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Geometric Computational Electrodynamics with Variational Integrators and Discrete Differential Forms

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Book cover Geometry, Mechanics, and Dynamics

Part of the book series: Fields Institute Communications ((FIC,volume 73))

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.

In memory of Jerry, our colleague, mentor, and friend.

— A.S., Y.T., and M.D.

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Acknowledgements

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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Authors and Affiliations

  1. Department of Mathematics, Washington University, St. Louis, MO, 63130, USA

    Ari Stern

  2. Department of Computer Science and Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Yiying Tong

  3. Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, 91125, USA

    Mathieu Desbrun

  4. California Institute of Technology, Pasadena, CA, 91125, USA

    Jerrold E. Marsden

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  1. Ari Stern
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Correspondence to Ari Stern .

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Editors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada

    Dong Eui Chang

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Darryl D. Holm

  3. Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

    George Patrick

  4. Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Tudor Ratiu

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Stern, A., Tong, Y., Desbrun, M., Marsden, J.E. (2015). Geometric Computational Electrodynamics with Variational Integrators and Discrete Differential Forms. In: Chang, D., Holm, D., Patrick, G., Ratiu, T. (eds) Geometry, Mechanics, and Dynamics. Fields Institute Communications, vol 73. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2441-7_19

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