Leapfrog integration

In numerical analysis, leapfrog integration is a method for numerically integrating differential equations of the form $${\displaystyle {\ddot {x}}={\frac {d^{2}x}{dt^{2}}}=A(x),}$$ or equivalently of the form $${\displaystyle {\dot {v}}={\frac {dv}{dt}}=A(x),\;{\dot {x}}={\frac {dx}{dt}}=v,}$$ 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 ${\displaystyle x(t)}$ and velocities ${\displaystyle v(t)={\dot {x}}(t)}$ 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 ${\displaystyle \Delta t}$ is constant, and ${\displaystyle \Delta t<2/\omega }$.^[1]

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