Splitting the Force

Streett and co-workers proposed a splitting of forces based on a distance parameter j-gpiit. In the potential energy formalism we write 

As a practical matter, particles will move in and out of the rspUt sphere for a given particle over the course of a simulation. To avoid discontinuities that result from a particle suddenly changing classification from the slow force component to the fast force component, the force can be decomposed into fast and slow components using a switching function S(r), 

The general plan for a multiple time-step numerical method is that will be evaluated at every step of the integration at time increments h, while F will be evaluated less frequently, typically at time increments xh where X 1 is an integer. The key question is How should be incorporated into the numerical dynamics In the original work of Streett et al., the slow force on particle i was approximated by a truncated Taylor series at each step 0 x, between updates at steps t and t + xh  

A natural simplification is to truncate the Taylor series after the constant term, resulting in a constant extrapolation of the slow force. The velocity Verlet method [6] can be easily modified to implement this constant extrapolation multiple time-step method  

The important feature to note is that the fast forces are computed at each step, while the slow forces are computed t times less frequently, with updates given by 

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This interaction splits the force constants, k of the individual oscillators into a larger one, k+ = k + 2q2/R5, and a smaller one, k = k - 2q2/R5. The oscillators then have modified frequencies ... 

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