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.

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

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

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:

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

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

$v(t+\Delta t)(predicted)=v(t)+{\frac {3}{2}}a(t)\Delta t-{\frac {1}{2}}a(t-\Delta t)\Delta t+O(\Delta t^{3})$

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

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

See also

• Velocity Verlet algorithm