On the Stability of Time-Domain Magnetic Field Integral Equation Using Laguerre Functions

The marching-on-in-degree (MOD) method has been proposed for solving time-domain integral equations (TDIEs), and it uses the associated Laguerre functions as temporal basis and testing functions. Although the MOD scheme is assumed to be stable and accurate, there exists little investigation into the accuracy of the integral operations involving the associated Laguerre functions of high degree. It is shown in this paper that the numerical quadrature for integrating the highly oscillatory Laguerre functions may not be accurate and this may result in instability for the MOD solvers. In this paper, the highly oscillatory nature of the high-degree associated Laguerre functions is investigated first. Based on that, the numerical procedures are described which result in the inaccuracy of the numerical quadrature associated with the highly oscillatory nature for the higher order Laguerre functions. This can produce instability for the MOD solvers when a solution requires a high degree of the Laguerre functions. Hence, a novel Filon-type quadrature method and a stabilized MOD technique for time-domain magnetic field integral equations (TD-MFIE-SMOD) are presented in this paper. Some numerical results are presented to illustrate the validity of the proposed Filon-type quadrature rule resulting in stable solution of the TD-MFIE for transient scattering problems.