A Family of Symplectic Integrators: Stability, Accuracy, and Molecular Dynamics Applications

The following integration methods for special second-order ordinary differential equations are studied: leapfrog, implicit midpoint, trapezoid, Störmer–Verlet, and Cowell–Numerov. We show that all are members, or equivalent to members, of a one-parameter family of schemes. Some methods have more than one common form, and we discuss a systematic enumeration of these forms. We also present a stability and accuracy analysis based on the idea of “modified equations” and a proof of symplecticness. It follows that Cowell–Numerov and “LIM2” (a method proposed by Zhang and Schlick) are symplectic. A different interpretation of the values used by these integrators leads to higher accuracy and better energy conservation. Hence, we suggest that the straightforward analysis of energy conservation is misleading.