Leapfrog Verlet algorithm

The simplest of the numerical techniques for the integration of equations of motion is leapfrog-Verlet algorithm (LFV), which is known to be symplectic and of second order. The name leapfrog steams from the fact that coordinates and velocities are calculated at different times. 

We specify a time step, during which the forces are assumed to remain constant. Then r, and v, are updated. There are several schemes for this to overcome problems associated with finite rather than infinitesimal time steps. The force (and thus the acceleration) is assumed to remain constant throughout the time step At. For example, in the Leapfrog Verlet algorithm (e.g. Allen and Tildesley, Further reading),... 

Each of these operators is unitary U —t) = U t). Updating a time step with the propagator Uf( At)U At)Uf At) yields the velocity-Verlet algorithm. Concatenating the force operator for successive steps yields the leapfrog algorithm ... 

Now we are able to substitute B/t) in Eq. 8 from Eq. 9. After replacing the acceleration Rj (t) with the force F/ (t) we finally obtain Eq. 6. There are several others algorithms to integrate the equations of motion (e.g., leapfrog, Verlet). The consequences of different equation of motion integration schemes with regard to AMD are discussed in the excellent review of Remler and Madden (54). 

There are various, essentially equivalent, versions of the Verlet algorithm, including the original method employed by Verlet [13,44] in his investigations of the properties of the Leimard-Jones fluid, and a leapfrog form [45]. Here we concentrate on the velocity Verlet algorithm [46], which may be written... 

Incorporating a half timestep velocity into the r(t + St), v(t + St), and a(t - - St) calculations effectively corrects the final velocity using the final acceleration. This algorithm has the advantage over a few others, such as the leapfrog Verlet [1], that at the end of each timestep r, v, and a are all known, which makes calculating thermodynamic quantities easier. 

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... 

In practice molecular dynamics is run with finite time steps. Using the equations above would therefore lead to the introduction of inaccuracies (Biesiadecki and Skeel 1993). A number of algorithms have been developed to overcome this difficulty. One of the most widely used is the Verlet Leapfrog Algorithm (VLA), modified from Verlet s original algorithm (Verlet 1967) which uses the velocity at the mid-step v( + /2 ). 

Two modifications of the Verlet scheme are of wide use. The first is the leapfrog algorithm [3] where positions and velocities are not ealeulated at the same time velocities are evaluated at half-integer time steps ... 

The Verlet velocity algorithm overcomes the out-of-synchrony shortcoming of the Verlet leapfrog method. The advantage here is that the positions, velocities, and accelerations are computed at the same time t. There is no compromise on precision. The Verlet velocity algorithm is as follows ... 

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