A Stabilized Marching-on-in-Degree Scheme for the Transient Solution of the Electric Field Integral Equation

Time-domain electric integral equation methods (TD-EFIE) suffer from the numerical stability problem. Numerous schemes have been proposed over the years for the purpose of taking care of this instability. Among them, the marching-on-in-degree (MOD) method is one of the newly proposed algorithms for calculating the various transient responses. The MOD solver employs a set of associated Laguerre functions and incorporates exact temporal Galerkin testing. Although the MOD scheme is assumed to be more stable when compared with marching-on-in-time (MOT), there are few numerical or theoretical investigations on the stability of the various numerical recursive operations for the MOD methodology. It has been shown that the time-domain magnetic field integral equation (TD-MFIE) using the MOD method can be stabilized by using a Filon-type quadrature formula to accurately integrate the highly oscillatory Laguerre functions which has caused instability for the MOD solvers using a higher degree of approximation. On the other hand, the analytical expression of temporal derivatives used in the MOD scheme is strongly affected by an unbounded initial value of the transient current. In this paper, a novel Filon-type radial integration method and a sequential constraint term for TD-EFIE-MOD are proposed, which make the stabilized MOD solver accurate and stable for the arbitrary degree of the Laguerre polynomials. Some numerical results are presented to illustrate the validity of the stabilized MOD solution of TD-EFIE for transient scattering problems.