Stability of time-marching numerical schemes for the electric field integral equation

Abstract

Time marching algorithms are often used to compute the current induced on a conductor by an incident pulsed electric field. They are obtained by discretizing the relevant equations on a uniform mesh whose spacing is determined by the frequency of the incident field; higher frequencies requiring a finer mesh. Unfortunately it is often observed that the error in the numerical solutions of time domain field integral equations grows exponentially when the incident field has high frequency. We show here that this happens when the numerical approximation used to solve the integral equation part of the field equations is unstable, and that it is just as important for this part of the scheme to be stable as it is to ensure that the PDE approximation satisfies a CFL type condition. We present a rigorous numerical stability analysis of time marching schemes for the electric field integral equation on a flat plate. This yields a stability test for such schemes which is relatively quick and simple to implement. We use numerical test results obtained for schemes derived by Rynne & Smith to illustrate the value of our theoretical predictions.