Verlet scheme

Backward Analysis In this type of analysis, the discrete solution is regarded as an exact solution of a perturbed problem. In particular, backward analysis of symplectic discretizations of Hamiltonian systems (such as the popular Verlet scheme) has recently achieved a considerable amount of attention (see [17, 8, 3]). Such discretizations give rise to the following feature the discrete solution of a Hamiltonian system is exponentially close to the exact solution of a perturbed Hamiltonian system, in which, for consistency order p and stepsize r, the perturbed Hamiltonian has the form [11, 3]... 

Fig. 1. The time evolution (top) and average cumulative difference (bottom) associated with the central dihedral angle of butane r (defined by the four carbon atoms), for trajectories differing initially in 10 , 10 , and 10 Angstoms of the Cartesian coordinates from a reference trajectory. The leap-frog/Verlet scheme at the timestep At = 1 fs is used in all cases, with an all-atom model comprised of bond-stretch, bond-angle, dihedral-angle, van der Waals, and electrostatic components, a.s specified by the AMBER force field within the INSIGHT/Discover program.

Fig. 4. The average end-to-end-distance of butane as a function of timestep (note logarithmic scale) for both single-timestep and triple-timestep Verlet schemes. The timestep used to define the data point for the latter is the outermost timestep At (the interval of updating the nonbonded forces), with the two smaller values used as Atj2 and At/A (for updating the dihedral-angle terms and the bond-length and angle terms, respectively).

We assume that A is a symmetric and positive semi-definite matrix. The case of interest is when the largest eigenvalue of A is significantly larger than the norm of the derivative of the nonlinear force f. A may be a constant matrix, or else A = A(y) is assumed to be slowly changing along solution trajectories, in which case A will be evaluated at the current averaged position in the numerical schemes below. In the standard Verlet scheme, which yields approximations y to y nAt) via... 

From the derivation of the method (4) it is obvious that the scheme is exact for constant-coefficient linear problems (3). Like the Verlet scheme, it is also time-reversible. For the special case A = 0 it reduces to the Verlet scheme. It is shown in [13] that the method has an 0 At ) error bound over finite time intervals for systems with bounded energy. In contrast to the Verlet scheme, this error bound is independent of the size of the eigenvalues Afc of A. 

A widely used variant of the Verlet scheme is its velocity version ... 

A different long-time-step method was previously proposed by Garci a-Archilla, Sanz-Serna, and Skeel [8]. Their mollified impulse method, which is based on the concept of operator splitting and also reduces to the Verlet scheme for A = 0 and admits second-order error estimates independently of the frequencies of A, reads as follows when applied to (1) ... 

Modifications to the basic Verlet scheme have been proposed to tackle the above deficiencies, particularly to improve the velocity evaluation. One of these modifications is the leap-frog algorithm, so called for its half-step scheme Velocities are evaluated at the midpoint of the position evaluation and vice versa [12,13]. The algorithm can be written as... 

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... 

Ryckaert et al. incorporated initially the basic Verlet integration algorithm, known also as the Stormer algorithm,into the method of undetermined parameters. In the basic Verlet scheme, the highest time derivative of the coordinates is of second order, and Eq. [37] with = 0 reduces to ... 

For the Verlet scheme with = 0, the are replaced by the (k = 1,. ..,/), and the method of undetermined parameters incorporating the basic Verlet scheme could equally well be termed, and is often referred to in the literature as, the method of undetermined (Lagrangian) multipliers. However, in general where > 0, the Lagrangian multipliers and their derivatives [X °>(tQ),. . ., computed, in addition to the undeter-... 

Inserting Eq. [49] into Eq. [46] shows that the coordinates given by the method of undetermined parameters are accurate to 0(ht ). The error in the coordinates (local error) is of the 0(bt ) present in the basic Verlet scheme. Therefore, consistent with the preceding error analysis, no additional error is introduced in the method of undetermined parameters, and the constraints are exactly satisfied at every time step. If the approximate forces of constraints are desired, they can be computed a posteriori as... 

The first approach for computing the undetermined parameters (7) is an application of the general method of solution, discussed in the preceding section, to the basic Verlet scheme. The total displacement in Eq- [54] containing... 

In the general case, the problem formulated in the previous sections has no analytical solution at each stage dierefore, numerical methods for solving are used, as a mle. In this chapter, for the first stages, the numerical integration of the equation of motion of the nanoparticle atoms in the relaxation process are used in accordance with Verlet scheme [26] ... 

More complicated symmetric second-order schemes can be devised by proceeding almost arbitrarily and maintaining a symmetric composition, but it is not found that alternative approaches improve on the two Verlet schemes, at least for molecular applications. The Verlet methods are seen as the gold-standard for molecular dynamics computations both require only one evaluation of VU q) per iteration (where the velocity Verlet scheme can reuse VU Q) for the next iteration), and offer a second-order symplectic evolution. 

The projected symplectic constrained method (4.20)-(4.24) is only first order accurate. We forego providing a detailed proof of this fact, but note that it could be demonstrated using standard methods [164]. Note that (4.20)-(4.24) reduces to the symplectic Euler method in the absence of constraints, and the projection of the momenta would not alter this fact. There are several constraint-preserving, second-order alternatives which generalize the Stormer-Verlet scheme. One of these is the SHAKE method [322]. The original derivation of the SHAKE method began from the position-only, two-step form of the Stormer rule for q = F(q)... 

As we have seen in the Chap. 3, when a symplectic method is applied to a molecular dynamics problem it induces a perturbed Hamiltonian (energy) function. For the Verlet scheme the modified Hamiltonian is... 

We compute the relative error in several average quantities, for each experiment shown in Fig. 7.6. These computed errors are relative to a baseline solution, computed using the Stochastic Position Verlet scheme, at y = 1/ps and h = 0.5 fs. The timestep for the baseline scheme is small enough that discretization effects will be negligible. 

The equations of motion are integrated using the Verlet scheme with time step At = O.OOSt where t = is the characteristic time of... 

Here, D is the self-consistent, optimized density matrix. Using a time-reversible Verlet scheme we get an explicit integration of the form... 

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 ... 

Although the Verlet scheme shown in equation (16) was derived from a discrete time representation of Hamilton s equations, it has the form of Newton s equation of motion, which is a second-order differential equation expressed, in terms of the Cartesian coordinates, as... 

In the absence of velocity-dependent forces this is just the Verlet scheme, but the solid-fluid boundary conditions (200) introduce a hydrodynamic force that depends linearly on the particle velocity [104,134],... 

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