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 second-order accuracy for the position and first-order 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.
method whose error is bounded is the half-step method.
In this method the average velocity during an interval is taken
to be the velocity in the middle of the interval. The half-step
method can be written as
Note that the half-step method is not self-starting, 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 half-step method is stable, it is a common textbook method.