A stability analysis of a time marching scheme for the general surface electric field integral equation
Approximating the electric field integral equation (EFIE) is a popular way of computing the current which is induced on the surface of a perfect conductor by an incident pulsed electric field. In the present paper a rigorous Fourier stability analysis of a general surface EFIE algorithm applied to a large flat plate is carried out. The analysis shows that the scheme has two different types of fast growing high frequency instabilities. One of these can be reduced by averaging the computed current in time, but the other is due to an interaction between the finite element basis functions and the (midpoint) quadrature rule used to approximate the retarded potential integral equations. The author explains why this is the case and shows how these unstable modes can be removed when the mesh is infinite. This can be done very efficiently, and is enough to make the resulting scheme effectively stable on any flat plate of computationally feasible size. The EFIE scheme analyzed here is essentially that due to \textit{B.~P.~Rynne} [Time domain scattering from arbitrary surfaces using the electric field equation, J. Electromagn. Waves Appl. 5, 93-112 (1991)]. The author choses this one scheme as an example since it is of comparable performance to many other EFIE approximation algorithms (and is more clearly described than most), but note that the techniques used here do not depend on this particular scheme. The rest of the paper is organized as follows. The author begins by describing the equations, basis functions and Rynne's algorithm (which is unstable on a fine mesh). The equations are a coupled system of partial differential equations and retarded potential integral equations (RPIEs), and it is the unusual (and often unfamiliar) properties of numerical schemes for RPIEs that make the problem interesting and challenging. The stability analysis requires the algorithm to be written in terms of shift operators on an infinite mesh, and this is described for a uniform equilateral triangulation. The analysis itself is given, and demonstrates the existence of fast growing high frequency unstable modes. It is shown that these can be removed.
scientific article; zbMATH DE number 1311325