A Runge-Kutta Discontinuous Galerkin Method For The Linear Nonlocal Drude Model

Nanopotonics is a subject that studies the interaction between light and material on the nanoscale and further the applications of these interactions.It is widely used in various fields such as science,technology and medical treatment.Particularly,metal nanomaterials have unique plasma effect,and thus have drawn researchers' attention.Because the physical experiments cost too much,and the information obtained by pure theoretical analysis is not enough,numerical methods are commonly used to simulate practical problems in research.In this thesis,a decoupled scheme with Runge-Kutta discontinuous Galerkin(RKDG)methods is designed for the linear nonlocal Drude model,to study the inherent law of the interaction between the nonlocal dispersion effect of light and the structure of metal nanomaterials.We analyze the energy stability and convergence of this numerical scheme,and meanwhile verify our theoretical analysis by numerical experiments.The dispersion problem of nanophotonics to be dealt with in this paper and the expression of equations of the linearized model considering its nonlocal resonance effect are briefly introduced first.Then the basic idea of the decoupled RKDG numerical scheme for the linear nonlocal dispersion model is discussed.Furthermore,we prove the unconditional energy stability of the fully implicit RK scheme and the DG spatial semidiscrete scheme,respectively.Moreover,we provide the analysis on the rate of convergence for the fully discrete scheme.Implicit RK temporal discretization is a general framework,and thus has a variety of forms.Therefore,in the analysis of fully discrete schemes,we specifically select one of these RK methods,which is combined with the DG method.The analysis shows that this decoupled RKDG numerical method has second order time accuracy and is locally stable.Finally,we utilize several simplified numerical examples of two-dimensional transverse electric mode to carry out our numerical experiments.The numerical results show that the convergence order and energy stability of the decoupled RKDG method are consistent with the theoretical analysis.Meanwhile,the results are compared with those properties of other numerical schemes,which shows the feasibility and advantages of ours.The main work of this thesis is to discretize the linear nonlocal Drude model by a decoupled RKDG method,which consists of the modified algebraically stable implicit RK temporal discretization method and the DG spatial discretization method.An appropriate numerical flux is adopted to construct the corresponding decoupled RKDG numerical scheme.We use the idea of "decoupled" to modify the fully implicit RKDG method,which is to take the current density to be explicit in the electric field equations,while the electric field is explicit in the current density equations.