Beeman's algorithm - Revision history

Dduque: Interesting: same positions as Verlet

2010-04-19T11:01:10Z

Interesting: same positions as Verlet

← Older revision		Revision as of 13:01, 19 April 2010
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	'''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.		'''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:

	:<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>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>

Nice and Tidy: Trivial tidy of LaTeX markup.

2010-04-19T10:32:48Z

Trivial tidy of LaTeX markup.

← Older revision		Revision as of 12:32, 19 April 2010
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	'''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.		'''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.

			:<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>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) + \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>
	:<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 <math>\Delta t</math> is the [[Time step\|time-step]].		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]].
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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==

Carl McBride: Added a couple of internal links.

2010-04-19T10:07:22Z

Added a couple of internal links.

← Older revision		Revision as of 12:07, 19 April 2010
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	'''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 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.


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	:<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>		:<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 <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:

Carl McBride: Added a reference

2010-04-19T08:52:33Z

Added a reference

← Older revision		Revision as of 10:52, 19 April 2010
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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 numerically integrating ordinary differential equations, generally position and velocity, which is closely related to Verlet integration.


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	==References==		==References==
			<references/>
			==External links==
	*[http://en.wikipedia.org/wiki/Beeman%27s_algorithm Beeman's algorithm entry on wikipedia]		*[http://en.wikipedia.org/wiki/Beeman%27s_algorithm Beeman's algorithm entry on wikipedia]
	[[category: Molecular dynamics]]		[[category: Molecular dynamics]]

Dduque: Link to wikipedia

2010-04-17T13:35:31Z

Link to wikipedia

← Older revision		Revision as of 15:35, 17 April 2010
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	==See also==		==See also==
	*[[Velocity Verlet algorithm]]		*[[Velocity Verlet algorithm]]

			==References==
			*[http://en.wikipedia.org/wiki/Beeman%27s_algorithm Beeman's algorithm entry on wikipedia]
	[[category: Molecular dynamics]]		[[category: Molecular dynamics]]

Dduque: common factor

2010-04-17T13:07:22Z

common factor

← Older revision		Revision as of 15:07, 17 April 2010
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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) ~~\Delta t^2~~ - \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) ~~\Delta t~~ + \frac{5}{6}a(t) ~~\Delta t~~ - \frac{1}{6}a(t - \Delta t) \Delta t + O(\Delta t^3)</math>		:<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.

Dduque: 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. :
2010-04-17T10:16:17Z

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...

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.

:<math>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) </math>
:<math>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)</math>

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:

:<math> 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).</math>

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>

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>

==See also==
*[[Velocity Verlet algorithm]]
[[category: Molecular dynamics]]