This book contains two applications of high-order accurate methods for solving Maxwell’s equations. The first is the time domain modeling of the electromagnetic wave propagation in photonic crystals to demonstrate the behavior and sensitivity of the frozen mode phenomenon in finite structures with anisotropic materials. Moreover, a PDE constrained nonlinear optimization technique is introduced to optimize the design of photonic crystals for the phenomenon. The second application is the development of thin layer approximations to compute the electromagnetic field with the thin layers of strong magnetic materials in multidimensional spaces without compromising efficiency and accuracy. Metal backed coatings and transmission layers with isotropic materials or with tangentially-oriented anisotropic materials are considered in one and two dimensions in the time domain.