Leapfrog integration

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 . Using Yoshida coefficients, applying the leapfrog integrator multiple times with the correct timesteps, a much higher order integrator can be generated. In leapfrog integration, the equations for updating position and velocity are where is position at step , is the velocity, or first derivative of , at step , is the acceleration, or second derivative of , at step , and is the size of each time step. These equations can be expressed in a form that gives velocity at integer steps as well: However, in this synchronized form, the time-step must be constant to maintain stability. The synchronised form can be re-arranged to the 'kick-drift-kick' form; which is primarily used where variable time-steps are required. The separation of the acceleration calculation onto the beginning and end of a step means that if time resolution is increased by a factor of two (), then only one extra (computationally expensive) acceleration calculation is required. One use of this equation is in gravity simulations, since in that case the acceleration depends only on the positions of the gravitating masses (and not on their velocities), although higher-order integrators (such as Runge–Kutta methods) are more frequently used.