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Stochastic Effects and Time-Filtered Leapfrog Schemes for Maxwell’s Equations

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Mathematics for Nonlinear Phenomena — Analysis and Computation (MNP2015 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 215))

Abstract

We consider Maxwell’s equations where the conductivity contains fast random fluctuations in time. Using an Ornstein–Uhlenbeck process, we study the effects of correlations between the random fluctuations of two different time scales, with one an order of magnitude smaller than the other. We show that this asymptotic regime gives rise to a limiting equation where the effects of the fluctuations in the conductivity are captured in additional terms containing deterministic and stochastic corrections. For deterministic dynamics, numerical solutions to the time dependent Maxwell’s equations using a new time stepping scheme are presented. This scheme, which is based on the leapfrog method and a fourth-order time filter, significantly reduces the short oscillations generated by numerical dispersion. It uses staggering in space only, allowing explicit treatment of the electric current density terms and application of numerical smoothers. Comparisons of simulation results where Maxwell’s equations are integrated in a presence of the scattering of an electromagnetic pulse by a perfectly conducting square and those obtained with the unfiltered leapfrog show that the developed method is robust and accurate.

Dedicated to Professor Yoshikazu Giga

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Acknowledgements

This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-15-1-0096.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

    Alex Mahalov & Austin McDaniel

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  1. Alex Mahalov
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  2. Austin McDaniel
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Correspondence to Alex Mahalov .

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Editors and Affiliations

  1. Mathematical Institute, Tohoku University, Sendai, Miyagi, Japan

    Yasunori Maekawa

  2. Department of Mathematics, Hokkaido University, Sapporo, Hokkaido, Japan

    Shuichi Jimbo

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Mahalov, A., McDaniel, A. (2017). Stochastic Effects and Time-Filtered Leapfrog Schemes for Maxwell’s Equations. In: Maekawa, Y., Jimbo, S. (eds) Mathematics for Nonlinear Phenomena — Analysis and Computation. MNP2015 2015. Springer Proceedings in Mathematics & Statistics, vol 215. Springer, Cham. https://doi.org/10.1007/978-3-319-66764-5_7

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