Beeman's algorithm
Beeman's algorithm
In its standard form, it produces the same trajectories as the Verlet algorithm, but the velocities are more accurate:
\[x(t+\Delta t) = x(t) + v(t) \Delta t + \left(\frac{2}{3}a(t) - \frac{1}{6} a(t - \Delta t) \right)\Delta t^2 + O( \Delta t^4) \] \[v(t + \Delta t) = v(t) + \left(\frac{1}{3}a(t + \Delta t) + \frac{5}{6}a(t) - \frac{1}{6}a(t - \Delta t) \right) \Delta t + O(\Delta t^3)\]
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:
\[ x(t+\Delta t) = x(t) + v(t) \Delta t + \frac{2}{3}a(t) \Delta t^2 - \frac{1}{6} a(t - \Delta t) \Delta t^2 + O( \Delta t^4).\]
The velocities at time \(t =t + \Delta t\) are then calculated from the positions.
\[ v(t + \Delta t)_{(\mathrm{predicted})} = v(t) + \frac{3}{2}a(t) \Delta t - \frac{1}{2}a(t - \Delta t) \Delta t + O( \Delta t^3)\]
The accelerations at time \(t =t + \Delta t\) are then calculated from the positions and predicted velocities.
\[ v(t + \Delta t)_{(\mathrm{corrected})} = v(t) + \frac{1}{3}a(t + \Delta t) \Delta t + \frac{5}{6}a(t) \Delta t - \frac{1}{6}a(t - \Delta t) \Delta t + O( \Delta t^3) \]
[edit] See also
[edit] References
- ↑ D. Beeman "Some multistep methods for use in molecular dynamics calculations", Journal of Computational Physics 20 pp. 130-139 (1976)
