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

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.