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In numerical analysis, leapfrog integration is a method for numerically integrating differential equations of the form
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 x {\displaystyle x} and velocities v = x ˙ {\displaystyle v={\dot {x}}} 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 Δ t {\displaystyle \Delta t} is constant, and Δ t ≤ 2 / ω {\displaystyle \Delta t\leq 2/\omega }.
Using Yoshida coefficients, applying the leapfrog integrator multiple times with the correct timesteps, a much higher order integrator can be generated.