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

A connection between the finite-difference time-domain (FDTD) method and the theory of dissipative systems is established. The FDTD equations for a rectangular region are interpreted as a dynamical system having the magnetic field on the boundary as input and the electric field on the boundary as output. 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-like 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, and embedded models) instead of the whole coupled setup. As an example of application, we derive a new subgridding scheme with support for 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.