Reduced-order Modeling in the Finite-difference Time-Domain Method

The finite-difference time-domain (FDTD) method is a well-known numerical algorithm. Despite its many advantages, FDTD becomes time-consuming when applied to multiscale problems, where sharp field variations or fine geometrical details impose a very fine grid, at least locally. Grid refinement affects computational cost in two ways: it increases the number of fields to update at each iteration, and it reduces the maximum timestep allowed by the Courant-Friedrichs-Lewy (CFL) stability limit. Our goal is to investigate how model order reduction (MOR) can be leveraged to address both computational issues associated with grid refinement, and devise accelerated FDTD schemes for multiscale problems. In the envisioned approach, multiscale objects are initially meshed with a locally-refined grid, while the rest of the domain is meshed with a coarse grid. Then, the FDTD update equations of each refined region are processed to generate a reduced-order model, which is finally coupled to the surrounding coarse grid. To achieve this vision, two main issues had to be addressed: i) how to generate FDTD-compatible reduced-order models, and ii) how to guarantee that the final FDTD scheme with embedded reduced models will be stable under a known CFL limit. Our first major contribution is to devise a new theoretical framework to create FDTD-like schemes with provable stability, based on the dissipativity theory. We show that most FDTD schemes can be seen as the connection of different subsystems, such as reduced-order models. We determine the conditions under which these FDTD-like subsystems are dissipative, and prove that, if each subsystem is dissipative, then the whole scheme will be dissipative, and thus stable. This theory provides a systematic way to create a variety of FDTD schemes with provable stability, from simple subgridding algorithms to the advanced FDTD schemes with embedded reduced-order models. Another major contribution is to generate reduced-order models compatible with FDTD simulations, and capable of accurately capturing the electromagnetic behaviour of a given volume enclosing arbitrary objects. The proposed models can be instantiated into a coarser FDTD grid and are compatible with leap-frogging. Through the dissipativity theory, we rigorously prove the stability of the proposed FDTD algorithms with reduced models.

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