In numerical analysis, leapfrog integration is a method for numerically integrating differential equations 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 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.

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

Algorithm

is the acceleration, or second derivative is the size of each time step. These equations can be expressed in a form that gives velocity at integer steps as well: must be constant to maintain stability.

), then only one extra (computationally expensive) acceleration calculation is required.

One use of this equation is in Newtonian gravity simulations, since in that case the acceleration depends only on the positions of the gravitating masses (and not on their velocities).

There are two primary strengths to leapfrog integration when applied to mechanics problems. The first is the time-reversibility of the Leapfrog method. One can integrate forward n steps, and then reverse the direction of integration and integrate backwards n steps to arrive at the same starting position. The second strength is its symplectic nature, which implies that it conserves the (slightly modified; see symplectic integrator) energy of a Hamiltonian dynamical system. This is especially useful when computing orbital dynamics, as many other integration schemes, such as the (order-4) Runge–Kutta method, do not conserve energy and allow the system to drift substantially over time.

Because of its time-reversibility, and because it is a symplectic integrator, leapfrog integration is also used in Hamiltonian Monte Carlo, a method for drawing random samples from a probability distribution whose overall normalization is unknown.

Yoshida algorithms

The leapfrog integrator can be converted into higher order integrators using techniques due to Haruo Yoshida. In this approach, the leapfrog is applied over a number of different timesteps. It turns out that when the correct timesteps are used in sequence, the errors cancel and far higher order integrators can be easily produced.

4th order Yoshida integrator

One step under the 4th order Yoshida integrator requires four intermediary steps. The position and velocity are computed at different times. Only three (computationally expensive) acceleration calculations are required.

The equations for the 4th order integrator to update position and velocity are

are the final position and velocity under one 4th order Yoshida step.

are derived in (see the equation (4.6))

See also

• Numerical methods for ordinary differential equations

• Symplectic integration

• Euler integration

• Verlet integration

• Runge–Kutta integration

References

External links

• , Drexel University Physics