Stability of FD–TD Schemes for Maxwell–Debye and Maxwell–Lorentz Equations

The stability of five finite difference–time domain (FD–TD) schemes coupling Maxwell equations to Debye or Lorentz models has been analyzed in [P. Petropoulos, Stability and phase error analysis of FD–TD in dispersive dielectrics, IEEE Trans. Antennas Propag., 42 (1994), pp. 62–69], where numerical evidence for specific media have been used. We use von Neumann analysis to give necessary and sufficient stability conditions for these schemes for any medium, in accordance with the partial results of [P. Petropoulos, Stability and phase error analysis of FD–TD in dispersive dielectrics, IEEE Trans. Antennas Propag., 42 (1994), pp. 62–69]. To make this approach tractable for two-dimensional and three-dimensional models, we have developed a computer algebra environment which has a wider range of applicability.