A family of symplectic integrators: Stability, accuracy, and molecular dynamics applications

Robert D. Skeel, Guihua Zhang, Tamar Schlick

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)203-222
Number of pages20
JournalSIAM Journal on Scientific Computing
Issue number1
StatePublished - Jan 1997


  • Cowell
  • Implicit midpoint
  • Leapfrog
  • Method of modified equations
  • Molecular dynamics
  • Numerov
  • Störmer
  • Symplectic integrator
  • Trapezoid
  • Verlet

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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