f(x,t)] is |
| introduced separately in the Jacobian. If ['df / dt = 0] simply fill `dfdt` with zeros. |
| |
| 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] |