Innumerical analysis,leapfrog integration is amethod for numerically integratingdifferential equations of the form$${\displaystyle \ddot{x}=\frac{d^{2}x}{d{t}^2}=A(x),}$$or equivalently of the form$${\displaystyle \dot{v}=\frac{dv}{dt}=A(x),\dot{x}=\frac{dx}{dt}=v,}$$particularly in the case of adynamical system ofclassical mechanics.

The method is known by different names in different disciplines. In particular, it is similar to thevelocity Verlet method, which is a variant ofVerlet 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 asecond-order method, in contrast toEuler 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${\displaystyle \mathrm{\Delta }t}$ is constant, and.^[1]

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${\displaystyle x_i}$ is position at step${\displaystyle i}$,${\displaystyle v_{i+1/2}}$ is the velocity, or first derivative of${\displaystyle x}$, at step${\displaystyle i+1/2}$,${\displaystyle a_i=A(x_i)}$ is the acceleration, or second derivative of${\displaystyle x}$, at step${\displaystyle i}$, and${\displaystyle \mathrm{\Delta }t}$ is the size of each time step. These equations can be expressed in a form that gives velocity at integer steps as well:^[2]However, in this synchronized form, the time-step${\displaystyle \mathrm{\Delta }t}$ must be constant to maintain stability.^[3]

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 (${\displaystyle \mathrm{\Delta }t\to \mathrm{\Delta }t/2}$), 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 thetime-reversibility of the Leapfrog method. One can integrate forwardn steps, and then reverse the direction of integration and integrate backwardsn steps to arrive at the same starting position. The second strength is itssymplectic nature, which implies that it conserves the (slightly modified; seesymplectic integrator) energy of a Hamiltonian dynamical system.^[4] 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 asymplectic integrator, leapfrog integration is also used inHamiltonian Monte Carlo, a method for drawing random samples from a probability distribution whose overall normalization is unknown.^[5]

The leapfrog integrator can be converted into higher order integrators using techniques due toHaruo 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.^[6][7]

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

where${\displaystyle x_i,v_i}$ are the starting position and velocity,${\displaystyle x_i^n,v_i^n}$ are intermediary position and velocity at intermediary step${\displaystyle n}$,${\displaystyle a(x_i^n)}$ is the acceleration at the position${\displaystyle x_i^n}$, and${\displaystyle x_{i+1},v_{i+1}}$ are the final position and velocity under one 4th order Yoshida step.

Coefficients${\displaystyle (c_1,c_2,c_3,c_4)}$ and${\displaystyle (d_1,d_2,d_3)}$ are derived in^[7] (see the equation (4.6))

All intermediary steps form one${\displaystyle \mathrm{\Delta }t}$ step which implies that coefficients sum up to one:$\sum _{i=1}^4{c}_i=1$ and$\sum _{i=1}^3{d}_i=1$. Note that position and velocity are computed at different times and some intermediary steps are backwards in time. To illustrate this, we give the numerical values of${\displaystyle c_n}$ coefficients:${\displaystyle c_1=0.6756}$,${\displaystyle c_2=-0.1756}$,${\displaystyle c_3=-0.1756}$,${\displaystyle c_4=\mathrm{0.6756.}}$

• Numerical methods for ordinary differential equations
• Symplectic integration
• Euler integration
• Verlet integration
• Runge–Kutta integration

• https://www.physics.drexel.edu/students/courses/Comp_Phys/Integrators/leapfrog/ (archived17 June 2010), Drexel University Physics

• Time reversible numerical differential equations

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