Stability of Time Marching Algorithms for the Electric Field Integral Equation

The stability of a simple numerical scheme for solving the time dependent form of the Electric Field Integral Equation is investigated in this paper. In this so-called "time marching” algorithm, the time step Δt times the speed of light c does not exceed the minimum spatial grid step Δ, and the surface current and charge distributions are found explicitly in terms of the distribution at earlier times. In order to obtain solutions which are stable in the long term, two points must be considered. (i) The product cΔt must not exceed Δ/√2. (Courant condition.) (ii) If the scatterer has internal resonances (i.e., is a closed body) or has a SEM pole close to the imaginary axis then, in addition to the Courant condition holding, an averaging procedure must be used. A four step averaging procedure is found to be necessary for closed bodies with internal resonances, such as the cube, whereas a three step procedure is found to be adequate for pair of parallel plates, which has no internal resonance, but does possess poles near the imaginary axis. These averaging processes are easily incorporated into the original time marching algorithm, and produce solutions which are stable in the long term and are of accuracy comparable to those produced by the unaveraged basic algorithm prior to the onset of instability.