Practical Issues in Propagation

One such method, first used by Verlet (1967), considers the sum of the Taylor expansions corresponding to forward and reverse time steps At. In that sum, all odd-order derivatives disappear since the odd powers of At have opposite sign in the two Taylor expansions. Rearranging terms and truncating at second order (which is equivalent to tmneating at third-order, since the third-order term has a coefficient of zero) yields 

for any particle, each subsequent position is determined by the current position, the previous position, and the particle s acceleration (determined from the forces on the particle and Eq. (3.13)). For the very first step (for which no position q(t — At) is available) one might use Eqs. (3.16) and (3.17). 

The Verlet scheme propagates the position vector with no reference to the particle velocities. Thus, it is particularly advantageous when the position coordinates of phase space are of more interest than the momentum coordinates, e.g., when one is interested in some property that is independent of momentum. However, often one wants to control the simulation temperature. This can be accomplished by scaling the particle velocities so that the temperature, as defined by Eq. (3.18), remains constant (or changes in some defined manner), as described in more detail in Section 3.6.3. To propagate the position and velocity vectors in a coupled fashion, a modification of Verlet s approach called the leapfrog algorithm has been proposed. In this case, Taylor expansions of the position vector truncated at second order 

Note that in die leapfrog method, position depends on the velocities as computed one-half time step out of phase, dins, scaling of the velocities can be accomplished to control temperature. Note also that no force-deld calculations actually take place for the fractional time steps. Eorces (and thus accelerations) in Eq. (3.24) are computed at integral time steps, halftime-step-forward velocities are computed therefrom, and these are then used in Eq. (3.23) to update the particle positions. The drawbacks of the leapfrog algorithm include ignoring third-order terms in the Taylor expansions and the half-time-step displacements of the position and velocity vectors - both of these features can contribute to decreased stability in numerical integration of the trajectoiy. 

A different method of increasing the time step without decreasing the numerical stability is to remove from the system those degrees of freedom having the highest frequency (assuming. 

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