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Published 2015 | public
Book Section - Chapter Open

Geometric Computational Electrodynamics with Variational Integrators and Discrete Differential Forms


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.

Additional Information

© 2015 Springer. We would like to thank several people for their inspiration and suggestions. First of all, Alain Bossavit for suggesting many years ago that we take the present DEC approach to computational electrodynamics, and for his excellent lectures at Caltech on the subject. Second, Michael Ortiz and Eva Kanso for their ongoing interactions on related topics and suggestions. We also thank Doug Arnold, Uri Ascher, Robert Kotiuga, Melvin Leok, Adrian Lew, and Matt West for their feedback and encouragement. In addition, the 3-D AVI simulations shown in Figure 11 were programmed and implemented by Patrick Xia, as part of a Summer Undergraduate Research Fellowship at Caltech supervised by M.D. and A.S. A.S. was partially supported by a Gordon and Betty Moore Foundation fellowship at Caltech, and by NSF grant CCF-0528101. Y.T. and M.D. were partially supported by NSF grants CCR-0133983 and DMS-0453145 and DOE contract DE-FG02-04ER25657. J.E.M. was partially supported by NSF grant CCF-0528101.

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August 20, 2023
August 20, 2023