On the Stability of the Finite-Difference Time-Domain Method

In this paper we give a necessary and sufficient condition for the stability of the finite-difference time-domain method (FDTD method). This is an explicit time stepping method that is used for solving transient electromagnetic field problems. A necessary (but not a sufficient) condition for its stability is usually obtained by requiring that discrete Fourier modes, defined on the FDTD grid, remain bounded as time stepping proceeds. Here we follow a different approach. We rewrite the basic FDTD equations in terms of an iteration matrix and study the eigenvalue problem for this matrix. From the analysis a necessary and sufficient condition for stability of the FDTD method follows. Moreover, we show that for a particular time step the 2-norm of the FDTD iteration matrix is equal to the golden ratio.