This work studies time-dependent electromagnetic scattering from obstacles whose interaction with the wave is fully determined by a nonlinear boundary condition. In particular, the boundary condition studied in this work enforces a power law type relation between the electric and magnetic field along the boundary. Based on time-dependent jump conditions of classical boundary operators, we derive a nonlinear system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. These fields can subsequently be computed at arbitrary points in the exterior domain by evaluating a time-dependent representation formula. Fully discrete schemes are obtained by discretising the nonlinear system of boundary integral equations with Runge--Kutta based convolution quadrature in time and Raviart--Thomas boundary elements in space. Error bounds with explicitly stated convergence rates are proven, under the assumption of sufficient regularity of the exact solution. The error analysis is conducted through novel techniques based on time-discrete transmission problems and the use of a new discrete partial integration inequality. Numerical experiments illustrate the use of the proposed method and provide empirical convergence rates.

翻译：本文研究了由非线性边界条件完全决定波与障碍物相互作用的时变电磁散射问题。具体而言，本文研究的边界条件在边界上建立了电场与磁场之间的幂律关系。基于经典边界算子的时变跳跃条件，我们推导了一个确定散射电场和磁场切向迹线的时变边界积分方程非线性系统。通过评估时变表示公式，可在外部区域任意点计算这些场。采用基于龙格-库塔的卷积求积法在时间方向、Raviart-Thomas边界元在空间方向对非线性边界积分方程系统进行离散，得到全离散格式。在假设精确解具有充分正则性的前提下，证明了具有显式收敛速度的误差界。误差分析通过基于时间离散传输问题的新技术以及新的离散分部积分不等式完成。数值实验展示了所提方法的实用性，并提供了经验收敛速度。