Leapfrog algorithm

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 ... 

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. 

After the momenta are selected from the distribution (8.39), the dynamics is propagated by a standard leapfrog algorithm (any symplectic and time-reversible integrator is suitable). The move is then accepted or rejected according to a criterion based on the detailed balance condition... 

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),... 

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). 

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. 

LEAPFROG algorithm (Sybyl) was used to screen virtual Selection of best monomers leading to MIPs with library of functional monomers high-binding capacity for the template... 

In general, boundary conditions are imposed at the end of each stage of (2.41) or the leapfrog time-step. Finally, in the case of absorbing boundary conditions, all derivatives are computed by the implicit algorithm across the entire domain including its interior and the absorber. Then, each system is updated by (2.41). 

An efficient way to circumvent this difficulty is to devise a gradually extending leapfrog algorithm, which can be reliably employed for any K value. For illustration, the fourth-order integrator is given by... 

This paragraph complements the analysis of Section 2.5 by presenting a technique for the construction of improved fourth-order spatial operators through the use of the discrete dispersion relation. Principally, the algorithm considers the ordinary leapfrog scheme for time marching, while it involves the parametric expression of (2.107) for spatial differentiation. By substituting plane-wave constituents in Maxwell s equations, the 2-D dispersion relation for an isotropic medium is... 

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... 

In practice, we use a more accurate computational scheme called the leapfrog algorithm. 

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