Velocity Verlet

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

The velocity Verlet algorithm may be derived by considering a standard approximate decomposition of the Liouville operator which preserves reversibility and is symplectic (which implies that volume in phase space is conserved). This approach [47] has had several beneficial consequences. 

A straightforward derivation (not reproduced here) shows that the effect of the diree successive steps embodied in equation (b3.3.7), with the above choice of operators, is precisely the velocity Verlet algorithm. This approach is particularly usefiil for generating multiple time-step methods. 

This actually translates into a fairly simple algorithm, based closely on the standard velocity Verlet method. Written in a Fortran-like pseudo-code, it is as follows. At tire start of the run we calculate both rapidly-varying (1) and slowly-varying (F) forces, then, in the main loop ... 

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 ... 

There is a number of algorithms to solve equations (1) and (2) that differ appreciably in their properties which are beyond the scope of the present article. In the discussion below we use the velocity Verlet algorithm. However, better approaches can be employed [2-5]. We define a rule - F X t), At) that modifies X t) to X t + At) and repeat the application of this rule as desired. For example the velocity Verlet algorithm ( rule ) is ... 

This procedure is then repeated after each time step. Comparison with Eq. (2) shows that the result is the velocity Verlet integrator and we have thus derived it from a split-operator technique which is not the way that it was originally derived. A simple interchange of the Ly and L2 operators yields an entirely equivalent integrator. 

Fig. 1. CPU times (in hours) for 1 ps MD runs for various proteins using three different methods, direct velocity Verlet with a time-step 0.5 fs, r-RESPA with direct evaluation of electrostatic forces and an overall time-step of 4.0 fs, and r-RESPA/TFMM with an overall time-step 4.0 fs (combination of (2,2,2,2) in force breakup).The energy conservation parameter log AE for the three methods are comparable. The CPU time (hours) is for RISC6000 /MODEL 590 computer.

A timestep of size At with the Verlet method ( velocity Verlet ) takes iqo,Po) to (51,pi) and can be divided into three steps (1) a kick ... 

The velocity Verlet method [Swope et al. 1982] gives positions, velocities and accelerations at the same time and does not compromise precision ... 

The velocity Verlet method is actually implemented as a three-stage procedure because, as can be seen from Equation (7.15), to calculate the new velocities requires the accelerations at both t and t + 8t. Thus in the first step the positions at f I- are calculated according to Equation (7.14) using the velocities and the accelerations at time t. The velocities at time t + 6t are then determined using ... 

The underlying theory of r-RESPA is somewhat involved, but the final result and const quent implementation is actually rather straightforward, being very closely related to th velocity Verlet integration scheme. For our four-way decomposition the algorithm woul... 

Our discussion so far has considered the use of SHAKE with the Verlet algorithm Versions have also been derived for other integration schemes, such as the leap-froj algorithm, the predictor-corrector methods and the velocity Verlet algorithm. In the cast of the velocity Verlet algorithm, the method has been named RATTLE [Anderson 1983]... 

The equations of motion are integrated using a modified velocity Verlet algorithm. The modification is required because the force depends upon the velocity the extra step involves... 

Dissipative particle dynamics (DPD) is a technique for simulating the motion of mesoscale beads. The technique is superficially similar to a Brownian dynamics simulation in that it incorporates equations of motion, a dissipative (random) force, and a viscous drag between moving beads. However, the simulation uses a modified velocity Verlet algorithm to ensure that total momentum and force symmetries are conserved. This results in a simulation that obeys the Navier-Stokes equations and can thus predict flow. In order to set up these equations, there must be parameters to describe the interaction between beads, dissipative force, and drag. 

If the magnitudes of the dissipative force, random noise, or the time step are too large, the modified velocity Verlet algorithm will not correctly integrate the equations of motion and thus give incorrect results. The values that are valid depend on the particle sizes being used. A system of reduced units can be defined in which these limits remain constant. 

As j approaches zero, this method reduces to the velocity Verlet algorithm ... 

Figure 1 A stepwise view of the Verlet integration algorithm and its variants, (a) The basic Verlet method, (b) Leap-frog integration, (c) Velocity Verlet integration. At each algorithm dark and light gray cells indicate the initial and calculating variables, respectively. The numbers in the cells represent the orders m the calculation procedures. The arrows point from the data that are used in the calculation of the variable that is being calculated at each step.

An even better handling of the velocities is obtained by another variant of the basic Verlet integrator, known as the velocity Verlet algorithm. This is a Verlet-type algorithm that stores positions, velocities, and accelerations all at the same time t and minimizes roundoff errors [14]. The velocity Verlet algorithm is written... 

Again, elimination of the velocities from these equations recovers the Verlet algorithm. In practice, the velocity Verlet algorithm consists of the following steps ... 

At this point, the kinetic energy at time t + At is available. Figure Ic gives a graphical representation of the steps involved in the velocity Verlet propagation. 

The first term m V is a function of x only. Let us assume that we are using the velocity Verlet time integrator, which is the most common. In that case, x is computed with local accuracy 0 dtA) and global accuracy 0(df2), and the velocity v at half-steps is computed with accuracy Oidt2 ) if the following approximation is used ... 

In this expression m5, V and a are functions of x and can computed at f and t+At. The velocity at the half-steps is directly provided by the velocity Verlet algorithm. To summarize, to calculate d(m )/df at time t + At/2 we need to collect data from time steps f - At, f, f + At, and f + 2At. Recall again that these equations are used to calculate the mean force along , not to advance the system in time. 

Algorithm 1 Velocity Verlet loop with ABF is the assignment operator. 

In tests using the moving ID Hamiltonian harmonic oscillator, (5.25), a velocity Verlet integrator [24] combined with ttapezoidal integration of W (/.) performed well when compared to the analytic solution. An interesting analysis of how... 

We have used various integrators (e.g., Runga-Kutta, velocity verlet, midpoint) to propagate the coupled set of first-order differential equations Eqs. (2.8) and (2.9) for the parameters of the Gaussian basis functions and Eq. (2.11) for the complex amplitudes. The specific choice is guided by the complexity of the problem and/or the stiffness of the differential equations. 

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