Convergence of a discontinuous Galerkin scheme for the mixed time-domain Maxwell's equations in dispersive media

Abstract

This study is concerned with the solution of the time domain Maxwell's equations in a dispersive propagation media by a Discontinuous Galerkin Time Domain (DGTD) method. The Debye model is used to describe the dispersive behaviour of the media. The resulting system of equations is solved using a centered flux discontinuous Galerkin formulation for the discretization in space and a second order leapfrog scheme for the integration in time. The numerical treatment of the dispersive model relies on an Auxiliary Differential Equation (ADE) approach similary to what is adopted in the Finite Difference Time Domain (FDTD) method. Stability estimates are derived through energy estimations and the convergence is proved for both the semi-discrete and the fully discrete case.

Analysis of a Generalized Dispersive Model Coupled to a DGTD Method with Application to Nanophotonics

“…The rest of the proof of existence goes along the same lines as in [LS13] and consists in a straightforward generalization of the result obtained in the latter reference. We will thus not detail the proof here, but simply state the final result.…”

“…To prove the convergence, we will furthermore suppose that initial conditions as stated in (3.4) are regular enough so that there exists s > 1 such that ϑ ∈ C 0 (0, T, H s (Ω) N ). The structure of the semidiscrete scheme is the same as in [LS13] up to the difference that the analogous expression of Kϑ, ϑ is negative in [LS13] and that here we allow for a more general definition of fluxes. Despite these notable differences, one can follow the key ideas of the latter reference to prove the semi-convergence of the scheme.…”

“…This first step reveals the key ingredients for a whole convergence proof relying on error energy principles and consistency type results, and we chose to rather sketch it since it would then readily follow a combination of e.g. [BEF10,LS13] and the semi-discrete convergence.…”

“…The latter reference propose to analyze a semi-discrete divergence free discontinuous Galerkin framework. Finally, in a previous work [LS13], the authors analyzed, for the Debye model, a fully discrete scheme based on a centered fluxes nodal Discontinuous Galerkin formulation and Leap frog discretization in time.…”

“…It extends the approach of [BEF10] that was valid for lower order Runge Kutta time schemes. The convergence result is quickly sketched since arguments then closely follow [BEF10,LS13]. We end this study with two tridimensional numerical tests related to nanophotonics: (i) the calculation of the reflection coefficient of an infinite dispersive slab which is used to confirm the good implementation of our method, and (ii) the absorption cross-section calculation of a gold-silver nano-shell to demonstrate the importance of a good description of the dispersive properties in metals for nanophotonics-related computations.…”