Beeman's algorithm: Difference between revisions
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(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...) |
(Interesting: same positions as Verlet) |
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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''' <ref>[http://dx.doi.org/10.1016/0021-9991(76)90059-0 D. Beeman "Some multistep methods for use in molecular dynamics calculations", Journal of Computational Physics '''20''' pp. 130-139 (1976)]</ref> is is a method for [[Integrators for molecular dynamics |numerically integrating ordinary differential equations]], generally position and velocity, which is closely related to Verlet integration. | ||
: | In its standard form, it produces the same trajectories as the Verlet algorithm, but the velocities are more accurate: | ||
where ''x'' is the position, ''v'' is the velocity, ''a'' is the acceleration, ''t'' is time, and | :<math>x(t+\Delta t) = x(t) + v(t) \Delta t + \left(\frac{2}{3}a(t) - \frac{1}{6} a(t - \Delta t) \right)\Delta t^2 + O( \Delta t^4) </math> | ||
:<math>v(t + \Delta t) = v(t) + \left(\frac{1}{3}a(t + \Delta t) + \frac{5}{6}a(t) - \frac{1}{6}a(t - \Delta t) \right) \Delta t + O(\Delta t^3)</math> | |||
where ''x'' is the position, ''v'' is the velocity, ''a'' is the acceleration, ''t'' is time, and <math>\Delta t</math> is the [[Time step|time-step]]. | |||
A predictor-corrector variant is useful when the forces are velocity-dependent: | A predictor-corrector variant is useful when the forces are velocity-dependent: | ||
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The velocities at time <math>t =t + \Delta t</math> are then calculated from the positions. | The velocities at time <math>t =t + \Delta t</math> are then calculated from the positions. | ||
:<math> 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)</math> | :<math> v(t + \Delta t)_{(\mathrm{predicted})} = v(t) + \frac{3}{2}a(t) \Delta t - \frac{1}{2}a(t - \Delta t) \Delta t + O( \Delta t^3)</math> | ||
The accelerations at time <math>t =t + \Delta t</math> are then calculated from the positions and predicted velocities. | The accelerations at time <math>t =t + \Delta t</math> are then calculated from the positions and predicted velocities. | ||
:<math> 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) </math> | :<math> v(t + \Delta t)_{(\mathrm{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) </math> | ||
==See also== | ==See also== | ||
*[[Velocity Verlet algorithm]] | *[[Velocity Verlet algorithm]] | ||
==References== | |||
<references/> | |||
==External links== | |||
*[http://en.wikipedia.org/wiki/Beeman%27s_algorithm Beeman's algorithm entry on wikipedia] | |||
[[category: Molecular dynamics]] | [[category: Molecular dynamics]] | ||
Latest revision as of 13:01, 19 April 2010
Beeman's algorithm [1] is is a method for numerically integrating ordinary differential equations, generally position and velocity, which is closely related to Verlet integration.
In its standard form, it produces the same trajectories as the Verlet algorithm, but the velocities are more accurate:
where x is the position, v is the velocity, a is the acceleration, t is time, and 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.