Arbitrary High Order Finite Difference Methods with Applications to Wave Propagation Modeled by Maxwell\u27s Equations

This dissertation investigates two different mathematical models based on the time-domain Maxwell\u27s equations: the Drude model for metamaterials and an equivalent Berenger\u27s perfectly matched layer (PML) model. We develop both an explicit high order finite difference scheme and a compact implicit scheme to solve both models. We develop a systematic technique to prove stability and error estimate for both schemes. Extensive numerical results supporting our analysis are presented. To our best knowledge, our convergence theory and stability results are novel and provide the first error estimate for the high-order finite difference methods for Maxwell\u27s equations