Midpoint and HalfStep Methods Perhaps the most obvious way to improve the Euler
method is to use the mean velocity during the interval to obtain
the new position. The corresponding midpoint method can
be written as Hence,
the midpoint method yields secondorder accuracy for the position
and firstorder accuracy for the velocity. Although the midpoint
approximation yields exact results for constant acceleration, it
usually does not yield much better results than the Euler method.
In fact, both methods are equally poor, because the error increases
with each time step.
A higherorder
method whose error is bounded is the halfstep method.
In this method the average velocity during an interval is taken
to be the velocity in the middle of the interval. The halfstep
method can be written as Note
that the halfstep method is not selfstarting, i.e., Eq. (10)
does not allow us to calculate
. The problem can be overcome by adopting the Euler algorithm
for the first half step:
Because
the halfstep method is stable, it is a common textbook method.


