Leapfrog integration

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In mathematics leapfrog integration is a method for numerically integrating differential equations of the form

${\displaystyle {\ddot {x}}=d^{2}x/dt^{2}=F(x)}$,

or equivalently of the form

${\displaystyle {\dot {v}}=dv/dt=F(x),\;{\dot {x}}=dx/dt=v}$,

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 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 ${\displaystyle \Delta t\leq 2/\omega }$.[1]

In leapfrog integration, the equations for updating position and velocity are

{\displaystyle {\begin{aligned}x_{i}&=x_{i-1}+v_{i-1/2}\,\Delta t,\\a_{i}&=F(x_{i}),\\v_{i+1/2}&=v_{i-1/2}+a_{i}\,\Delta t,\end{aligned}}}

where ${\displaystyle x_{i}}$ is position at step ${\displaystyle i}$, ${\displaystyle v_{i+1/2\,}}$ is the velocity, or first derivative of ${\displaystyle x}$, at step ${\displaystyle i+1/2\,}$, ${\displaystyle a_{i}=F(x_{i})}$ is the acceleration, or second derivative of ${\displaystyle x}$, at step ${\displaystyle i}$, and ${\displaystyle \Delta t}$ is the size of each time step. These equations can be expressed in a form that gives velocity at integer steps as well:[2]

{\displaystyle {\begin{aligned}x_{i+1}&=x_{i}+v_{i}\,\Delta t+{\tfrac {1}{2}}\,a_{i}\,\Delta t^{\,2},\\v_{i+1}&=v_{i}+{\tfrac {1}{2}}(a_{i}+a_{i+1})\,\Delta t.\end{aligned}}}

However, even in this synchronized form, the time-step ${\displaystyle \Delta t}$ must be constant to maintain stability.[3]

The synchronised form can be re-arranged to the 'kick-drift-kick' form;

{\displaystyle {\begin{aligned}v_{i+1/2}&=v_{i}+a_{i}{\frac {\Delta t}{2}},\\x_{i+1}&=x_{i}+v_{i+1/2}\Delta t,\\v_{i+1}&=v_{i+1/2}+a_{i+1}{\frac {\Delta t}{2}},\end{aligned}}}

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 (${\displaystyle \Delta t\rightarrow \Delta t/2}$), 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.

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) energy of dynamical systems. 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.[4]