  A second order explicit one-step numerical method for the initial value
problem of the general ordinary differential equation is proposed. It is
obtained by natural modifications of the well-known leapfrog method, which is a
second order, two-step, explicit method. According to the latter method, the
input data for an integration step are two system states, which refer to
different times. The usage of two states instead of a single one can be seen as
the reason for the robustness of the method. Since the time step size thus is
part of the step input data, it is complicated to change this size during the
computation of a discrete trajectory. This is a serious drawback when one needs
to implement automatic time step control.
  The proposed modification transforms one of the two input states into a
velocity and thus gets rid of the time step dependency in the step input data.
For these new step input data, the leapfrog method gives a unique prescription
how to evolve them stepwise.
  The stability properties of this modified method are the same as for the
original one: the set of absolute stability is the interval [-i,+i] on the
imaginary axis. This implies exponential growth of trajectories in situations
where the exact trajectory has an asymptote.
  By considering new evolution steps that are composed of two consecutive old
evolution steps we can average over the velocities of the sub-steps and get an
integrator with a much larger set of absolute stability, which is immune to the
asymptote problem.
  The method is exemplified with the equation of motion of a one-dimensional
non-linear oscillator describing the radial motion in the Kepler problem.
