Leapfrog integration

In mathematics 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 ${\displaystyle x(t)}$ and velocities ${\displaystyle v(t)={\dot {x}}(t)}$ at 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 ${\displaystyle \Delta t}$ is constant, and ${\displaystyle \Delta t\leq 2/\omega }$.

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