Beeman's algorithm is a method for numerically integrating ordinary differential equations of order 2, more specifically Newton's equations of motion
It was designed to allow high numbers of particles in simulations of molecular dynamics.
There is a direct or explicit and an implicit variant of the method.
The direct variant was published by Schofield in 1973[1] as a personal communication from Beeman.
It is a variant of the Verlet integration method.
It produces identical positions, but uses a different formula for the velocities.
The formula used to compute the positions at time
in the full predictor-corrector[2] scheme is: Using only the predictor formula and the corrector for the velocities one obtains a direct or explicit method[1] which is a variant of the Verlet integration method:[3] This is the variant that is usually understood as Beeman's method.
Beeman[2] also proposed to alternatively replace the velocity update in the last equation by the second order Adams–Moulton method: where In systems where the forces are a function of velocity in addition to position, the above equations need to be modified into a predictor–corrector form whereby the velocities at time
are predicted and the forces calculated, before producing a corrected form of the velocities.
As shown above, the local error term is
In exchange for greater accuracy, Beeman's algorithm is moderately computationally more expensive.
The simulation must keep track of position, velocity, acceleration and previous acceleration vectors per particle (though some clever workarounds for storing the previous acceleration vector are possible), keeping its memory requirements on par with velocity Verlet and slightly more expensive than the original Verlet method.