Verlet method

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 ... 

Fig. 2. The time evolution of the total energy of four water molecules (potential-energy details are given in [48]) as propagated by the symplectic Verlet method (solid) and the nonsymplectic fourth-order Runge-Kutta method (dashed pattern) for Newtonian dynamics at two timestep values.

The popular Verlet method is recovered by setting 7 and fi to zero in the above propagation formulas. 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

The standard discretization for the equations (9) in molecular dynamics is the (explicit) Verlet method. Stability considerations imply that the Verlet method must be applied with a step-size restriction k < e = j2jK,. Various methods have been suggested to avoid this step-size barrier. The most popular is to replace the stiff spring by a holonomic constraint, as in (4). For our first model problem, this leads to the equations d... 

The constrained equations of motion in cartesian eoordinates can be solved by the SHAKE or (the essentially equivalent) RATTLE method (see [8]) which requires the solution of a non-linear system of equations in the Lagrange multiplier funetion A. The equivalent formulation in local coordinates ean still be integrated by using the explicit Verlet method. 

The application in [24] is to celestial mechanics, in which the reduced problem for consists of the Keplerian motion of planets around the sun and in which the impulses account for interplanetary interactions. Application to MD is explored in [14]. It is not easy to find a reduced problem that can be integrated analytically however. The choice /f = 0 is always possible and this yields the simple but effective leapfrog/Stormer/Verlet method, whose use according to [22] dates back to at least 1793 [5]. This connection should allay fears concerning the quality of an approximation using Dirac delta functions. 

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 standard numerical integrators for the constrained system (12) are the SHAKE scheme [23], which extends the Verlet method (2),... 

Letting m/M 0 in the numerical method, it can be shown that the solution given by (21) tends to a small perturbation of the Verlet method formally applied to that equation ... 

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 ... 

Beeman s algorithm [Beeman 1976] is also related to the Verlet method ... 

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.

In the Verlet method, this equation is written by using central finite differences (see Interpolation and Finite Differences ). Note that the accelerations do not depend upon the velocities. 

The dWi are Gaussian white noise processes, and their strength a is related to the kinetic friction y through the fluctuation-dissipation relation.72 When deriving integrators for these methods, one has to be careful to take into account the special character of the random forces employed in these simulations.73 A variant of the velocity Verlet method, including a stochastic dynamics treatment of constraints, can be found in Ref. 74. The stochastic... 

G. Vanden Berghe and M. Van Daele, Exponentially-fitted Stormer/Verlet methods, JNAIAM J. Numer. Anal. Indust. Appl. Math., 2006, 1(3), 241—255. 

The classical equations of motion are solved with the velocity Verlet method [66]. The simulation is performed in the microcanonical ensemble (N,V,E) in a cubic box with length 15 A. Periodic boundary conditions are applied. The system is equilibrated by performing a constant temperature simulation [67] for 10 ps to achieve a mean temperature of 1000 K. The temperature control is then turned off to gain constant energy conditions. 

In this scheme, v and r refer to one of the 3N velocities or positions, respectively. Note that the different types of force are calculated throughout the algorithm. It can be readily seen that the method reduces to the standard velocity Verlet method if Mi, M2 and M3 are set equal to 1. 

The r-RESPA method has been applied to a variety of systems, including simple model systems [Tuckerman et al. 1992] but also organic molecules [Watanabe and Karplus 1993], fullerene crystals [Procacci and Berne 1994] and also proteins [Humphreys et al. 1994, 1996]. In these studies the reduction in computational time compared with the standard velocity Verlet method varied between 4-5 and 20-40, depending upon the size of the system, without any noticeable loss in accuracy. Other developments of the r-RESPA algorithm include its coupling to the fast multipole method (see Section 6.8.3) [Zhou and Berne 1995]. 

The Verlet method (also known as leapfrog or Stormer-Verlet) is a second order method that is popular for molecular simulation. It is specialized to problems that can be expressed in the former = v,Mv = F( ), with even dimensional phase space which includes constant energy molecular dynamics. Some generalizations exist for other classes of Hamiltonian systems. 

The Verlet method is a numerical method that respects certain conservation principles associated to the continuous time ordinary differential equations, i.e. it is a geometric integrator. Maintaining these conservation properties is essential in molecular simulation as they play a key role in maintaining the physical environment. As a prelude to a more general discussion of this topic, we demonstrate here that it is possible to derive the Verlet method from the variational principle. This is not the case for every convergent numerical method. The Verlet method is thus a special type of numerical method that provides a discrete model for classical mechanics. 

The method (2.4) is commonly referred to as StOrmer s rule. It was used by the mathematician Stormer for calculations in the first decade of the 1900s. In molecular dynamics this method is referred to as the Verlet method since it was used by Verlet in his important 1967 paper [387]. 

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