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

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Abstract
The following integration methods for special second-order ordinary differential equations are studied: leapfrog, implicit midpoint, trapezoid, Stormer-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.

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