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

## TL;DR

This paper introduces a dissipation-based framework for 3-D FDTD that ensures stability in complex setups, enabling modular design of stable schemes like a new subgridding method with low reflections and flexible refinement.

## Contribution

It presents a novel dissipation theory for 3-D FDTD that guarantees stability by design, simplifying the creation of stable, modular electromagnetic simulation schemes.

## Key findings

- Proposes a dissipation framework inspired by energy conservation.
- Derives a stable subgridding scheme with low reflections.
- Enables modular stability enforcement for complex FDTD setups.

## Abstract

The finite-difference time-domain (FDTD) algorithm is a popular numerical method for solving electromagnetic problems. FDTD simulations can suffer from instability due to the explicit nature of the method. Stability enforcement can be particularly challenging in scenarios where a setup is composed of multiple components, such as grids of different resolution, advanced boundary conditions, reduced-order models, and lumped elements. We propose a dissipation theory for 3-D FDTD inspired by the principle of energy conservation. We view the FDTD update equations for a 3-D region as a dynamical system, and show under which conditions the system is dissipative. By requiring each component of an FDTD-like scheme to be dissipative, the stability of the overall coupled scheme follows by construction. The proposed framework enables the creation of provably stable schemes in an easy and modular fashion, since conditions are imposed on the individual components, rather than on the overall coupled scheme as in existing approaches. With the proposed framework, we derive a new subgridding scheme with guaranteed stability, low reflections, support for material traverse and arbitrary (odd) grid refinement ratio.