Conservation of Power in the Galerkin Approximation of the Electric Field Integral Equation

Abstract

The problem of transient scattering from an arbitrary surface is an extremely important one with many potential applications. In order to solve this problem recourse must generally be made to a numerical method. One such approach which has received considerable attention recently is that of a time-marching algorithm based on the electric field integral equation (EFIE) (Rynne, 1991; Rao and Wilton, 1991). This approach has several advantages over other numerical methods, however, the solution obtained is prone to numerical instability. In order to suppress the onset of this instability Rynne and Smith (1990) introduced a scheme for time-averaging the solution for the current density obtained from the EFIE. These averaging schemes are reminiscent of Crank-Nicholson schemes for a numerical solution of the wave equation. Other considerations have also been found to affect the stability of numerical solutions. Rynne (1985) noted that the use of centred differences to approximate time derivatives tended to suppress the onset of solution instability. Whilst such methods have been shown to suppress (although not eliminate) the numerical instabilities observed in time marching solutions of the EFIE, a clear physical interpretation of these results has not been given.