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.

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

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

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t =t + \Delta t} are then calculated from the positions.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

References

• Beeman's algorithm entry on wikipedia