08-197 Ulrich Mutze

An asynchronous leap-frog method (336K, pdf) 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.

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