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.

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

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

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:

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

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

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

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

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

See also

• Velocity Verlet algorithm

References

• Beeman's algorithm entry on wikipedia