Beeman's algorithm: Difference between revisions
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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]] | ||
Revision as of 10:52, 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.
- 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) - \frac{1}{6} a(t - \Delta t) )\Delta t^2 + O( \Delta t^4) }
- 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) = 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.
- 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) (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) }