[/============================================================================
  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)
=============================================================================/]



[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]]

['d S[subl 2] / dt = S[subl 1]]


In order to efficiently solve stiff systems numerically the Jacobian

['J = d f[subl i] / d x[subl j]]

is needed. Here is the definition of the above example

[stiff_system_definition]

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]

Now you can use `stiff_system` in combination with `std::vector` or
`std::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:

[integrate_stiff_system]

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.

[integrate_stiff_system_alternative]

Note, that we have used __boost_phoenix, a great functional programming library, to create and compose the observer.

The full example can be found here: [github_link examples/stiff_system.cpp stiff_system.cpp]


[endsect]
