Revisiting the Stability of Crank–Nicolson and ADI-FDTD

The stability properties of Crank-Nicolson (CN) and alternating direction implicit finite-difference time-domain ADI-FDTD methods to solve Maxwell's equations are investigated. Since the ADI-FDTD scheme can be formulated in many cases as an approximation to a CN-FDTD scheme, we start by analyzing the stability of CN-FDTD and, from it, we try to derive conditions on ADI-FDTD. We first recall the von Neumann analysis, and then we present a matrix-norm analysis. We show that the matrix-norm stability of CN-FDTD is a necessary condition to find matrix-norm stable conditions for ADI-FDTD.