Understanding the physics of strong light-matter interactions is difficult without realistic simulations, especially in non-resonant optical excitation in solid-state and gaseous materials, where the numerical solution of Maxwell's equations is one of the necessary components. Despite the fact that a number of different numerical methods work very well for the the Maxwell system, experiments at the intersection between nonlinear optics, strong-field-, attosecond-, and condensed matter physics continue to pose new challenges that motivate further research into accurate numerical methods suitable for highly nonlinear Maxwell systems. Including a brief overview of different approaches to electromagnetic propagation equations in the context of nonlinear optics, this talk will concentrate on the techniques that make it possible to obtain high-fidelity solutions in the presence of complex linear and nonlinear material properties.