A New Hybrid Implicit–Explicit FDTD Method for Local Subgridding in Multiscale 2-D TE Scattering Problems

Abstract

Abstract-The conventional Finite-Difference Time-Domain (FDTD) method with staggered Yee scheme does not easily allow including thin material layers, especially so if these layers are highly conductive. This paper proposes a novel subgridding technique for 2D problems, based on a Hybrid Implicit-Explicit (HIE) scheme, that efficiently copes with this problem. In the subgrid, the new method collocates field components such that the thin layer boundaries are defined unambiguously. Moreover, aspect ratios of more than a million do not impair the stability of the method and allow for very accurate predictions of the skin effect. The new method retains the Courant limit of the coarse Yee grid and is easily incorporated into existing FDTD codes. A number of illustrative examples, including scattering by a metal grating, demonstrate the accuracy and stability of the new method.

Explicit and provably stable spatiotemporal FDTD refinement

“…(1) is derived under the assumption of a standard Yee grid filled with a uniform medium. More general but not fundamentally different stability conditions are known in the case of non-Yee grids and non-uniform media, such as (9).…”

“…Features which are small in just one direction (e.g. thin layers of conducting material) can be included using partially implicit approaches, which do not impose global time step reduction nor globally denser meshes [9,20]. In fact, local spatial refinement with no global time step reduction is always possible provided one is willing to use Crank-Nicolson (implicit) updates in the refined region [19,20,21], or by filtering out unstable modes in the refined grid [22], but this reduces both the ease of implementation, and the ease of parallelization.…”

“…We must stress that (8) and (9) do not assume homogeneity: C depends on the (possibly inhomogeneous) ε r and µ r . (8) and (9) are not limited to uniform grids either, a fact of which we will make use to construct the spatial part of our spatiotemporal refinement, in section 5.…”

Full-Wave Analysis of the Shielding Effectiveness of Thin Graphene Sheets with the 3D Unidirectionally Collocated HIE-FDTD Method

“…To resolve both problems at once, the unidirectionally collocated hybrid implicit-explicit (UCHIE) FDTD method was proposed in 2D in [1] for good conductors in a multiscale environment and is extended here to 3D problems including graphene sheets. This 3D UCHIE-FDTD method implicitizes the direction perpendicular to the graphene boundary, hereby eliminating the small cell sizes from the time step limit and, consequently, speeding up the simulation.…”

“…Hence, different cell sizes can be adopted inside and outside the graphene sheet without any form of accuracy trade-off. In contrast to [1], where the UCHIE method was confined to a small part in the interior of the simulation domain, open-space simulations now require the construction of a perfectly matched layer (PML) for the UCHIE method. Moreover, the dispersive properties of graphene necessitate an auxiliary differential equation (ADE) formulation of the UCHIE method.…”