Qualitative study of the symplectic Störmer–Verlet integrator

Abstract

Symplectic numerical integrators, such as the Störmer–Verlet method, are useful in preserving properties that are not preserved by conventional numerical integrators. This paper analyzes the Störmer–Verlet method as applied to the simple harmonic model, whose generalization is an important model for molecular dynamics simulations. Restricting our attention to the one -dimensional case, both the exact solution and the Störmer–Verlet solution to this model are expressed as functions of the number of time steps taken, and then both of these functions are interpreted geometrically. The paper shows the existence of an upper bound on the error from the Störmer–Verlet method, and then an example is worked to demonstrate the closeness of this bound.