pyCBD/doc/examples/ContinuousTime.rst

Continuous Time Simulation
==========================
Given that continuous time simulation will always have to be discretized when
executed, it is important to ask *how* this discretization happens. In the
:doc:`SinGen` example, this was done by reducing the delta inbetween multiple
steps. However, this may compute too often for what is required in a given
simulation. Assume we have the following data (from a sine wave):

.. figure:: ../_figures/stepsize/sine.png
   :width: 600

Fixed Step Size
---------------
With the previously discussed technique, the data could be plot using a fixed
step size, where the new data is being computed every :code:`dt` time. The
smaller this :code:`dt` value is, the more precise the plot. Below, the same
function has been plot twice, for :code:`dt = 0.5` and :code:`dt = 0.1`, where
the marked points on the plot highlight *when* the data has been recomputed. It
is clear from these plots that a smaller step size identifies more accurate
results.

.. figure:: ../_figures/stepsize/sine-5.png
   :width: 600

.. figure:: ../_figures/stepsize/sine-1.png
   :width: 600

This behaviour is the default in the simulator. To explicitly set it, make use of
the :class:`CBD.stepsize.Fixed` class as follows (assuming :code:`sim` is the
simulator object):

.. code-block:: python

   sim.setStepSize(Fixed(0.5))
   #  OR, EQUIVALENTLY
   sim.setStepSize(0.5)

Yet, when the step size logic has been untouched since the creation of the
simulator instance, the normal :func:`CBD.simulator.Simulator.setDeltaT` can be
used as well.
This was done for academic reasons, as it is much easier to explain CBDs with a
fixed step size, as compared to varying step sizes.

Euler 2-Step
------------
Variable step size algorithms continuously increases or decreases the :code:`dt`
to allow for less steps to be taken. Whenever the simulation fluctuates a lot,
:code:`dt` will be small. However, if there are only small changes, it will be
large.

The `Euler 2-Step` does this by (approximately) halving/doubling the current
:code:`dt`, trying to get the best match for the given data. More information on
the actual algorithm can be found in the documentation for
:class:`CBD.stepsize.Euler2` and `professor Joel Feldman's notes on variable step
size algorithms <http://www.math.ubc.ca/~feldman/math/vble.pdf>`_.
It can be set on the simulator object as follows:

.. code-block:: python

   sim.setStepSize(Euler2(0.1, 0.005))

For the sine wave example, with any starting delta and an acceptance error epsilon
of :code:`0.005`, the following plot is obtained. Note that a decrease of the
epsilon will make the plot more precise.

.. figure:: ../_figures/stepsize/sine-euler2.png
   :width: 600