A discrete action principle for electrodynamics and the construction of explicit symplectic integrators for linear, non-dispersive media

Jeffrey M. McMahon, Stephen K. Gray, George C. Schatz

Research output: Contribution to journalArticle

5 Citations (Scopus)


In this work, we derive a discrete action principle for electrodynamics that can be used to construct explicit symplectic integrators for Maxwell's equations. Different integrators are constructed depending on the choice of discrete Lagrangian used to approximate the action. By combining discrete Lagrangians in an explicit symplectic partitioned Runge-Kutta method, an integrator capable of achieving any order of accuracy is obtained. Using the von Neumann stability analysis, we show that the integrators greatly increase the numerical stability and reduce the numerical dispersion compared to other methods. For practical purposes, we demonstrate how to implement the integrators using many features of the finite-difference time-domain method. However, our approach is also applicable to other spatial discretizations, such as those used in finite element methods. Using this implementation, numerical examples are presented that demonstrate the ability of the integrators to efficiently reduce and maintain a minimal amount of numerical dispersion, particularly when the time-step is less than the stability limit. The integrators are therefore advantageous for modeling large, inhomogeneous computational domains.

Original languageEnglish
Pages (from-to)3421-3432
Number of pages12
JournalJournal of Computational Physics
Issue number9
Publication statusPublished - May 20 2009



  • Dispersion
  • Electrodynamics
  • FDTD
  • Lagrangian
  • Runge-Kutta
  • Stability
  • Symplectic integrator

ASJC Scopus subject areas

  • Numerical Analysis
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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