# Difference between revisions of "Beeman's algorithm"

(New page: '''Beeman's algorithm''' is is a method for numerically integrating ordinary differential equations, generally position and velocity, which is closely related to Verlet integration. :<mat...) |
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'''Beeman's algorithm''' is is a method for numerically integrating ordinary differential equations, generally position and velocity, which is closely related to Verlet integration. | '''Beeman's algorithm''' is is a method for numerically integrating ordinary differential equations, generally position and velocity, which is closely related to Verlet integration. | ||

− | :<math>x(t+\Delta t) = x(t) + v(t) \Delta t + \frac{2}{3}a(t) | + | |

− | :<math>v(t + \Delta t) = v(t) + \frac{1}{3}a(t + \Delta t) | + | :<math>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) </math> |

+ | :<math>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)</math> | ||

where ''x'' is the position, ''v'' is the velocity, ''a'' is the acceleration, ''t'' is time, and ''\Delta t'' is the time-step. | where ''x'' is the position, ''v'' is the velocity, ''a'' is the acceleration, ''t'' is time, and ''\Delta t'' is the time-step. |

## Revision as of 14:07, 17 April 2010

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

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:

The velocities at time are then calculated from the positions.

The accelerations at time are then calculated from the positions and predicted velocities.