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：这项工作研究从与波浪的相互作用完全由非线性边界条件决定的障碍中随时间产生的电磁散射。 特别是, 这项工作所研究的边界条件使沿边界的电磁场之间建立起一种权力法型关系。 根据古典边界操作者的时间性跳跃条件, 我们从一个非线性系统得出一个非线性系统, 确定分散的电磁场的相近痕迹。 这些领域随后可以通过评价一个时间性代表公式, 在外部任意点上计算。 通过将非线性边界整体方程式系统与基于Runge- Kutta的卷变二次和空间Raviart- Thomas的边界要素分离, 获得完全独立的计划。 在假定精确解决办法足够正常的情况下, 证明与明确表明的汇合率的错误界限。 错误分析是通过基于时间性分解传输问题和使用新的离性部分融合不平等的新技术进行的。 数字实验显示了拟议方法的使用情况,并提供经验性汇合率。