- 08-197 Ulrich Mutze
- An asynchronous leap-frog method
Oct 24, 08
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Abstract. 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
leap-frog 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
(we employ the terminology of dynamical systems).
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 leap-frog method gives a unique
prescription how to evolve them stepwise.
The method is exemplified with the equation of motion of a one-dimensional
non-linear oscillator describing the radial motion in the Kepler problem.
For this equation the modified leap-frog method is shown to be
significantly more accurate than the original method.
As a result, we have a second order explicit method that, just as the
simple explicit Euler method, needs only one evaluation of the
right-hand side of the differential equation per integration step,
and allows to change the time step without any additional computational
burden after each integration step.
Unlike the Euler method and the explicit Runge-Kutta methods it is robust
in the sense that it allows us to reliably model the dynamics of a
wide variety of physical systems over extended periods of time.