Leapfrog integration

In numerical analysis, leapfrog integration is a method for numerically integrating differential equations of the form

{\ddot {x}}={\frac {d^{2}x}{dt^{2}}}=A(x),

or equivalently of the form

{\dot {v}}={\frac {dv}{dt}}=A(x),\;{\dot {x}}={\frac {dx}{dt}}=v,

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

Using Yoshida coefficients, applying the leapfrog integrator multiple times with the correct timesteps, a much higher order integrator can be generated.

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