On the Stability of the RK-FDTD Method for Graphene Modeling

The Runge–Kutta finite-difference time-domain (RK-FDTD) method is an extension of the conventional finite-difference time-domain (FDTD) technique to include graphene sheets. According to this method, the relationship between the current density and the electric field for graphene is discretized by applying an explicit second-order Runge–Kutta (RK) scheme. It has recently been concluded that the RK-FDTD method is subject to the same Courant–Friedrichs–Lewy (CFL) stability limit as the conventional FDTD method. This communication revisits the stability analysis of the RK-FDTD method. To this end, the von Neumann method is combined with the Routh–Hurwitz (RH) criterion. As a result, closed-form stability conditions are obtained. It is shown that in addition to the CFL stability limit, the RK-FDTD method must also satisfy new conditions involving graphene parameters. Unfortunately, the RK-FDTD method becomes unstable for commonly used values of these parameters. The theoretical results are confirmed with numerical examples.