Discontinuous Galerkin Methods For Some Time-Domain Electromagnetic Wave Propagation Problems

Nanophotonics has broad applications in practical scientific and engineering fields.This thesis concentrates on the electromagnetic wave propagation problems in nonlinear optical nanostructures.It is known to all that,the propagation of electromagnetic waves is governed by classical Maxwell’s equations which admit two versions: the microscopic version based on the current density J and the macroscopic version based on the polarization P.This thesis consists of two parts: the first part mainly concerns the development of efficient numerical method for nonlinear microscopic Maxwell’s equations,with particular applications in numerical investigation of some complex optical phenomena in metallic nanostructures;the second part is devoted to the design of high order energy stable numerical schemes for the nonlinear macroscopic Maxwell’s equations in optical media.To simulate the high order harmonic generations and nonlocal effects from metallic nanostructures,we borrow the Euler equations from fluid dynamics to describe the motion of electron gas in metallic nanostructures.By coupling to the microscopic Maxwell’s equations,we obtain the self-consistent hydrodynamic Drude model.We apply high order discontinuous Galerkin method to solve this model.Considering the large amount of dissipation of Lax-Friedrichs(LF)numerical flux,we use less dissipative Harten-Lax-van Leer(HLL)numerical flux instead for the nonlinear Euler equations.We numerically demonstrate the outperformance of HLL numerical flux over LF numerical flux for the hydrodynamical Drude model in terms of accuracy.With this proposed discontinuous Galerkin scheme,we successfully simulate high order harmonic generations and nonlocal effects from different metallic nanostructures.For the metallic nanostructure coupled with non-metalic nanostructure,we use a polarization model to take the bound electrons into account and numerically study their enhancing effect on the high order harmonic generations for selected experimental setting.In addition,in order to investigate the influence of nonlineartities on the nonlocality,we use a switch-on-andoff numerical analysis to confirm that only the quantum pressure term is responsible for the nonlocal effects.Using nonlinear macroscopic model,we study the behaviour of light in optical media in the presence of linear Lorentz,nonlinear instantaneous Kerr effect and retarded Raman effect.By a careful treatment of nonlinearities,we propose high order provable energy stable discontinuous Galerkin methods for this model in mixed order form.Error estimate to the semi-discrete discontinuous Galerkin scheme and energy stability analysis for the fully discrete schemes are provided.Thanks to the less variables in mixed order form,our proposed schemes save a lot of computational cost compared with the corresponding schemes in first order form.For the soliton-like open wave propagation,we incorporate the super-grid-scale method into our discontinuous Galerkin schemes with semi-discrete stability to compress the nonphysical reflections on the artificial boundary.Besides,we extend our energy stable discontinuous Galerkin schemes to two dimensional case.Several one-dimensional numerical examples are presented to demonstrate the accuracy,energy stable property,computational efficiency as well as effectiveness in absorbing outgoing wave of our schemes.Finally,we test the accuracy of our two dimensional discontinuous Galerkin schemes,with which,we further simulate the spatial soliton-like wave propagation and air-hole scattering.