A Stabilized Solution of Time-Domain Combined Field Integral Equation using Associated Laguerre Functions

The time domain integral equation (TDIE) using associated Laguerre functions [1] is one of the promising transient and broadband computational electromagnetic methods. It is a marching-on-in-degree (MOD) method for solving various electromagnetic compatibility and electromagnetic interference (EMC & EMI) problems. Over the past decades, the instability of time-domain integral equation remains one of the main obstacles to improve the quality of the TDIE solutions. The MOD method adopts the associated Laguerre functions as temporal basis functions. As the associated Laguerre polynomials decays exponentially in time, the MOD scheme is immune to the late-time unstable behavior of marching-on-in-time methods (MOT). Although the MOD solver is assumed to be stable, there still exists an instabilty when the TDIE-MOD scheme marches on to very high degree of the Laguerre polynomials [2]. The main reason is that MOD iterates with respect to the polynomial degrees rather than the time steps. The instability related to the MOD solution of time-domain magnetic field integral equation (TD-MFIE-MOD) and time-domain electric integral equation (TD-EFIE-MOD) are eliminated by a Filon-type quadrature for associated Laguerre polynomials, and a novel Filon-type radial integration method for oscillatory quadrature on triangular patches, which are proposed in [2], [3]. On the other hand, the exact temporal Galerkin testing procedure in the MOD solution is affected by an unbounded initial value of the transient current density, which allows numerical errors to accumulate and affect the ultimate stability. Therefore, a sequential constraint term needs to be added without destroying the causality of the MOD method.