Editing Beeman's algorithm
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'''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. | ||
:<math>x(t+\Delta t) = x(t) + v(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) + | :<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 | 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: | 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) | :<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. | The accelerations at time <math>t =t + \Delta t</math> are then calculated from the positions and predicted velocities. | ||
:<math> v(t + \Delta t) | :<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== | ==See also== | ||
*[[Velocity Verlet algorithm]] | *[[Velocity Verlet algorithm]] | ||
[[category: Molecular dynamics]] | [[category: Molecular dynamics]] |