A New State-space-based Algorithm to Assess the Stability of the Finite-difference Time-domain Method for 3D Finite Inhomogeneous Problems

The finite-difference time-domain (FDTD) method is an explicit time discretization scheme for Maxwell’s equations. In this context it is well-known that explicit time discretization schemes have a stability induced time step restriction. In this paper, we recast the spatial discretization of Maxwell’s equations, initially without time discretization, into a more convenient format, called the FDTD state-space system. This in turn allows us to derive a new algorithm in order to determine the stability limit of FDTD for lossy, inhomogeneous finite problems. It is shown that a crucial parameter is the spectral norm of the matrix resulting from the spatial discretization of the curl operator. In a rectangular simulation domain the time step upper bound can be calculated in closed form and results in a time step limit less stringent than the Courant condition. Finally, the validity of the technique is illustrated by means of some pertinent numerical examples.