1A Stable Higher Order Space-Time Galerkin Scheme for Time Domain Integral Equations

Abstract—Stability of time domain integral equation (TDIE) solvers has remained an elusive goal for many years. Advance-ment of this research has largely progressed on four fronts: (1) Exact integration, (2) Lubich quadrature, (3) smooth temporal basis functions, and (4) Space-time separation of convolutions with the retarded potential. The latter method was explored in [1]. This method’s efficacy in stabilizing solutions to the time domain electric field integral equation (TD-EFIE) was demonstrated on first order surface descriptions (flat elements) in tandem with 0th order functions as the temporal basis. In this work, we develop the methodology necessary to extend to higher order surface descriptions as well as to enable its use with higher order temporal basis functions. These higher order temporal basis functions are used in a Galerkin framework. A number of results that demonstrate convergence, stability, and applicability are presented. Index Terms—Time-domain integral equations, higher order temporal basis, stability, time-domain analysis, marching-on-in-time, space-time Galerkin method. I