A Dissipation Theory for Three-Dimensional FDTD with Application to Stability Analysis and Subgridding

The finite-difference time-domain algorithm is a popular numerical method for solving electromagnetic problems. FDTD simulations can suffer from instability due to the explicit nature of the method. Enforcement of stability can be particularly challenging in scenarios where a setup is composed of multiple components, such as differently refined FDTD grids, macromodels, advanced boundary conditions, and others. Moreover, the lack of modularity in existing stability analysis methods can make small modifications to the system burdensome. We propose a 3-D dissipativity framework that can be used to enforce stability of novel FDTD schemes in a modular fashion, which is an extension of previous developments in 2-D [1]. With this framework one can analyze parts of the system separately, which simplifies stability proofs and avoids the need to revise the entire proof when a single part of a system is changed. With these results, we derive a 3-D subgridding scheme with guaranteed stability, good accuracy, and support for any odd grid refinement ratios. Subgridding boundary is allowed to traverse different materials, producing meaningful results with no instability.