Integral equations Verlet approximation

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]. 

One of the most common numerical methods used in molecular dynamics to solve Newton s equations of motions is the Velocity Verlet integrator. This is typically implemented as a second order method, and we find that it can become numerically tmstable dtuing the course of hyperthermal collision events, where the atom velocities are often far from equilibrium. As an alternative, we have implemented a fifth/ sixth order predictor-corrector scheme for our calculations. Specifically, the driver we chose utilizes the Adams-Bashforth predictor method together with the Adams-Moulton corrector method for approximating the solution to the equations of motion. 

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