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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

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 (US)
Pages (from-to)3421-3432
Number of pages12
JournalJournal of Computational Physics
Volume228
Issue number9
DOIs
StatePublished - May 20 2009

Funding

G.C.S. and J.M.M. were supported by the US Department of Energy under Grant No. DEFG02-03-ER15487 and the Northwestern Materials Research Center, sponsored by the National Science Foundation (DMR-0520513). The work at Argonne National Laboratory was supported by the US Department of Energy, Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. The authors thank David Masiello for many fruitful discussions.

Keywords

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

ASJC Scopus subject areas

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

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