Maxwell's equations are a system of vector partial differential equations (PDEs) that govern the behavior of electromagnetic fields in materials. Simulating the interaction of electromagnetic waves with different materials allows us to understand and design materials that have novel optical properties. Such materials often have complicated micro structures and require substantial computational effort to simulate. Thus, an efficient numerical simulation bridging the microscopic structures and macroscopic optical phenomena is critical for advancing material design and engineering.

In this thesis, we will introduce a robust multiscale modeling framework based on HMM and operator splitting methods that not only addresses current limitations in simulating electromagnetic phenomena but also opens new avenues for optimal material design and broader applications in scientific computing. Specifically, we study the development of a HMM for simulating electromagnetic phenomenon in multiscale optical materials that exhibit electric polarization. The mathematical model for such optical materials is Maxwell's equations combined with constitutive laws based on ordinary differential equations (ODEs) for modeling the polarization in the material which is presented as a mixture of dispersive and non-dispersive dielectrics. We use the Lorentz model, a second order ODE, for the dispersive dielectric. A key contribution of this thesis is the integration of HMM with the finite-difference time-domain (FDTD) numerical method to numerically discretize the Maxwell-Lorentz system. This proves to be challenging due to the staggered grid nature of the FDTD method.

Operator splitting techniques are introduced to address the challenges that arise. Operator splitting allows us to decompose a complex problem into smaller, more manageable sub-problems that can be solved sequentially. By combining operator splitting techniques with HMM, we not only address the challenges that hinder the integration of HMM and FDTD but also open the door to new possibilities. While HMM has been successfully implemented for PDEs modeling wave propagation problems and ordinary differential equations separately, operator splitting allows us to merge these approaches, enabling both theoretical advancements and practical implementations.