We present two strategies for designing passivity preserving higher order discretization methods for Maxwell's equations in nonlinear Kerr-type media. Both approaches are based on variational approximation schemes in space and time. This allows to rigorously prove energy conservation or dissipation, and thus passivity, on the fully discrete level. For linear media, the proposed methods coincide with certain combinations of mixed finite element and implicit Runge-Kutta schemes. The order optimal convergence rates, which can thus be expected for linear problems, are also observed for nonlinear problems in the numerical tests.

翻译：我们提出了两种战略,用于在非线性Kerr型介质中为Maxwell的方程式设计高顺序离散方法,两种方法都以空间和时间的变近法为基础,从而能够严格证明节能或消散,从而在完全离散的层次上是被动的。对于线性介质,建议的方法与混合的有限元素和隐含的龙格-库塔计划的某些组合相吻合。对于线性问题,也观察到了在数字测试中的非线性问题,因此可以预计的顺序最佳趋同率。