Verlet algorithms

The Verlet algorithm uses positions and accelerations at time t and the positions from time t - 5t to calculate new positions at time t + 5t. No explicit velocities are used. It is straightforward and modest on the storage requirements, but the precision is moderate. 

The fact that the velocities are not explicitly represented in the Verlet algorithm is one of the drawbacks to this method in that no velocities are available until the positions have been determined at the next time step. Also, in order to calculate the position of particles at t = d t, it is necessary to determine the positions at f = -St since the algorithm requires the position at time t- St to calculate the position at time t + St. Often, this drawback is overcome by using 

In an attempt to improve upon the original Verlet algorithm, several variations have been developed. The leap-frog algorithm (Hockney 1970) is one of the variations that uses the following equations to update the positions  

The velocity Verlet method (Swope et al. 1982), which is a variation of the standard Verlet method, calculates the positions, velocities, and accelerations at the same time by using the following equations  

Molecular Dynamics Simulation From Ab lnitio to Coarse Grained  

The velocity Verlet method is a three-stage algorithm because the calculation of the new velocities (O Eq. 7.28) requires both the acceleration at time t and at time t+8t. Therefore, first, the positions att+St re calculated using O Eq. 7.27 and the velocities and accelerations at time t. The velocities at time t + i d i are then calculated using 

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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. 

But the methods have not really changed. The Verlet algorithm to solve Newton s equations, introduced by Verlet in 1967 [7], and it s variants are still the most popular algorithms today, possibly because they are time-reversible and symplectic, but surely because they are simple. The force field description was then, and still is, a combination of Lennard-Jones and Coulombic terms, with (mostly) harmonic bonds and periodic dihedrals. Modern extensions have added many more parameters but only modestly more reliability. The now almost universal use of constraints for bonds (and sometimes bond angles) was already introduced in 1977 [8]. That polarisability would be necessary was realized then [9], but it is still not routinely implemented today. Long-range interactions are still troublesome, but the methods that now become popular date back to Ewald in 1921 [10] and Hockney and Eastwood in 1981 [11]. 

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 ... 

Helmut Grubmuller, Helmut Heller, Andreas Windemuth, and Klaus Schulten. Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions. Mol. Sim., 6 121-142, 1991. 

For future reference, the Verlet algorithm [18] can be generalized to iticlude the friction and stochastic terms above, and is typically used in the following form described by Brooks, Briinger and Karplus, known as BBK [23, 37] ... 

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 ... 

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. 

Grubmiiller, H., Heller, H., Windemuth, A., Schulten, K. Generalized Verlet Algorithm for Efficient Molecular Dynamics Simulations with Long-range Interactions. Molecular Simulation 6 (1991) 121-142... 

Using the symmetric Verlet algorithm for integrating exp(rL ) yields ... 

The Beeman integration scheme uses a more accurate expression for the velocity. As a consequence it often gives better energy conservation, because the kinetic energy is calculated directly from the velocities. However, the expressions used are more complex than those of the Verlet algorithm and so it is computationally more expensive. 

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 ... 

The Verlet algorithm is not self-starting. A lower order Taylor expansion [e.g., Eq. (13)] is often used to initiate the propagation. 

This algorithm is computationally a little more expensive than the Verlet algorithm. 

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 ... 

There are many algorithms in the literature, many of which date from the early days of the science of numerical analysis. I simply haven t space to review them all, so I will end this section with the famous Verlet algorithm. 

Adding these two equations gives the Verlet algorithm, which is used to advance the position vector r from its value at time t to time f + Sf... 

This is the Verlet algorithm for solving Newton s equation numerically. Notice that the term involving the change in acceleration (b) disappears, i.e. the equation is correct to third order in At. At the initial point the previous positions are not available, but may be estimated from a first-order approximation of eq. (16.29). 

The Verlet algorithm has the numerical disadvantage that the new positions are obtained by adding a term proportional to Af to a difference in positions (2r — r, i). Since At is a small number and (2r, — r, i) is a difference between two large numbers, this may lead to truncation errors due to finite precision. The Verlet furthermore has the disadvantage that velocities do not appear explicitly, which is a problem in connection with generating ensembles with constant temperature, as discussed below. 

The numerical aspect, and the lack of explicit velocities, in the Verlet algorithm can be remedied by the leap-frog algorithm. Performing expansions analogous to eqs. (16.28) and (16.29) with half a time step followed by subtraction gives... 

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