### 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 language | English (US) |
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Title of host publication | Mathematics for Nonlinear Phenomena—Analysis and Computation - In Honor of Yoshikazu Giga’s 60th Birthday |

Publisher | Springer New York LLC |

Pages | 131-160 |

Number of pages | 30 |

Volume | 215 |

ISBN (Print) | 9783319667621 |

DOIs | |

State | Published - Jan 1 2017 |

Event | International 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 2015 → Aug 18 2015 |

### Other

Other | International Conference on Mathematics for Nonlinear Phenomena: Analysis and Computation in Honor of Professor Yoshikazu Giga on his 60th Birthday, MNP 2015 |
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Country | Japan |

City | Sapporo |

Period | 8/16/15 → 8/18/15 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - Stochastic effects and time-filtered leapfrog schemes for Maxwell’s equations

AU - Mahalov, Alex

AU - McDaniel, Austin

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

AB - 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.

KW - Fast numerical schemes

KW - Maxwell equations

KW - Stochastic effects

KW - Wave propagation

UR - http://www.scopus.com/inward/record.url?scp=85034238261&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85034238261&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-66764-5_7

DO - 10.1007/978-3-319-66764-5_7

M3 - Conference contribution

AN - SCOPUS:85034238261

SN - 9783319667621

VL - 215

SP - 131

EP - 160

BT - Mathematics for Nonlinear Phenomena—Analysis and Computation - In Honor of Yoshikazu Giga’s 60th Birthday

PB - Springer New York LLC

ER -