Time-dependent electromagnetic scattering from thin layers

TL;DR

This paper develops and analyzes a stable, convergent numerical method for simulating time-dependent electromagnetic scattering from thin layers using boundary integral equations and advanced discretization techniques.

Contribution

It introduces a novel approach combining boundary integral equations with Runge-Kutta convolution quadrature and Raviart-Thomas elements for accurate time-dependent scattering simulations.

Findings

The method is proven to be stable and convergent with explicit rates.

Optimal convergence order is achieved away from the boundary.

Numerical experiments confirm theoretical predictions.

Abstract

The scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell's equations. The time-dependent boundary integral equationis discretized with Runge--Kutta based convolution quadrature in time and Raviart--Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge--Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments.