Beeman's algorithm

Beeman's algorithm is is a method for numerically integrating ordinary differential equations, generally position and velocity, which is closely related to Verlet integration.

x (t + Δ t) = x (t) + v (t) Δ t + \frac{2}{3} a (t) Δ t^{2} - \frac{1}{6} a (t - Δ t) Δ t^{2} + O (Δ t^{4})

v (t + Δ t) = v (t) + \frac{1}{3} a (t + Δ t) Δ t + \frac{5}{6} a (t) Δ t - \frac{1}{6} a (t - Δ t) Δ t + O (Δ t^{3})

where x is the position, v is the velocity, a is the acceleration, t is time, and \Delta t is the time-step.

A predictor-corrector variant is useful when the forces are velocity-dependent:

x (t + Δ t) = x (t) + v (t) Δ t + \frac{2}{3} a (t) Δ t^{2} - \frac{1}{6} a (t - Δ t) Δ t^{2} + O (Δ t^{4}) .

The velocities at time $t = t + Δ t$ are then calculated from the positions.

v (t + Δ t) (p r e d i c t e d) = v (t) + \frac{3}{2} a (t) Δ t - \frac{1}{2} a (t - Δ t) Δ t + O (Δ t^{3})

The accelerations at time $t = t + Δ t$ are then calculated from the positions and predicted velocities.

v (t + Δ t) (c o r r e c t e d) = v (t) + \frac{1}{3} a (t + Δ t) Δ t + \frac{5}{6} a (t) Δ t - \frac{1}{6} a (t - Δ t) Δ t + O (Δ t^{3})

Beeman's algorithm

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Beeman's algorithm

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