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Prior theoretical studies and experience confirm that the stability of marching-on-in-time (MOT) solvers pertinent to the analysis of scattering from free-standing three-dimensional perfect electrically conducting surfaces hinges on the accurate evaluation of MOT matrix elements resulting from a Galerkin discretization of the underlying time domain integral equation (TDIE). Unfortunately, the accurate evaluation of the four-dimensional spatial integrals involved in the expressions for these matrix elements is prohibitively expensive when performed by computational means. Here, a method that permits the quasi-exact evaluation of MOT matrix elements is presented. Specifically, the proposed method permits the analytical evaluation of three out of the four spatial integrations, leaving only one integral to be evaluated numerically. Since the latter has finite range and a piecewise smooth integrand, it can be evaluated to very high accuracy using standard quadrature rules. As a result, the proposed method permits the fast evaluation of MOT matrix elements with arbitrary (user-specified) accuracy. Extensive numerical experiments show that an MOT solver for the electric field TDIE that uses the proposed quasi-exact method is stable for a very wide range of time step sizes and yields solutions that decay exponentially after the excitation vanishes.Index Terms-Exact integration, late time instability, marching-on-in-time (MOT), time domain integral equations.

A time-domain, finite element-boundary integral (FE-BI) method is presented for analyzing electromagnetic (EM) scattering from two-dimensional (2-D) inhomogeneous objects. The scheme's finite-element component expands transverse fields in terms of a pair of orthogonal vector basis functions and is coupled to its boundary integral component in such a way that the resultant finite element mass matrix is diagonal, and more importantly, the method delivers solutions that are free of spurious modes. The boundary integrals are computed using the multilevel plane-wave time-domain algorithm to enable the simulation of large-scale scattering phenomena. Numerical results demonstrate the capabilities and accuracy of the proposed hybrid scheme.