Stochastic effects and time-filtered leapfrog schemes for Maxwell’s equations

Alex Mahalov, Austin McDaniel

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

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 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.

Original languageEnglish (US)
Title of host publicationMathematics for Nonlinear Phenomena—Analysis and Computation - In Honor of Yoshikazu Giga’s 60th Birthday
PublisherSpringer New York LLC
Pages131-160
Number of pages30
Volume215
ISBN (Print)9783319667621
DOIs
StatePublished - Jan 1 2017
EventInternational Conference on Mathematics for Nonlinear Phenomena: Analysis and Computation in Honor of Professor Yoshikazu Giga on his 60th Birthday, MNP 2015 - Sapporo, Japan
Duration: Aug 16 2015Aug 18 2015

Other

OtherInternational Conference on Mathematics for Nonlinear Phenomena: Analysis and Computation in Honor of Professor Yoshikazu Giga on his 60th Birthday, MNP 2015
CountryJapan
CitySapporo
Period8/16/158/18/15

Fingerprint

Maxwell's equations
Fluctuations
Conductivity
Numerical Dispersion
Limiting Equations
Electric Current
Time Stepping
Term
Fourth Order
Time Scales
Scattering
Numerical Solution
Oscillation
Filter
Simulation

Keywords

  • Fast numerical schemes
  • Maxwell equations
  • Stochastic effects
  • Wave propagation

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Mahalov, A., & McDaniel, A. (2017). Stochastic effects and time-filtered leapfrog schemes for Maxwell’s equations. In Mathematics for Nonlinear Phenomena—Analysis and Computation - In Honor of Yoshikazu Giga’s 60th Birthday (Vol. 215, pp. 131-160). Springer New York LLC. https://doi.org/10.1007/978-3-319-66764-5_7

Stochastic effects and time-filtered leapfrog schemes for Maxwell’s equations. / Mahalov, Alex; McDaniel, Austin.

Mathematics for Nonlinear Phenomena—Analysis and Computation - In Honor of Yoshikazu Giga’s 60th Birthday. Vol. 215 Springer New York LLC, 2017. p. 131-160.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Mahalov, A & McDaniel, A 2017, Stochastic effects and time-filtered leapfrog schemes for Maxwell’s equations. in Mathematics for Nonlinear Phenomena—Analysis and Computation - In Honor of Yoshikazu Giga’s 60th Birthday. vol. 215, Springer New York LLC, pp. 131-160, International Conference on Mathematics for Nonlinear Phenomena: Analysis and Computation in Honor of Professor Yoshikazu Giga on his 60th Birthday, MNP 2015, Sapporo, Japan, 8/16/15. https://doi.org/10.1007/978-3-319-66764-5_7
Mahalov A, McDaniel A. Stochastic effects and time-filtered leapfrog schemes for Maxwell’s equations. In Mathematics for Nonlinear Phenomena—Analysis and Computation - In Honor of Yoshikazu Giga’s 60th Birthday. Vol. 215. Springer New York LLC. 2017. p. 131-160 https://doi.org/10.1007/978-3-319-66764-5_7
Mahalov, Alex ; McDaniel, Austin. / Stochastic effects and time-filtered leapfrog schemes for Maxwell’s equations. Mathematics for Nonlinear Phenomena—Analysis and Computation - In Honor of Yoshikazu Giga’s 60th Birthday. Vol. 215 Springer New York LLC, 2017. pp. 131-160
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