Beeman's algorithm

Beeman's Algorithm is a method for numerically integrating ordinary differential equations, generally position and velocity, which is closely related to verlet integration. It is generally more accurate in both position and velocity than most verlet schemes. It's most commonly seen in molecular dynamics simulations.

Equation

$r(t+\Delta t)=r(t)+v(t)\Delta t+{\frac {2}{3}}a(t)\Delta t^{2}-{\frac {1}{6}}a(t-\Delta t)\Delta t^{2}+O(\Delta t^{4})$

$v(t+\Delta t)=v(t)+v(t)\Delta t+{\frac {1}{3}}a(t)\Delta t+{\frac {5}{6}}a(t)\Delta t-{\frac {1}{6}}a(t-\Delta t)\Delta t+O(\Delta t^{3})$

where

• $t$ is present time (ie: independant variable)
• $\Delta t$ is the time step size
• $r(t)$ is the position at time t
• $v(t)$ is the velocity at time t
• $a(t)$ is the acceleration at time t

Error term

As shown above, the error term is $O(\Delta t^{4})$ for position and $O(\Delta t^{3})$ for velocity. In comparison, Verlet is $O(\Delta t^{4})$ for position and $O(\Delta t^{2})$ for velocity; Velocity Verlet method is $O(\Delta t^{3})$ for both position and velocity. In exchange for greater accuracy, Beeman's algorithm is moderately computationally more expensive.

Memory Requirements

The simulation must keep track of position, velocity, acceleration and previous acceleration vectors per particle, keeping it's memory requirements on par with velocity verlet and slightly more expensive than the original verlet method.

References

• Molecular Dynamics Simulations (near bottom of page)
• Integration Algorithms