Determination of optimal step size
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## Determination of optimal step size

There are three possible outcomes of the step size comparison in the
numerical integration:

- the step size should be reduced
- the step size should be increased
- the step size was acceptable

In practice, the step size is always either increased or decreased.
There is a limit to how much it can be increased or decreased,
however. It will not be increased by a factor of more than four in one
iteration. It will not be decreased by a factor of more than ten in
one iteration. If the step size needs to be decreased by more than a
factor of ten, then the current iteration is considered to have
failed, and the * current* step size is decreased. The calculations
are then started over again.

Another test is made prior to integrating. This test either passes or
fails the current step size. A constraint is made that when two particles
pass ``through'' each other at the centre of mass, that they do it
with a small step size. The reasons for this will be explained in the
section . This constraint is maintained by failing the current
step size (reducing it by half, and causing the calculations to be
restarted) if the current step size times the velocity, causes the
particle to pass its partner.

overshoot3Potential as a function of time, showing
overshoot problem
There is a third control on the step size that can be seen in the main
loop above; the simulation is made to approach a change in pairing at
a slow pace. In effect, the simulation is performing a binary search
of the position space, to determine precisely where the sharp critical
point in the potential function is. By approaching this point slowly,
and passing through it with a very small step size, the error due to
overshooting is minimized. Figure shows two
overshoots that can happen. It would be fortunate if the two types
cancelled, but this can not really be expected.

*mcr@ccs.carleton.ca*