Editing Beeman's algorithm
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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 | '''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. | ||
In its standard form, it produces the same trajectories as the Verlet algorithm, but the velocities are more accurate: | In its standard form, it produces the same trajectories as the Verlet algorithm, but the velocities are more accurate: | ||
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:<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) + \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 | 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: |