Beemans Algorithm

This algorithm is closely related to the Verlet algorithm  

It provides a more accurate expression for the velocities and better energy conservation but is 

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Beeman s algorithm [Beeman 1976] is also related to the Verlet method ... 

The Beeman integration scheme uses a more accurate expression for the velocity. As a consequence it often gives better energy conservation, because the kinetic energy is calculated directly from the velocities. However, the expressions used are more complex than those of the Verlet algorithm and so it is computationally more expensive. 

In the past decades, a large number of methods have been proposed to achieve better energy conservation for example, the Gear family of algorithms and the family of Verlet algorithms (Frenkel and Smit, 1996). In our 3D code, we have incorporated yet another type of method developed by Beeman, which has a somewhat better energy conservation than the Verlet algorithm (Frenkel and Smit, 1996). In the Beeman method, the position and velocity of particle a are calculated via... 

Another scheme which is sometimes used in molecular dynamics is Beeman s Algorithm [30, 331],... 

Example 2.8 (Beeman s Algorithm) The method treats the positions and momenta differently, updating these from the formulas... 

This method requires that the positions (and forces) be known at two successive points h apart in time in order to initialize the iteration. These might be generated by using the Verlet method or some other self-starting scheme. Beeman s algorithm is explicit since, given q , q - andp , one directly obtains q + and then, q i, and thus p +i, with only one new force evaluation. Because it is a partitioned multistep method, its analysis is more involved than for the one-step methods, and, in particular its qualitative features are difficult to relate to those of the flow map. The order of accuracy of the scheme above can be shown to be three. 

V is a function of the position of all the atoms in the system. Due to the complicated nature of this function, there is no analytical solution of the equations of motion. It can be solved numerically using different algorithms such as the Verlet, leap-frog, or Beeman s procedure (Cramer 2002). 

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