Stormer-Verlet

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. 

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 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 interest is usually in conserving some more complicated first integrals, for example the energy of the system. None of the schemes we have considered so far (Euler s method, Stormer-Verlet, etc.) conserves this quantity exactly, even in the... 

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)... 

The symmetry suggests that this will be second order. In order to understand the symplectic property associated to this method, we need to define the updates for both positions and momenta. A natural choice is to consider the phase space formulation of the Stormer-Verlet method for q = M p,p = F q), then replace F by F —... 

Recall that the Stormer-Verlet method could be constructed by composing steps using Symplectic Euler and its adjoint method. Using more complicated methods it is possible to build higher order schemes. It seems natural that a similar procedure should be possible in the constrained setting. But what, precisely, is the adjoint method in the case of (4.20)-(4.24) ... 

Yet this is not the only constrained composition method we could have designed in analogy with Stormer-Verlet. Recall that Stdrmer-Verlet is also the symmetric composition of the flow maps of the two Hamiltonians K = p M p/2 and U = U q). It would be natural to consider a composition of steps resolving the constrained flows on K and U separately. For the potential energy term, an exact solution is given by... 

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 ... 

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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