Leapfrog integration
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In numerical analysis, leapfrog integration is a method for numerically integrating differential equations of the form
or equivalently of the form
particularly in the case of a dynamical system of classical mechanics.
The method is known by different names in different disciplines. In particular, it is similar to the velocity Verlet method, which is a variant of Verlet integration. Leapfrog integration is equivalent to updating positions and velocities at different interleaved time points, staggered in such a way that they "leapfrog" over each other.
Leapfrog integration is a second-order method, in contrast to Euler integration, which is only first-order, yet requires the same number of function evaluations per step. Unlike Euler integration, it is stable for oscillatory motion, as long as the time-step is constant, and .[1]
Using Yoshida coefficients, applying the leapfrog integrator multiple times with the correct timesteps, a much higher order integrator can be generated.