Geometry, Mechanics, and Dynamics pp 437-475

Part of the Fields Institute Communications book series (FIC, volume 73) | Cite as

Geometric Computational Electrodynamics with Variational Integrators and Discrete Differential Forms

  • Ari Stern
  • Yiying Tong
  • Mathieu Desbrun
  • Jerrold E. Marsden

Abstract

In this paper, we develop a structure-preserving discretization of the Lagrangian framework for electrodynamics, combining the techniques of variational integrators and discrete differential forms. This leads to a general family of variational, multisymplectic numerical methods for solving Maxwell’s equations that automatically preserve key symmetries and invariants. In doing so, we show that Yee’s finite-difference time-domain (FDTD) scheme and its variants are multisymplectic and derive from a discrete Lagrangian variational principle. We also generalize the Yee scheme to unstructured meshes, not just in space but in 4-dimensional spacetime, which relaxes the need to take uniform time steps or even to have a preferred time coordinate. Finally, as an example of the type of methods that can be developed within this general framework, we introduce a new asynchronous variational integrator (AVI) for solving Maxwell’s equations. These results are illustrated with some prototype simulations that show excellent numerical behavior and absence of spurious modes, even for an irregular mesh with asynchronous time stepping.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Ari Stern
    • 1
  • Yiying Tong
    • 2
  • Mathieu Desbrun
    • 3
  • Jerrold E. Marsden
    • 4
  1. 1.Department of MathematicsWashington UniversitySt. LouisUSA
  2. 2.Department of Computer Science and EngineeringMichigan State UniversityEast LansingUSA
  3. 3.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA
  4. 4.California Institute of TechnologyPasadenaUSA

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