A Dissipative Systems Theory for FDTD with Application to Stability Analysis and Subgridding

TL;DR

This paper links FDTD to dissipative systems theory, providing a new framework for stability analysis and introducing a stable, easy-to-implement subgridding method with material traverse and arbitrary refinement.

Contribution

It develops a dissipative systems framework for FDTD, simplifying stability proofs and enabling modular analysis of components, leading to a novel stable subgridding technique.

Findings

FDTD system is dissipative under a generalized CFL condition.

The new subgridding method is stable, easy to implement, and handles material traverse.

Numerical results confirm low reflections and stability of the proposed method.

Abstract

This paper establishes a far-reaching connection between the Finite-Difference Time-Domain method (FDTD) and the theory of dissipative systems. The FDTD equations for a rectangular region are written as a dynamical system having the magnetic and electric fields on the boundary as inputs and outputs. Suitable expressions for the energy stored in the region and the energy absorbed from the boundaries are introduced, and used to show that the FDTD system is dissipative under a generalized Courant-Friedrichs-Lewy condition. Based on the concept of dissipation, a powerful theoretical framework to investigate the stability of FDTD methods is devised. The new method makes FDTD stability proofs simpler, more intuitive, and modular. Stability conditions can indeed be given on the individual components (e.g. boundary conditions, meshes, embedded models) instead of the whole coupled setup. As an example of application, we derive a new subgridding method with material traverse, arbitrary grid refinement, and guaranteed stability. The method is easy to implement and has a straightforward stability proof. Numerical results confirm its stability, low reflections, and ability to handle material traverse.