  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.
